nips nips2012 nips2012-190 knowledge-graph by maker-knowledge-mining

190 nips-2012-Learning optimal spike-based representations

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Author: Ralph Bourdoukan, David Barrett, Sophie Deneve, Christian K. Machens

Abstract: How can neural networks learn to represent information optimally? We answer this question by deriving spiking dynamics and learning dynamics directly from a measure of network performance. We ﬁnd that a network of integrate-and-ﬁre neurons undergoing Hebbian plasticity can learn an optimal spike-based representation for a linear decoder. The learning rule acts to minimise the membrane potential magnitude, which can be interpreted as a representation error after learning. In this way, learning reduces the representation error and drives the network into a robust, balanced regime. The network becomes balanced because small representation errors correspond to small membrane potentials, which in turn results from a balance of excitation and inhibition. The representation is robust because neurons become self-correcting, only spiking if the representation error exceeds a threshold. Altogether, these results suggest that several observed features of cortical dynamics, such as excitatory-inhibitory balance, integrate-and-ﬁre dynamics and Hebbian plasticity, are signatures of a robust, optimal spike-based code. A central question in neuroscience is to understand how populations of neurons represent information and how they learn to do so. Usually, learning and information representation are treated as two different functions. From the outset, this separation seems like a good idea, as it reduces the problem into two smaller, more manageable chunks. Our approach, however, is to study these together. This allows us to treat learning and information representation as two sides of a single mechanism, operating at two different timescales. Experimental work has given us several clues about the regime in which real networks operate in the brain. Some of the most prominent observations are: (a) high trial-to-trial variability—a neuron responds differently to repeated, identical inputs [1, 2]; (b) asynchronous ﬁring at the network level—spike trains of different neurons are at most very weakly correlated [3, 4, 5]; (c) tight balance of excitation and inhibition—every excitatory input is met by an inhibitory input of equal or greater size [6, 7, 8] and (4) spike-timing-dependent plasticity (STDP)—the strength of synapses change as a function of presynaptic and postsynaptic spike times [9]. Previously, it has been shown that observations (a)–(c) can be understood as signatures of an optimal, spike-based code [10, 11]. The essential idea is to derive spiking dynamics from the assumption that neurons only ﬁre if their spike improves information representation. Information in a network may ∗ Authors contributed equally 1 originate from several possible sources: external sensory input, external neural network input, or alternatively, it may originate within the network itself as a memory, or as a computation. Whatever the source, this initial assumption leads directly to the conclusion that a network of integrate-and-ﬁre neurons can optimally represent a signal while exhibiting properties (a)–(c). A major problem with this framework is that network connectivity must be completely speciﬁed a priori, and requires the tuning of N 2 parameters, where N is the number of neurons in the network. Although this is feasible mathematically, it is unclear how a real network could tune itself into this optimal regime. In this work, we solve this problem using a simple synaptic learning rule. The key insight is that the plasticity rule can be derived from the same basic principle as the spiking rule in the earlier work—namely, that any change should improve information representation. Surprisingly, this can be achieved with a local, Hebbian learning rule, where synaptic plasticity is proportional to the product of presynaptic ﬁring rates with post-synaptic membrane potentials. Spiking and synaptic plasticity then work hand in hand towards the same goal: the spiking of a neuron decreases the representation error on a fast time scale, thereby giving rise to the actual population representation; synaptic plasticity decreases the representation error on a slower time scale, thereby improving or maintaining the population representation. For a large set of initial connectivities and spiking dynamics, neural networks are driven into a balanced regime, where excitation and inhibition cancel each other and where spike trains are asynchronous and irregular. Furthermore, the learning rule that we derive reproduces the main features of STDP (property (d) above). In this way, a network can learn to represent information optimally, with synaptic, neural and network dynamics consistent with those observed experimentally. 1 Derivation of the learning rule for a single neuron We begin by deriving a learning rule for a single neuron with an autapse (a self-connection) (Fig. 1A). Our approach is to derive synaptic dynamics for the autapse and spiking dynamics for the neuron such that the neuron learns to optimally represent a time-varying input signal. We will derive a learning rule for networks of neurons later, after we have developed the fundamental concepts for the single neuron case. Our ﬁrst step is to derive optimal spiking dynamics for the neuron, so that we have a target for our learning rule. We do this by making two simple assumptions [11]. First, we assume that the neuron can provide an estimate or read-out x(t) of a time-dependent signal x(t) by ﬁltering its spike train ˆ o(t) as follows: ˙ x(t) = −ˆ(t) + Γo(t), ˆ x (1) where Γ is a ﬁxed read-out weight, which we will refer to as the neuron’s “output kernel” and the spike train can be written as o(t) = i δ(t − ti ), where {ti } are the spike times. Next, we assume that the neuron only produces a spike if that spike improves the read-out, where we measure the read-out performance through a simple squared-error loss function: 2 L(t) = x(t) − x(t) . ˆ (2) With these two assumptions, we can now derive optimal spiking dynamics. First, we observe that if the neuron produces an additional spike at time t, the read-out increases by Γ, and the loss function becomes L(t|spike) = (x(t) − (x(t) + Γ))2 . This allows us to restate our spiking rule as follows: ˆ the neuron should only produce a spike if L(t|no spike) > L(t|spike), or (x(t) − x(t))2 > (x(t) − ˆ (x(t) + Γ))2 . Now, squaring both sides of this inequality, deﬁning V (t) ≡ Γ(x(t) − x(t)) and ˆ ˆ deﬁning T ≡ Γ2 /2 we ﬁnd that the neuron should only spike if: V (t) > T. (3) We interpret V (t) to be the membrane potential of the neuron, and we interpret T as the spike threshold. This interpretation allows us to understand the membrane potential functionally: the voltage is proportional to a prediction error—the difference between the read-out x(t) and the actual ˆ signal x(t). A spike is an error reduction mechanism—the neuron only spikes if the error exceeds the spike threshold. This is a greedy minimisation, in that the neuron ﬁres a spike whenever that action decreases L(t) without considering the future impact of that spike. Importantly, the neuron does not require direct access to the loss function L(t). 2 To determine the membrane potential dynamics, we take the derivative of the voltage, which gives ˙ ˙ us V = Γ(x − x). (Here, and in the following, we will drop the time index for notational brevity.) ˙ ˆ ˙ Now, using Eqn. (1) we obtain V = Γx − Γ(−x + Γo) = −Γ(x − x) + Γ(x + x) − Γ2 o, so that: ˙ ˆ ˆ ˙ ˙ V = −V + Γc − Γ2 o, (4) where c = x + x is the neural input. This corresponds exactly to the dynamics of a leaky integrate˙ and-ﬁre neuron with an inhibitory autapse1 of strength Γ2 , and a feedforward connection strength Γ. The dynamics and connectivity guarantee that a neuron spikes at just the right times to optimise the loss function (Fig. 1B). In addition, it is especially robust to noise of different forms, because of its error-correcting nature. If x is constant in time, the voltage will rise up to the threshold T at which point a spike is ﬁred, adding a delta function to the spike train o at time t, thereby producing a read-out x that is closer to x and causing an instantaneous drop in the voltage through the autapse, ˆ by an amount Γ2 = 2T , effectively resetting the voltage to V = −T . We now have a target for learning—we know the connection strength that a neuron must have at the end of learning if it is to represent information optimally, for a linear read-out. We can use this target to derive synaptic dynamics that can learn an optimal representation from experience. Speciﬁcally, we consider an integrate-and-ﬁre neuron with some arbitrary autapse strength ω. The dynamics of this neuron are given by ˙ V = −V + Γc − ωo. (5) This neuron will not produce the correct spike train for representing x through a linear read-out (Eqn. (1)) unless ω = Γ2 . Our goal is to derive a dynamical equation for the synapse ω so that the spike train becomes optimal. We do this by quantifying the loss that we are incurring by using the suboptimal strength, and then deriving a learning rule that minimises this loss with respect to ω. The loss function underlying the spiking dynamics determined by Eqn. (5) can be found by reversing the previous membrane potential analysis. First, we integrate the differential equation for V , assuming that ω changes on time scales much slower than the membrane potential. We obtain the following (formal) solution: V = Γx − ω¯, o (6) ˙ where o is determined by o = −¯ + o. The solution to this latter equation is o = h ∗ o, a convolution ¯ ¯ o ¯ of the spike train with the exponential kernel h(τ ) = θ(τ ) exp(−τ ). As such, it is analogous to the instantaneous ﬁring rate of the neuron. Now, using Eqn. (6), and rewriting the read-out as x = Γ¯, we obtain the loss incurred by the ˆ o sub-optimal neuron, L = (x − x)2 = ˆ 1 V 2 + 2(ω − Γ2 )¯ + (ω − Γ2 )2 o2 . o ¯ Γ2 (7) We observe that the last two terms of Eqn. (7) will vanish whenever ω = Γ2 , i.e., when the optimal reset has been found. We can therefore simplify the problem by deﬁning an alternative loss function, 1 2 V , (8) 2 which has the same minimum as the original loss (V = 0 or x = x, compare Eqn. (2)), but yields a ˆ simpler learning algorithm. We can now calculate how changes to ω affect LV : LV = ∂LV ∂V ∂o ¯ =V = −V o − V ω ¯ . (9) ∂ω ∂ω ∂ω We can ignore the last term in this equation (as we will show below). Finally, using simple gradient descent, we obtain a simple Hebbian-like synaptic plasticity rule: τω = − ˙ ∂LV = V o, ¯ ∂ω (10) where τ is the learning time constant. 1 This contribution of the autapse can also be interpreted as the reset of an integrate-and-ﬁre neuron. Later, when we generalise to networks of neurons, we shall employ this interpretation. 3 This synaptic learning rule is capable of learning the synaptic weight ω that minimises the difference between x and x (Fig. 1B). During learning, the synaptic weight changes in proportion to the postˆ synaptic voltage V and the pre-synaptic ﬁring rate o (Fig. 1C). As such, this is a Hebbian learning ¯ rule. Of course, in this single neuron case, the pre-synaptic neuron and post-synaptic neuron are the same neuron. The synaptic weight gradually approaches its optimal value Γ2 . However, it never completely stabilises, because learning never stops as long as neurons are spiking. Instead, the synapse oscillates closely about the optimal value (Fig. 1D). This is also a “greedy” learning rule, similar to the spiking rule, in that it seeks to minimise the error at each instant in time, without regard for the future impact of those changes. To demonstrate that the second term in Eqn. (5) can be neglected we note that the equations for V , o, and ω deﬁne a system ¯ of coupled differential equations that can be solved analytically by integrating between spikes. This results in a simple recurrence relation for changes in ω from the ith to the (i + 1)th spike, ωi+1 = ωi + ωi (ωi − 2T ) . τ (T − Γc − ωi ) (11) This iterative equation has a single stable ﬁxed point at ω = 2T = Γ2 , proving that the neuron’s autaptic weight or reset will approach the optimal solution. 2 Learning in a homogeneous network We now generalise our learning rule derivation to a network of N identical, homogeneously connected neurons. This generalisation is reasonably straightforward because many characteristics of the single neuron case are shared by a network of identical neurons. We will return to the more general case of heterogeneously connected neurons in the next section. We begin by deriving optimal spiking dynamics, as in the single neuron case. This provides a target for learning, which we can then use to derive synaptic dynamics. As before, we want our network to produce spikes that optimally represent a variable x for a linear read-out. We assume that the read-out x is provided by summing and ﬁltering the spike trains of all the neurons in the network: ˆ ˙ x = −ˆ + Γo, ˆ x (12) 2 where the row vector Γ = (Γ, . . . , Γ) contains the read-out weights of the neurons and the column vector o = (o1 , . . . , oN ) their spike trains. Here, we have used identical read-out weights for each neuron, because this indirectly leads to homogeneous connectivity, as we will demonstrate. Next, we assume that a neuron only spikes if that spike reduces a loss-function. This spiking rule is similar to the single neuron spiking rule except that this time there is some ambiguity about which neuron should spike to represent a signal. Indeed, there are many different spike patterns that provide exactly the same estimate x. For example, one neuron could ﬁre regularly at a high rate (exactly like ˆ our previous single neuron example) while all others are silent. To avoid this ﬁring rate ambiguity, we use a modiﬁed loss function, that selects amongst all equivalent solutions, those with the smallest neural ﬁring rates. We do this by adding a ‘metabolic cost’ term to our loss function, so that high ﬁring rates are penalised: ¯ L = (x − x)2 + µ o 2 , ˆ (13) where µ is a small positive constant that controls the cost-accuracy trade-off, akin to a regularisation parameter. Each neuron in the optimal network will seek to reduce this loss function by ﬁring a spike. Speciﬁcally, the ith neuron will spike whenever L(no spike in i) > L(spike in i). This leads to the following spiking rule for the ith neuron: Vi > Ti (14) where Vi ≡ Γ(x − x) − µoi and Ti ≡ Γ2 /2 + µ/2. We can naturally interpret Vi as the membrane ˆ potential of the ith neuron and Ti as the spiking threshold of that neuron. As before, we can now derive membrane potential dynamics: ˙ V = −V + ΓT c − (ΓT Γ + µI)o, 2 (15) The read-out weights must scale as Γ ∼ 1/N so that ﬁring rates are not unrealistically small in large networks. We can see this by calculating the average ﬁring rate N oi /N ≈ x/(ΓN ) ∼ O(N/N ) ∼ O(1). i=1 ¯ 4 where I is the identity matrix and ΓT Γ + µI is the network connectivity. We can interpret the selfconnection terms {Γ2 +µ} as voltage resets that decrease the voltage of any neuron that spikes. This optimal network is equivalent to a network of identical integrate-and-ﬁre neurons with homogeneous inhibitory connectivity. The network has some interesting dynamical properties. The voltages of all the neurons are largely synchronous, all increasing to the spiking threshold at about the same time3 (Fig. 1F). Nonetheless, neural spiking is asynchronous. The ﬁrst neuron to spike will reset itself by Γ2 + µ, and it will inhibit all the other neurons in the network by Γ2 . This mechanism prevents neurons from spik- x 3 The ﬁrst neuron to spike will be random if there is some membrane potential noise. V (A) (B) x x ˆ x 10 1 0.1 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 1 D 0.5 V V 0 ˆ x V ˆ x (C) 1 0 1 2 0 0.625 25 25.625 (D) start of learning 1 V 50 200.625 400 400.625 1 2.4 O 1.78 ω 1.77 25 neuron$ 0 1 2 !me$ 3 4 25 1 5 V 400.625 !me$ (F) 25 1 2.35 1.05 1.049 400 25.625 !me$ (E) neuron$ 100.625 200 end of learning 1.4 1.35 ω 100 !me$ 1 V 1 O 50.625 0 1 2 !me$ 3 4 5 V !me$ !me$ Figure 1: Learning in a single neuron and a homogeneous network. (A) A single neuron represents an input signal x by producing an output x. (B) During learning, the single neuron output x (solid red ˆ ˆ line, top panel) converges towards the input x (blue). Similarly, for a homogeneous network the output x (dashed red line, top panel) converges towards x. Connectivity also converges towards optimal ˆ connectivity in both the single neuron case (solid black line, middle panel) and the homogeneous net2 2 work case (dashed black line, middle panel), as quantiﬁed by D = maxi,j ( Ωij − Ωopt / Ωopt ) ij ij at each point in time. Consequently, the membrane potential reset (bottom panel) converges towards the optimal reset (green line, bottom panel). Spikes are indicated by blue vertical marks, and are produced when the membrane potential reaches threshold (bottom panel). Here, we have rescaled time, as indicated, for clarity. (C) Our learning rule dictates that the autapse ω in our single neuron (bottom panel) changes in proportion to the membrane potential (top panel) and the ﬁring rate (middle panel). (D) At the end of learning, the reset ω ﬂuctuates weakly about the optimal value. (E) For a homogeneous network, neurons spike regularly at the start of learning, as shown in this raster plot. Membrane potentials of different neurons are weakly correlated. (F) At the end of learning, spiking is very irregular and membrane potentials become more synchronous. 5 ing synchronously. The population as a whole acts similarly to the single neuron in our previous example. Each neuron ﬁres regularly, even if a different neuron ﬁres in every integration cycle. The design of this optimal network requires the tuning of N (N − 1) synaptic parameters. How can an arbitrary network of integrate-and-ﬁre neurons learn this optimum? As before, we address this question by using the optimal network as a target for learning. We start with an arbitrarily connected network of integrate-and-ﬁre neurons: ˙ V = −V + ΓT c − Ωo, (16) where Ω is a matrix of connectivity weights, which includes the resets of the individual neurons. Assuming that learning occurs on a slow time scale, we can rewrite this equation as V = ΓT x − Ω¯ . o (17) Now, repeating the arguments from the single neuron derivation, we modify the loss function to obtain an online learning rule. Speciﬁcally, we set LV = V 2 /2, and calculate the gradient: ∂LV = ∂Ωij Vk k ∂Vk =− ∂Ωij Vk δki oj − ¯ k Vk Ωkl kl ∂ ol ¯ . ∂Ωij (18) We can simplify this equation considerably by observing that the contribution of the second summation is largely averaged out under a wide variety of realistic conditions4 . Therefore, it can be neglected, and we obtain the following local learning rule: ∂LV ˙ = V i oj . ¯ τ Ωij = − ∂Ωij (19) This is a Hebbian plasticity rule, whereby connectivity changes in proportion to the presynaptic ﬁring rate oj and post-synaptic membrane potential Vi . We assume that the neural thresholds are set ¯ to a constant T and that the neural resets are set to their optimal values −T . In the previous section we demonstrated that these resets can be obtained by a Hebbian plasticity rule (Eqn. (10)). This learning rule minimises the difference between the read-out and the signal, by approaching the optimal recurrent connection strengths for the network (Fig. 1B). As in the single neuron case, learning does not stop, so the connection strengths ﬂuctuate close to their optimal value. During learning, network activity becomes progressively more asynchronous as it progresses towards optimal connectivity (Fig. 1E, F). 3 Learning in the general case Now that we have developed the fundamental concepts underlying our learning rule, we can derive a learning rule for the more general case of a network of N arbitrarily connected leaky integrateand-ﬁre neurons. Our goal is to understand how such networks can learn to optimally represent a ˙ J-dimensional signal x = (x1 , . . . , xJ ), using the read-out equation x = −x + Γo. We consider a network with the following membrane potential dynamics: ˙ V = −V + ΓT c − Ωo, (20) where c is a J-dimensional input. We assume that this input is related to the signal according to ˙ c = x + x. This assumption can be relaxed by treating the input as the control for an arbitrary linear dynamical system, in which case the signal represented by the network is the output of such a computation [11]. However, this further generalisation is beyond the scope of this work. As before, we need to identify the optimal recurrent connectivity so that we have a target for learning. Most generally, the optimal recurrent connectivity is Ωopt ≡ ΓT Γ + µI. The output kernels of the individual neurons, Γi , are given by the rows of Γ, and their spiking thresholds by Ti ≡ Γi 2 /2 + 4 From the deﬁnition of the membrane potential we can see that Vk ∼ O(1/N ) because Γ ∼ 1/N . Therefore, the size of the ﬁrst term in Eqn. (18) is k Vk δki oj = Vi oj ∼ O(1/N ). Therefore, the second term can ¯ ¯ be ignored if kl Vk Ωkl ∂ ol /∂Ωij ¯ O(1/N ). This happens if Ωkl O(1/N 2 ) as at the start of learning. It also happens towards the end of learning if the terms {Ωkl ∂ ol /∂Ωij } are weakly correlated with zero mean, ¯ or if the membrane potentials {Vi } are weakly correlated with zero mean. 6 µ/2. With these connections and thresholds, we ﬁnd that a network of integrate-and-ﬁre neurons ˆ ¯ will produce spike trains in such a way that the loss function L = x − x 2 + µ o 2 is minimised, ˆ where the read-out is given by x = Γ¯ . We can show this by prescribing a greedy5 spike rule: o a spike is ﬁred by neuron i whenever L(no spike in i) > L(spike in i) [11]. The resulting spike generation rule is Vi > Ti , (21) ˆ where Vi ≡ ΓT (x − x) − µ¯i is interpreted as the membrane potential. o i 5 Despite being greedy, this spiking rule can generate ﬁring rates that are practically identical to the optimal solutions: we checked this numerically in a large ensemble of networks with randomly chosen kernels. (A) x1 … x … 1 1 (B) xJJ x 10 L 10 T T 10 4 6 8 1 Viii V D ˆˆ ˆˆ x11 xJJ x x F 0.5 0 0.4 … … 0.2 0 0 2000 4000 !me (C) x V V 1 x 10 x 3 ˆ x 8 0 x 10 1 2 3 !me 4 5 4 0 1 4 0 1 8 V (F) Ρ(Δt) E-‐I input 0.4 ˆ x 0 3 0 1 x 10 1.3 0.95 x 10 ˆ x 4 V (E) 1 x 0 end of learning 50 neuron neuron 50 !me 2 0 ˆ x 0 0.5 ISI Δt 1 2 !me 4 5 4 1.5 1.32 3 2 0.1 Ρ(Δt) x E-‐I input (D) start of learning 0 2 !me 0 0 0.5 ISI Δt 1 Figure 2: Learning in a heterogeneous network. (A) A network of neurons represents an input ˆ signal x by producing an output x. (B) During learning, the loss L decreases (top panel). The difference between the connection strengths and the optimal strengths also decreases (middle panel), as 2 2 quantiﬁed by the mean difference (solid line), given by D = Ω − Ωopt / Ωopt and the maxi2 2 mum difference (dashed line), given by maxi,j ( Ωij − Ωopt / Ωopt ). The mean population ﬁring ij ij rate (solid line, bottom panel) also converges towards the optimal ﬁring rate (dashed line, bottom panel). (C, E) Before learning, a raster plot of population spiking shows that neurons produce bursts ˆ of spikes (upper panel). The network output x (red line, middle panel) fails to represent x (blue line, middle panel). The excitatory input (red, bottom left panel) and inhibitory input (green, bottom left panel) to a randomly selected neuron is not tightly balanced. Furthermore, a histogram of interspike intervals shows that spiking activity is not Poisson, as indicated by the red line that represents a best-ﬁt exponential distribution. (D, F) At the end of learning, spiking activity is irregular and ˆ Poisson-like, excitatory and inhibitory input is tightly balanced and x matches x. 7 How can we learn this optimal connection matrix? As before, we can derive a learning rule by minimising the cost function LV = V 2 /2. This leads to a Hebbian learning rule with the same form as before: ˙ τ Ωij = Vi oj . ¯ (22) Again, we assume that the neural resets are given by −Ti . Furthermore, in order for this learning rule to work, we must assume that the network input explores all possible directions in the J-dimensional input space (since the kernels Γi can point in any of these directions). The learning performance does not critically depend on how the input variable space is sampled as long as the exploration is extensive. In our simulations, we randomly sample the input c from a Gaussian white noise distribution at every time step for the entire duration of the learning. We ﬁnd that this learning rule decreases the loss function L, thereby approaching optimal network connectivity and producing optimal ﬁring rates for our linear decoder (Fig. 2B). In this example, we have chosen connectivity that is initially much too weak at the start of learning. Consequently, the initial network behaviour is similar to a collection of unconnected single neurons that ignore each other. Spike trains are not Poisson-like, ﬁring rates are excessively large, excitatory and inhibitory ˆ input is unbalanced and the decoded variable x is highly unreliable (Fig. 2C, E). As a result of learning, the network becomes tightly balanced and the spike trains become asynchronous, irregular and Poisson-like with much lower rates (Fig. 2D, F). However, despite this apparent variability, the population representation is extremely precise, only limited by the the metabolic cost and the discrete nature of a spike. This learnt representation is far more precise than a rate code with independent Poisson spike trains [11]. In particular, shufﬂing the spike trains in response to identical inputs drastically degrades this precision. 4 Conclusions and Discussion In population coding, large trial-to-trial spike train variability is usually interpreted as noise [2]. We show here that a deterministic network of leaky integrate-and-ﬁre neurons with a simple Hebbian plasticity rule can self-organise into a regime where information is represented far more precisely than in noisy rate codes, while appearing to have noisy Poisson-like spiking dynamics. Our learning rule (Eqn. (22)) has the basic properties of STDP. Speciﬁcally, a presynaptic spike occurring immediately before a post-synaptic spike will potentiate a synapse, because membrane potentials are positive immediately before a postsynaptic spike. Furthermore, a presynaptic spike occurring immediately after a post-synaptic spike will depress a synapse, because membrane potentials are always negative immediately after a postsynaptic spike. This is similar in spirit to the STDP rule proposed in [12], but different to classical STDP, which depends on post-synaptic spike times [9]. This learning rule can also be understood as a mechanism for generating a tight balance between excitatory and inhibitory input. We can see this by observing that membrane potentials after learning can be interpreted as representation errors (projected onto the read-out kernels). Therefore, learning acts to minimise the magnitude of membrane potentials. Excitatory and inhibitory input must be balanced if membrane potentials are small, so we can equate balance with optimal information representation. Previous work has shown that the balanced regime produces (quasi-)chaotic network dynamics, thereby accounting for much observed cortical spike train variability [13, 14, 4]. Moreover, the STDP rule has been known to produce a balanced regime [16, 17]. Additionally, recent theoretical studies have suggested that the balanced regime plays an integral role in network computation [15, 13]. In this work, we have connected these mechanisms and functions, to conclude that learning this balance is equivalent to the development of an optimal spike-based population code, and that this learning can be achieved using a simple Hebbian learning rule. Acknowledgements We are grateful for generous funding from the Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (CKM) and the Chaire d’excellence of the Agence National de la Recherche (CKM, DB), as well as a James Mcdonnell Foundation Award (SD) and EU grants BACS FP6-IST-027140, BIND MECT-CT-20095-024831, and ERC FP7-PREDSPIKE (SD). 8 References [1] Tolhurst D, Movshon J, and Dean A (1982) The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Res 23: 775–785. [2] Shadlen MN, Newsome WT (1998) The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. J Neurosci 18(10): 3870–3896. [3] Zohary E, Newsome WT (1994) Correlated neuronal discharge rate and its implication for psychophysical performance. Nature 370: 140–143. [4] Renart A, de la Rocha J, Bartho P, Hollender L, Parga N, Reyes A, & Harris, KD (2010) The asynchronous state in cortical circuits. Science 327, 587–590. [5] Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK, Tolias AS (2010) Decorrelated neuronal ﬁring in cortical microcircuits. Science 327: 584–587. [6] Okun M, Lampl I (2008) Instantaneous correlation of excitation and inhibition during ongoing and sensory-evoked activities. Nat Neurosci 11, 535–537. [7] Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrent balanced cortical activity. Nature 423, 288–293. [8] Gentet LJ, Avermann M, Matyas F, Staiger JF, Petersen CCH (2010) Membrane potential dynamics of GABAergic neurons in the barrel cortex of behaving mice. Neuron 65: 422–435. [9] Caporale N, Dan Y (2008) Spike-timing-dependent plasticity: a Hebbian learning rule. Annu Rev Neurosci 31: 25–46. [10] Boerlin M, Deneve S (2011) Spike-based population coding and working memory. PLoS Comput Biol 7, e1001080. [11] Boerlin M, Machens CK, Deneve S (2012) Predictive coding of dynamic variables in balanced spiking networks. under review. [12] Clopath C, B¨ sing L, Vasilaki E, Gerstner W (2010) Connectivity reﬂects coding: a model of u voltage-based STDP with homeostasis. Nat Neurosci 13(3): 344–352. [13] van Vreeswijk C, Sompolinsky H (1998) Chaotic balanced state in a model of cortical circuits. Neural Comput 10(6): 1321–1371. [14] Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory neurons. J Comput Neurosci 8, 183–208. [15] Vogels TP, Rajan K, Abbott LF (2005) Neural network dynamics. Annu Rev Neurosci 28: 357–376. [16] Vogels TP, Sprekeler H, Zenke F, Clopath C, Gerstner W. (2011) Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. Science 334(6062):1569– 73. [17] Song S, Miller KD, Abbott LF (2000) Competitive Hebbian learning through spike-timingdependent synaptic plasticity. Nat Neurosci 3(9): 919–926. 9

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We answer this question by deriving spiking dynamics and learning dynamics directly from a measure of network performance. [sent-14, score-0.646]

2 We ﬁnd that a network of integrate-and-ﬁre neurons undergoing Hebbian plasticity can learn an optimal spike-based representation for a linear decoder. [sent-15, score-0.557]

3 The learning rule acts to minimise the membrane potential magnitude, which can be interpreted as a representation error after learning. [sent-16, score-0.568]

4 The network becomes balanced because small representation errors correspond to small membrane potentials, which in turn results from a balance of excitation and inhibition. [sent-18, score-0.657]

5 The representation is robust because neurons become self-correcting, only spiking if the representation error exceeds a threshold. [sent-19, score-0.494]

6 The essential idea is to derive spiking dynamics from the assumption that neurons only ﬁre if their spike improves information representation. [sent-29, score-0.995]

7 Information in a network may ∗ Authors contributed equally 1 originate from several possible sources: external sensory input, external neural network input, or alternatively, it may originate within the network itself as a memory, or as a computation. [sent-30, score-0.493]

8 Whatever the source, this initial assumption leads directly to the conclusion that a network of integrate-and-ﬁre neurons can optimally represent a signal while exhibiting properties (a)–(c). [sent-31, score-0.418]

9 A major problem with this framework is that network connectivity must be completely speciﬁed a priori, and requires the tuning of N 2 parameters, where N is the number of neurons in the network. [sent-32, score-0.453]

10 The key insight is that the plasticity rule can be derived from the same basic principle as the spiking rule in the earlier work—namely, that any change should improve information representation. [sent-35, score-0.7]

11 Surprisingly, this can be achieved with a local, Hebbian learning rule, where synaptic plasticity is proportional to the product of presynaptic ﬁring rates with post-synaptic membrane potentials. [sent-36, score-0.649]

12 For a large set of initial connectivities and spiking dynamics, neural networks are driven into a balanced regime, where excitation and inhibition cancel each other and where spike trains are asynchronous and irregular. [sent-38, score-0.975]

13 In this way, a network can learn to represent information optimally, with synaptic, neural and network dynamics consistent with those observed experimentally. [sent-40, score-0.407]

14 1 Derivation of the learning rule for a single neuron We begin by deriving a learning rule for a single neuron with an autapse (a self-connection) (Fig. [sent-41, score-1.339]

15 Our approach is to derive synaptic dynamics for the autapse and spiking dynamics for the neuron such that the neuron learns to optimally represent a time-varying input signal. [sent-43, score-1.737]

16 We will derive a learning rule for networks of neurons later, after we have developed the fundamental concepts for the single neuron case. [sent-44, score-0.826]

17 Our ﬁrst step is to derive optimal spiking dynamics for the neuron, so that we have a target for our learning rule. [sent-45, score-0.431]

18 Next, we assume that the neuron only produces a spike if that spike improves the read-out, where we measure the read-out performance through a simple squared-error loss function: 2 L(t) = x(t) − x(t) . [sent-48, score-1.312]

19 First, we observe that if the neuron produces an additional spike at time t, the read-out increases by Γ, and the loss function becomes L(t|spike) = (x(t) − (x(t) + Γ))2 . [sent-50, score-0.896]

20 This allows us to restate our spiking rule as follows: ˆ the neuron should only produce a spike if L(t|no spike) > L(t|spike), or (x(t) − x(t))2 > (x(t) − ˆ (x(t) + Γ))2 . [sent-51, score-1.254]

21 Now, squaring both sides of this inequality, deﬁning V (t) ≡ Γ(x(t) − x(t)) and ˆ ˆ deﬁning T ≡ Γ2 /2 we ﬁnd that the neuron should only spike if: V (t) > T. [sent-52, score-0.858]

22 (3) We interpret V (t) to be the membrane potential of the neuron, and we interpret T as the spike threshold. [sent-53, score-0.805]

23 This interpretation allows us to understand the membrane potential functionally: the voltage is proportional to a prediction error—the difference between the read-out x(t) and the actual ˆ signal x(t). [sent-54, score-0.463]

24 A spike is an error reduction mechanism—the neuron only spikes if the error exceeds the spike threshold. [sent-55, score-1.331]

25 This is a greedy minimisation, in that the neuron ﬁres a spike whenever that action decreases L(t) without considering the future impact of that spike. [sent-56, score-0.889]

26 Importantly, the neuron does not require direct access to the loss function L(t). [sent-57, score-0.48]

27 This corresponds exactly to the dynamics of a leaky integrate˙ and-ﬁre neuron with an inhibitory autapse1 of strength Γ2 , and a feedforward connection strength Γ. [sent-62, score-0.81]

28 The dynamics and connectivity guarantee that a neuron spikes at just the right times to optimise the loss function (Fig. [sent-63, score-0.771]

29 We now have a target for learning—we know the connection strength that a neuron must have at the end of learning if it is to represent information optimally, for a linear read-out. [sent-67, score-0.508]

30 We can use this target to derive synaptic dynamics that can learn an optimal representation from experience. [sent-68, score-0.381]

31 Speciﬁcally, we consider an integrate-and-ﬁre neuron with some arbitrary autapse strength ω. [sent-69, score-0.585]

32 The dynamics of this neuron are given by ˙ V = −V + Γc − ωo. [sent-70, score-0.559]

33 (5) This neuron will not produce the correct spike train for representing x through a linear read-out (Eqn. [sent-71, score-0.889]

34 Our goal is to derive a dynamical equation for the synapse ω so that the spike train becomes optimal. [sent-73, score-0.526]

35 The loss function underlying the spiking dynamics determined by Eqn. [sent-75, score-0.392]

36 The solution to this latter equation is o = h ∗ o, a convolution ¯ ¯ o ¯ of the spike train with the exponential kernel h(τ ) = θ(τ ) exp(−τ ). [sent-79, score-0.447]

37 3 This synaptic learning rule is capable of learning the synaptic weight ω that minimises the difference between x and x (Fig. [sent-94, score-0.51]

38 During learning, the synaptic weight changes in proportion to the postˆ synaptic voltage V and the pre-synaptic ﬁring rate o (Fig. [sent-96, score-0.407]

39 Of course, in this single neuron case, the pre-synaptic neuron and post-synaptic neuron are the same neuron. [sent-99, score-1.326]

40 2 Learning in a homogeneous network We now generalise our learning rule derivation to a network of N identical, homogeneously connected neurons. [sent-109, score-0.539]

41 This generalisation is reasonably straightforward because many characteristics of the single neuron case are shared by a network of identical neurons. [sent-110, score-0.617]

42 We begin by deriving optimal spiking dynamics, as in the single neuron case. [sent-112, score-0.752]

43 We assume that the read-out x is provided by summing and ﬁltering the spike trains of all the neurons in the network: ˆ ˙ x = −ˆ + Γo, ˆ x (12) 2 where the row vector Γ = (Γ, . [sent-115, score-0.668]

44 Next, we assume that a neuron only spikes if that spike reduces a loss-function. [sent-123, score-0.915]

45 This spiking rule is similar to the single neuron spiking rule except that this time there is some ambiguity about which neuron should spike to represent a signal. [sent-124, score-2.092]

46 Indeed, there are many different spike patterns that provide exactly the same estimate x. [sent-125, score-0.416]

47 For example, one neuron could ﬁre regularly at a high rate (exactly like ˆ our previous single neuron example) while all others are silent. [sent-126, score-0.937]

48 Each neuron in the optimal network will seek to reduce this loss function by ﬁring a spike. [sent-129, score-0.668]

49 Speciﬁcally, the ith neuron will spike whenever L(no spike in i) > L(spike in i). [sent-130, score-1.274]

50 This leads to the following spiking rule for the ith neuron: Vi > Ti (14) where Vi ≡ Γ(x − x) − µoi and Ti ≡ Γ2 /2 + µ/2. [sent-131, score-0.396]

51 We can naturally interpret Vi as the membrane ˆ potential of the ith neuron and Ti as the spiking threshold of that neuron. [sent-132, score-1.04]

52 As before, we can now derive membrane potential dynamics: ˙ V = −V + ΓT c − (ΓT Γ + µI)o, 2 (15) The read-out weights must scale as Γ ∼ 1/N so that ﬁring rates are not unrealistically small in large networks. [sent-133, score-0.367]

53 We can interpret the selfconnection terms {Γ2 +µ} as voltage resets that decrease the voltage of any neuron that spikes. [sent-136, score-0.733]

54 This optimal network is equivalent to a network of identical integrate-and-ﬁre neurons with homogeneous inhibitory connectivity. [sent-137, score-0.72]

55 The voltages of all the neurons are largely synchronous, all increasing to the spiking threshold at about the same time3 (Fig. [sent-139, score-0.428]

56 The ﬁrst neuron to spike will reset itself by Γ2 + µ, and it will inhibit all the other neurons in the network by Γ2 . [sent-142, score-1.272]

57 This mechanism prevents neurons from spik- x 3 The ﬁrst neuron to spike will be random if there is some membrane potential noise. [sent-143, score-1.382]

58 me$ Figure 1: Learning in a single neuron and a homogeneous network. [sent-169, score-0.502]

59 (A) A single neuron represents an input signal x by producing an output x. [sent-170, score-0.54]

60 (B) During learning, the single neuron output x (solid red ˆ ˆ line, top panel) converges towards the input x (blue). [sent-171, score-0.54]

61 Connectivity also converges towards optimal ˆ connectivity in both the single neuron case (solid black line, middle panel) and the homogeneous net2 2 work case (dashed black line, middle panel), as quantiﬁed by D = maxi,j ( Ωij − Ωopt / Ωopt ) ij ij at each point in time. [sent-173, score-0.891]

62 Consequently, the membrane potential reset (bottom panel) converges towards the optimal reset (green line, bottom panel). [sent-174, score-0.598]

63 (C) Our learning rule dictates that the autapse ω in our single neuron (bottom panel) changes in proportion to the membrane potential (top panel) and the ﬁring rate (middle panel). [sent-177, score-1.041]

64 (E) For a homogeneous network, neurons spike regularly at the start of learning, as shown in this raster plot. [sent-179, score-0.72]

65 (F) At the end of learning, spiking is very irregular and membrane potentials become more synchronous. [sent-181, score-0.619]

66 The population as a whole acts similarly to the single neuron in our previous example. [sent-183, score-0.495]

67 Each neuron ﬁres regularly, even if a different neuron ﬁres in every integration cycle. [sent-184, score-0.884]

68 o (17) Now, repeating the arguments from the single neuron derivation, we modify the loss function to obtain an online learning rule. [sent-190, score-0.48]

69 ¯ τ Ωij = − ∂Ωij (19) This is a Hebbian plasticity rule, whereby connectivity changes in proportion to the presynaptic ﬁring rate oj and post-synaptic membrane potential Vi . [sent-194, score-0.749]

70 In the previous section we demonstrated that these resets can be obtained by a Hebbian plasticity rule (Eqn. [sent-196, score-0.369]

71 This learning rule minimises the difference between the read-out and the signal, by approaching the optimal recurrent connection strengths for the network (Fig. [sent-198, score-0.491]

72 As in the single neuron case, learning does not stop, so the connection strengths ﬂuctuate close to their optimal value. [sent-200, score-0.553]

73 During learning, network activity becomes progressively more asynchronous as it progresses towards optimal connectivity (Fig. [sent-201, score-0.4]

74 3 Learning in the general case Now that we have developed the fundamental concepts underlying our learning rule, we can derive a learning rule for the more general case of a network of N arbitrarily connected leaky integrateand-ﬁre neurons. [sent-203, score-0.411]

75 We consider a network with the following membrane potential dynamics: ˙ V = −V + ΓT c − Ωo, (20) where c is a J-dimensional input. [sent-208, score-0.478]

76 The output kernels of the individual neurons, Γi , are given by the rows of Γ, and their spiking thresholds by Ti ≡ Γi 2 /2 + 4 From the deﬁnition of the membrane potential we can see that Vk ∼ O(1/N ) because Γ ∼ 1/N . [sent-214, score-0.57]

77 It also happens towards the end of learning if the terms {Ωkl ∂ ol /∂Ωij } are weakly correlated with zero mean, ¯ or if the membrane potentials {Vi } are weakly correlated with zero mean. [sent-219, score-0.495]

78 With these connections and thresholds, we ﬁnd that a network of integrate-and-ﬁre neurons ˆ ¯ will produce spike trains in such a way that the loss function L = x − x 2 + µ o 2 is minimised, ˆ where the read-out is given by x = Γ¯ . [sent-221, score-0.851]

79 We can show this by prescribing a greedy5 spike rule: o a spike is ﬁred by neuron i whenever L(no spike in i) > L(spike in i) [11]. [sent-222, score-1.69]

80 The resulting spike generation rule is Vi > Ti , (21) ˆ where Vi ≡ ΓT (x − x) − µ¯i is interpreted as the membrane potential. [sent-223, score-0.853]

81 o i 5 Despite being greedy, this spiking rule can generate ﬁring rates that are practically identical to the optimal solutions: we checked this numerically in a large ensemble of networks with randomly chosen kernels. [sent-224, score-0.469]

82 95 x 10 ˆ x 4 V (E) 1 x 0 end of learning 50 neuron neuron 50 ! [sent-233, score-0.884]

83 (A) A network of neurons represents an input ˆ signal x by producing an output x. [sent-242, score-0.434]

84 (C, E) Before learning, a raster plot of population spiking shows that neurons produce bursts ˆ of spikes (upper panel). [sent-246, score-0.538]

85 The excitatory input (red, bottom left panel) and inhibitory input (green, bottom left panel) to a randomly selected neuron is not tightly balanced. [sent-248, score-0.791]

86 (D, F) At the end of learning, spiking activity is irregular and ˆ Poisson-like, excitatory and inhibitory input is tightly balanced and x matches x. [sent-250, score-0.62]

87 Furthermore, in order for this learning rule to work, we must assume that the network input explores all possible directions in the J-dimensional input space (since the kernels Γi can point in any of these directions). [sent-255, score-0.376]

88 We ﬁnd that this learning rule decreases the loss function L, thereby approaching optimal network connectivity and producing optimal ﬁring rates for our linear decoder (Fig. [sent-258, score-0.64]

89 As a result of learning, the network becomes tightly balanced and the spike trains become asynchronous, irregular and Poisson-like with much lower rates (Fig. [sent-264, score-0.791]

90 This learnt representation is far more precise than a rate code with independent Poisson spike trains [11]. [sent-267, score-0.51]

91 In particular, shufﬂing the spike trains in response to identical inputs drastically degrades this precision. [sent-268, score-0.507]

92 4 Conclusions and Discussion In population coding, large trial-to-trial spike train variability is usually interpreted as noise [2]. [sent-269, score-0.5]

93 We show here that a deterministic network of leaky integrate-and-ﬁre neurons with a simple Hebbian plasticity rule can self-organise into a regime where information is represented far more precisely than in noisy rate codes, while appearing to have noisy Poisson-like spiking dynamics. [sent-270, score-0.964]

94 Speciﬁcally, a presynaptic spike occurring immediately before a post-synaptic spike will potentiate a synapse, because membrane potentials are positive immediately before a postsynaptic spike. [sent-273, score-1.286]

95 Furthermore, a presynaptic spike occurring immediately after a post-synaptic spike will depress a synapse, because membrane potentials are always negative immediately after a postsynaptic spike. [sent-274, score-1.286]

96 This is similar in spirit to the STDP rule proposed in [12], but different to classical STDP, which depends on post-synaptic spike times [9]. [sent-275, score-0.575]

97 This learning rule can also be understood as a mechanism for generating a tight balance between excitatory and inhibitory input. [sent-276, score-0.381]

98 We can see this by observing that membrane potentials after learning can be interpreted as representation errors (projected onto the read-out kernels). [sent-277, score-0.376]

99 Excitatory and inhibitory input must be balanced if membrane potentials are small, so we can equate balance with optimal information representation. [sent-279, score-0.669]

100 Previous work has shown that the balanced regime produces (quasi-)chaotic network dynamics, thereby accounting for much observed cortical spike train variability [13, 14, 4]. [sent-280, score-0.82]

similar papers computed by tfidf model

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same-paper 1 0.99999994 190 nips-2012-Learning optimal spike-based representations

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2 0.32417318 239 nips-2012-Neuronal Spike Generation Mechanism as an Oversampling, Noise-shaping A-to-D converter

Author: Dmitri B. Chklovskii, Daniel Soudry

Abstract: We test the hypothesis that the neuronal spike generation mechanism is an analog-to-digital (AD) converter encoding rectified low-pass filtered summed synaptic currents into a spike train linearly decodable in postsynaptic neurons. Faithful encoding of an analog waveform by a binary signal requires that the spike generation mechanism has a sampling rate exceeding the Nyquist rate of the analog signal. Such oversampling is consistent with the experimental observation that the precision of the spikegeneration mechanism is an order of magnitude greater than the cut -off frequency of low-pass filtering in dendrites. Additional improvement in the coding accuracy may be achieved by noise-shaping, a technique used in signal processing. If noise-shaping were used in neurons, it would reduce coding error relative to Poisson spike generator for frequencies below Nyquist by introducing correlations into spike times. By using experimental data from three different classes of neurons, we demonstrate that biological neurons utilize noise-shaping. Therefore, the spike-generation mechanism can be viewed as an oversampling and noise-shaping AD converter. The nature of the neural spike code remains a central problem in neuroscience [1-3]. In particular, no consensus exists on whether information is encoded in firing rates [4, 5] or individual spike timing [6, 7]. On the single-neuron level, evidence exists to support both points of view. On the one hand, post-synaptic currents are low-pass-filtered by dendrites with the cut-off frequency of approximately 30Hz [8], Figure 1B, providing ammunition for the firing rate camp: if the signal reaching the soma is slowly varying, why would precise spike timing be necessary? On the other hand, the ability of the spike-generation mechanism to encode harmonics of the injected current up to about 300Hz [9, 10], Figure 1B, points at its exquisite temporal precision [11]. Yet, in view of the slow variation of the somatic current, such precision may seem gratuitous and puzzling. The timescale mismatch between gradual variation of the somatic current and high precision of spike generation has been addressed previously. Existing explanations often rely on the population nature of the neural code [10, 12]. Although this is a distinct possibility, the question remains whether invoking population coding is necessary. Other possible explanations for the timescale mismatch include the possibility that some synaptic currents (for example, GABAergic) may be generated by synapses proximal to the soma and therefore not subject to low-pass filtering or that the high frequency harmonics are so strong in the pre-synaptic spike that despite attenuation, their trace is still present. Although in some cases, these explanations could apply, for the majority of synaptic inputs to typical neurons there is a glaring mismatch. The perceived mismatch between the time scales of somatic currents and the spike-generation mechanism can be resolved naturally if one views spike trains as digitally encoding analog somatic currents [13-15], Figure 1A. Although somatic currents vary slowly, information that could be communicated by their analog amplitude far exceeds that of binary signals, such as all- or-none spikes, of the same sampling rate. Therefore, faithful digital encoding requires sampling rate of the digital signal to be much higher than the cut-off frequency of the analog signal, socalled over-sampling. Although the spike generation mechanism operates in continuous time, the high temporal precision of the spikegeneration mechanism may be viewed as a manifestation of oversampling, which is needed for the digital encoding of the analog signal. Therefore, the extra order of magnitude in temporal precision available to the spike-generation mechanism relative to somatic current, Figure 1B, is necessary to faithfully encode the amplitude of the analog signal, thus potentially reconciling the firing rate and the spike timing points of view [13-15]. Figure 1. Hybrid digital-analog operation of neuronal circuits. A. Post-synaptic currents are low-pass filtered and summed in dendrites (black) to produce a somatic current (blue). This analog signal is converted by the spike generation mechanism into a sequence of all-or-none spikes (green), a digital signal. Spikes propagate along an axon and are chemically transduced across synapses (gray) into post-synatpic currents (black), whose amplitude reflects synaptic weights, thus converting digital signal back to analog. B. Frequency response function for dendrites (blue, adapted from [8]) and for the spike generation mechanism (green, adapted from [9]). Note one order of magnitude gap between the cut off frequencies. C. Amplitude of the summed postsynaptic currents depends strongly on spike timing. If the blue spike arrives just 5ms later, as shown in red, the EPSCs sum to a value already 20% less. Therefore, the extra precision of the digital signal may be used to communicate the amplitude of the analog signal. In signal processing, efficient AD conversion combines the principle of oversampling with that of noise-shaping, which utilizes correlations in the digital signal to allow more accurate encoding of the analog amplitude. This is exemplified by a family of AD converters called modulators [16], of which the basic one is analogous to an integrate-and-fire (IF) neuron [13-15]. The analogy between the basic modulator and the IF neuron led to the suggestion that neurons also use noise-shaping to encode incoming analog current waveform in the digital spike train [13]. However, the hypothesis of noise-shaping AD conversion has never been tested experimentally in biological neurons. In this paper, by analyzing existing experimental datasets, we demonstrate that noise-shaping is present in three different classes of neurons from vertebrates and invertebrates. This lends support to the view that neurons act as oversampling and noise-shaping AD converters and accounts for the mismatch between the slowly varying somatic currents and precise spike timing. Moreover, we show that the degree of noise-shaping in biological neurons exceeds that used by basic  modulators or IF neurons and propose viewing more complicated models in the noise-shaping framework. This paper is organized as follows: We review the principles of oversampling and noise-shaping in Section 2. In Section 3, we present experimental evidence for noise-shaping AD conversion in neurons. In Section 4 we argue that rectification of somatic currents may improve energy efficiency and/or implement de-noising. 2 . Oversampling and noise-shaping in AD converters To understand how oversampling can lead to more accurate encoding of the analog signal amplitude in a digital form, we first consider a Poisson spike encoder, whose rate of spiking is modulated by the signal amplitude, Figure 2A. Such an AD converter samples an analog signal at discrete time points and generates a spike with a probability given by the (normalized) signal amplitude. Because of the binary nature of spike trains, the resulting spike train encodes the signal with a large error even when the sampling is done at Nyquist rate, i.e. the lowest rate for alias-free sampling. To reduce the encoding error a Poisson encoder can sample at frequencies, fs , higher than Nyquist, fN – hence, the term oversampling, Figure 2B. When combined with decoding by lowpass filtering (down to Nyquist) on the receiving end, this leads to a reduction of the error, which can be estimated as follows. The number of samples over a Nyquist half-period (1/2fN) is given by the oversampling ratio: . As the normalized signal amplitude, , stays roughly constant over the Nyquist half-period, it can be encoded by spikes generated with a fixed probability, x. For a Poisson process the variance in the number of spikes is equal to the mean, . Therefore, the mean relative error of the signal decoded by averaging over the Nyquist half-period: , (1) indicating that oversampling reduces transmission error. However, the weak dependence of the error on the oversampling frequency indicates diminishing returns on the investment in oversampling and motivates one to search for other ways to lower the error. Figure 2. Oversampling and noise-shaping in AD conversion. A. Analog somatic current (blue) and its digital code (green). The difference between the green and the blue curves is encoding error. B. Digital output of oversampling Poisson encoder over one Nyquist half-period. C. Error power spectrum of a Nyquist (dark green) and oversampled (light green) Poisson encoder. Although the total error power is the same, the fraction surviving low-pass filtering during decoding (solid green) is smaller in oversampled case. D. Basic  modulator. E. Signal at the output of the integrator. F. Digital output of the  modulator over one Nyquist period. G. Error power spectrum of the  modulator (brown) is shifted to higher frequencies and low-pass filtered during decoding. The remaining error power (solid brown) is smaller than for Poisson encoder. To reduce encoding error beyond the ½ power of the oversampling ratio, the principle of noiseshaping was put forward [17]. To illustrate noise-shaping consider a basic AD converter called  [18], Figure 2D. In the basic  modulator, the previous quantized signal is fed back and subtracted from the incoming signal and then the difference is integrated in time. Rather than quantizing the input signal, as would be done in the Poisson encoder,  modulator quantizes the integral of the difference between the incoming analog signal and the previous quantized signal, Figure 2F. One can see that, in the oversampling regime, the quantization error of the basic  modulator is significantly less than that of the Poisson encoder. As the variance in the number of spikes over the Nyquist period is less than one, the mean relative error of the signal is at most, , which is better than the Poisson encoder. To gain additional insight and understand the origin of the term noise-shaping, we repeat the above analysis in the Fourier domain. First, the Poisson encoder has a flat power spectrum up to the sampling frequency, Figure 2C. Oversampling preserves the total error power but extends the frequency range resulting in the lower error power below Nyquist. Second, a more detailed analysis of the basic  modulator, where the dynamics is linearized by replacing the quantization device with a random noise injection [19], shows that the quantization noise is effectively differentiated. Taking the derivative in time is equivalent to multiplying the power spectrum of the quantization noise by frequency squared. Such reduction of noise power at low frequencies is an example of noise shaping, Figure 2G. Under the additional assumption of the white quantization noise, such analysis yields: , (2) which for R >> 1 is significantly better performance than for the Poisson encoder, Eq.(1). As mentioned previously, the basic  modulator, Figure 2D, in the continuous-time regime is nothing other than an IF neuron [13, 20, 21]. In the IF neuron, quantization is implemented by the spike generation mechanism and the negative feedback corresponds to the after-spike reset. Note that resetting the integrator to zero is strictly equivalent to subtraction only for continuous-time operation. In discrete-time computer simulations, the integrator value may exceed the threshold, and, therefore, subtraction of the threshold value rather than reset must be used. Next, motivated by the -IF analogy, we look for the signs of noise-shaping AD conversion in real neurons. 3 . Experimental evidence of noise-shaping AD conversion in real neurons In order to determine whether noise-shaping AD conversion takes place in biological neurons, we analyzed three experimental datasets, where spike trains were generated by time-varying somatic currents: 1) rat somatosensory cortex L5 pyramidal neurons [9], 2) mouse olfactory mitral cells [22, 23], and 3) fruit fly olfactory receptor neurons [24]. In the first two datasets, the current was injected through an electrode in whole-cell patch clamp mode, while in the third, the recording was extracellular and the intrinsic somatic current could be measured because the glial compartment included only one active neuron. Testing the noise-shaping AD conversion hypothesis is complicated by the fact that encoded and decoded signals are hard to measure accurately. First, as somatic current is rectified by the spikegeneration mechanism, only its super-threshold component can be encoded faithfully making it hard to know exactly what is being encoded. Second, decoding in the dendrites is not accessible in these single-neuron recordings. In view of these difficulties, we start by simply computing the power spectrum of the reconstruction error obtained by subtracting a scaled and shifted, but otherwise unaltered, spike train from the somatic current. The scaling factor was determined by the total weight of the decoding linear filter and the shift was optimized to maximize information capacity, see below. At the frequencies below 20Hz the error contains significantly lower power than the input signal, Figure 3, indicating that the spike generation mechanism may be viewed as an AD converter. Furthermore, the error power spectrum of the biological neuron is below that of the Poisson encoder, thus indicating the presence of noise-shaping. For dataset 3 we also plot the error power spectrum of the IF neuron, the threshold of which is chosen to generate the same number of spikes as the biological neuron. 4 somatic current biological neuron error Poisson encoder error I&F; neuron error 10 1 10 0 Spectral power, a.u. Spectral power, a.u. 10 3 10 -1 10 -2 10 -3 10 2 10 -4 10 0 10 20 30 40 50 60 Frequency [Hz] 70 80 90 0 10 20 30 40 50 60 70 80 90 100 Frequency [Hz] Figure 3. Evidence of noise-shaping. Power spectra of the somatic current (blue), difference between the somatic current and the digital spike train of the biological neuron (black), of the Poisson encoder (green) and of the IF neuron (red). Left: datset 1, right: dataset 3. Although the simple analysis presented above indicates noise-shaping, subtracting the spike train from the input signal, Figure 3, does not accurately quantify the error when decoding involves additional filtering. An example of such additional encoding/decoding is predictive coding, which will be discussed below [25]. To take such decoding filter into account, we computed a decoded waveform by convolving the spike train with the optimal linear filter, which predicts the somatic current from the spike train with the least mean squared error. Our linear decoding analysis lends additional support to the noise-shaping AD conversion hypothesis [13-15]. First, the optimal linear filter shape is similar to unitary post-synaptic currents, Figure 4B, thus supporting the view that dendrites reconstruct the somatic current of the presynaptic neuron by low-pass filtering the spike train in accordance with the noise-shaping principle [13]. Second, we found that linear decoding using an optimal filter accounts for 60-80% of the somatic current variance. Naturally, such prediction works better for neurons in suprathreshold regime, i.e. with high firing rates, an issue to which we return in Section 4. To avoid complications associated with rectification for now we focused on neurons which were in suprathreshold regime by monitoring that the relationship between predicted and actual current is close to linear. 2 10 C D 1 10 somatic current biological neuron error Poisson encoder error Spectral power, a.u. Spectral power, a.u. I&F; neuron error 3 10 0 10 -1 10 -2 10 -3 10 2 10 -4 0 10 20 30 40 50 60 Frequency [Hz] 70 80 90 10 0 10 20 30 40 50 60 70 80 90 100 Frequency [Hz] Figure 4. Linear decoding of experimentally recorded spike trains. A. Waveform of somatic current (blue), resulting spike train (black), and the linearly decoded waveform (red) from dataset 1. B. Top: Optimal linear filter for the trace in A, is representative of other datasets as well. Bottom: Typical EPSPs have a shape similar to the decoding filter (adapted from [26]). C-D. Power spectra of the somatic current (blue), the decdoding error of the biological neuron (black), the Poisson encoder (green), and IF neuron (red) for dataset 1 (C) dataset 3 (D). Next, we analyzed the spectral distribution of the reconstruction error calculated by subtracting the decoded spike train, i.e. convolved with the computed optimal linear filter, from the somatic current. We found that at low frequencies the error power is significantly lower than in the input signal, Figure 4C,D. This observation confirms that signals below the dendritic cut-off frequency of 20-30Hz can be efficiently communicated using spike trains. To quantify the effect of noise-shaping we computed information capacity of different encoders: where S(f) and N(f) are the power spectra of the somatic current and encoding error correspondingly and the sum is computed only over the frequencies for which S(f) > N(f). Because the plots in Figure 4C,D use semi-logrithmic scale, the information capacity can be estimated from the area between a somatic current (blue) power spectrum and an error power spectrum. We find that the biological spike generation mechanism has higher information capacity than the Poisson encoder and IF neurons. Therefore, neurons act as AD converters with stronger noise-shaping than IF neurons. We now return to the predictive nature of the spike generation mechanism. Given the causal nature of the spike generation mechanism it is surprising that the optimal filters for all three datasets carry most of their weight following a spike, Figure 4B. This indicates that the spike generation mechanism is capable of making predictions, which are possible in these experiments because somatic currents are temporally correlated. We note that these observations make delay-free reconstruction of the signal possible, thus allowing fast operation of neural circuits [27]. The predictive nature of the encoder can be captured by a  modulator embedded in a predictive coding feedback loop [28], Figure 5A. We verified by simulation that such a nested architecture generates a similar optimal linear filter with most of its weight in the time following a spike, Figure 5A right. Of course such prediction is only possible for correlated inputs implying that the shape of the optimal linear filter depends on the statistics of the inputs. The role of predictive coding is to reduce the dynamic range of the signal that enters , thus avoiding overloading. A possible biological implementation for such integrating feedback could be Ca2+ 2+ concentration and Ca dependent potassium channels [25, 29]. Figure 5. Enhanced  modulators. A.  modulator combined with predictive coder. In such device, the optimal decoding filter computed for correlated inputs has most of its weight following a spike, similar to experimental measurements, Figure 4B. B. Second-order  modulator possesses stronger noise-shaping properties. Because such circuit contains an internal state variable it generates a non-periodic spike train in response to a constant input. Bottom trace shows a typical result of a simulation. Black – spikes, blue – input current. 4 . Possible reasons for current rectification: energy efficiency and de-noising We have shown that at high firing rates biological neurons encode somatic current into a linearly decodable spike train. However, at low firing rates linear decoding cannot faithfully reproduce the somatic current because of rectification in the spike generation mechanism. If the objective of spike generation is faithful AD conversion, why would such rectification exist? We see two potential reasons: energy efficiency and de-noising. It is widely believed that minimizing metabolic costs is an important consideration in brain design and operation [30, 31]. Moreover, spikes are known to consume a significant fraction of the metabolic budget [30, 32] placing a premium on their total number. Thus, we can postulate that neuronal spike trains find a trade-off between the mean squared error in the decoded spike train relative to the input signal and the total number of spikes, as expressed by the following cost function over a time interval T: , (3) where x is the analog input signal, s is the binary spike sequence composed of zeros and ones, and is the linear filter. To demonstrate how solving Eq.(3) would lead to thresholding, let us consider a simplified version taken over a Nyquist period, during which the input signal stays constant: (4) where and normalized by w. Minimizing such a cost function reduces to choosing the lowest lying parabola for a given , Figure 6A. Therefore, thresholding is a natural outcome of minimizing a cost function combining the decoding error and the energy cost, Eq.(3). In addition to energy efficiency, there may be a computational reason for thresholding somatic current in neurons. To illustrate this point, we note that the cost function in Eq. (3) for continuous variables, st, may be viewed as a non-negative version of the L1-norm regularized linear regression called LASSO [33], which is commonly used for de-noising of sparse and Laplacian signals [34]. Such cost function can be minimized by iteratively applying a gradient descent and a shrinkage steps [35], which is equivalent to thresholding (one-sided in case of non-negative variables), Figure 6B,C. Therefore, neurons may be encoding a de-noised input signal. Figure 6. Possible reasons for rectification in neurons. A. Cost function combining encoding error squared with metabolic expense vs. input signal for different values of the spike number N, Eq.(4). Note that the optimal number of spikes jumps from zero to one as a function of input. B. Estimating most probable “clean” signal value for continuous non-negative Laplacian signal and Gaussian noise, Eq.(3) (while setting w = 1). The parabolas (red) illustrate the quadratic loglikelihood term in (3) for different values of the measurement, s, while the linear function (blue) reflects the linear log-prior term in (3). C. The minimum of the combined cost function in B is at zero if s , and grows linearly with s, if s >. 5 . Di scu ssi on In this paper, we demonstrated that the neuronal spike-generation mechanism can be viewed as an oversampling and noise-shaping AD converter, which encodes a rectified low-pass filtered somatic current as a digital spike train. Rectification by the spike generation mechanism may subserve both energy efficiency and de-noising. As the degree of noise-shaping in biological neurons exceeds that in IF neurons, or basic , we suggest that neurons should be modeled by more advanced  modulators, e.g. Figure 5B. Interestingly,  modulators can be also viewed as coders with error prediction feedback [19]. Many publications studied various aspects of spike generation in neurons yet we believe that the framework [13-15] we adopt is different and discuss its relationship to some of the studies. Our framework is different from previous proposals to cast neurons as predictors [36, 37] because a different quantity is being predicted. The possibility of perfect decoding from a spike train with infinite temporal precision has been proven in [38]. Here, we are concerned with a more practical issue of how reconstruction error scales with the over-sampling ratio. Also, we consider linear decoding which sets our work apart from [39]. Finally, previous experiments addressing noiseshaping [40] studied the power spectrum of the spike train rather than that of the encoding error. Our work is aimed at understanding biological and computational principles of spike-generation and decoding and is not meant as a substitute for the existing phenomenological spike-generation models [41], which allow efficient fitting of parameters and prediction of spike trains [42]. Yet, the theoretical framework [13-15] we adopt may assist in building better models of spike generation for a given somatic current waveform. First, having interpreted spike generation as AD conversion, we can draw on the rich experience in signal processing to attack the problem. Second, this framework suggests a natural metric to compare the performance of different spike generation models in the high firing rate regime: a mean squared error between the injected current waveform and the filtered version of the spike train produced by a model provided the total number of spikes is the same as in the experimental data. The AD conversion framework adds justification to the previously proposed spike distance obtained by subtracting low-pass filtered spike trains [43]. As the framework [13-15] we adopt relies on viewing neuronal computation as an analog-digital hybrid, which requires AD and DA conversion at every step, one may wonder about the reason for such a hybrid scheme. Starting with the early days of computers, the analog mode is known to be advantageous for computation. For example, performing addition of many variables in one step is possible in the analog mode simply by Kirchhoff law, but would require hundreds of logical gates in the digital mode [44]. However, the analog mode is vulnerable to noise build-up over many stages of computation and is inferior in precisely communicating information over long distances under limited energy budget [30, 31]. 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Abstract: We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of empirical to population measures, which, unlike existing bounds, are applicable to a wide class of measures. 1 Introduction and Motivation In this paper we study the problem of learning from random samples a probability distribution supported on a manifold, when the learning error is measured using transportation metrics. The problem of learning a probability distribution is classic in statistics, and is typically analyzed for distributions in X = Rd that have a density with respect to the Lebesgue measure, with total variation, and L2 among the common distances used to measure closeness of two densities (see for instance [10, 32] and references therein.) The setting in which the data distribution is supported on a low dimensional manifold embedded in a high dimensional space has only been considered more recently. In particular, kernel density estimators on manifolds have been described in [36], and their pointwise consistency, as well as convergence rates, have been studied in [25, 23, 18]. A discussion on several topics related to statistics on a Riemannian manifold can be found in [26]. Interestingly, the problem of approximating measures with respect to transportation distances has deep connections with the ﬁelds of optimal quantization [14, 16], optimal transport [35] and, as we point out in this work, with unsupervised learning (see Sec. 4.) In fact, as described in the sequel, some of the most widely-used algorithms for unsupervised learning, such as k-means (but also others such as PCA and k-ﬂats), can be shown to be performing exactly the task of estimating the data-generating measure in the sense of the 2-Wasserstein distance. This close relation between learning theory, and optimal transport and quantization seems novel and of interest in its own right. Indeed, in this work, techniques from the above three ﬁelds are used to derive the new probabilistic bounds described below. Our technical contribution can be summarized as follows: (a) we prove uniform lower bounds for the distance between a measure and estimates based on discrete sets (such as the empirical measure or measures derived from algorithms such as kmeans); (b) we provide new probabilistic bounds for the rate of convergence of empirical to population measures which, unlike existing probabilistic bounds, hold for a very large class of measures; 1 (c) we provide probabilistic bounds for the rate of convergence of measures derived from k-means to the data measure. The structure of the paper is described at the end of Section 2, where we discuss the exact formulation of the problem as well as related previous works. 2 Setup and Previous work Consider the problem of learning a probability measure ρ supported on a space M, from an i.i.d. sample Xn = (x1 , . . . , xn ) ∼ ρn of size n. We assume M to be a compact, smooth d-dimensional manifold of bounded curvature, with C 1 metric and volume measure λM , embedded in the unit ball of a separable Hilbert space X with inner product ·, · , induced norm · , and distance d (for d instance M = B2 (1) the unit ball in X = Rd .) Following [35, p. 94], let Pp (M) denote the Wasserstein space of order 1 ≤ p < ∞: Pp (M) := x p dρ(x) < ∞ ρ ∈ P (M) : M of probability measures P (M) supported on M, with ﬁnite p-th moment. The p-Wasserstein distance 1/p Wp (ρ, µ) = inf [E X − Y p ] : Law(X) = ρ, Law(Y ) = µ (1) X,Y where the random variables X and Y are distributed according to ρ and µ respectively, is the optimal expected cost of transporting points generated from ρ to those generated from µ, and is guaranteed to be ﬁnite in Pp (M) [35, p. 95]. The space Pp (M) with the Wp metric is itself a complete separable metric space [35]. We consider here the problem of learning probability measures ρ ∈ P2 (M), where the performance is measured by the distance W2 . There are many possible choices of distances between probability measures [13]. Among them, Wp metrizes weak convergence (see [35] theorem 6.9), that is, in Pp (M), a sequence (µi )i∈N of measures converges weakly to µ iff Wp (µi , µ) → 0 and their p-th order moments converge to that of µ. There are other distances, such as the L´ vy-Prokhorov, or the weak-* distance, that also metrize e weak convergence. However, as pointed out by Villani in his excellent monograph [35, p. 98], 1. “Wasserstein distances are rather strong, [...]a deﬁnite advantage over the weak-* distance”. 2. “It is not so difﬁcult to combine information on convergence in Wasserstein distance with some smoothness bound, in order to get convergence in stronger distances.” Wasserstein distances have been used to study the mixing and convergence of Markov chains [22], as well as concentration of measure phenomena [20]. To this list we would add the important fact that existing and widely-used algorithms for unsupervised learning can be easily extended (see Sec. 4) to compute a measure ρ that minimizes the distance W2 (ˆn , ρ ) to the empirical measure ρ n ρn := ˆ 1 δx , n i=1 i a fact that will allow us to prove, in Sec. 5, bounds on the convergence of a measure induced by k-means to the population measure ρ. The most useful versions of Wasserstein distance are p = 1, 2, with p = 1 being the weaker of the two (by H¨ lder’s inequality, p ≤ q ⇒ Wp ≤ Wq .) In particular, “results in W2 distance are usually o stronger, and more difﬁcult to establish than results in W1 distance” [35, p. 95]. A discussion of p = ∞ would take us out of topic, since its behavior is markedly different. 2.1 Closeness of Empirical and Population Measures By the strong law of large numbers, the empirical measure converges almost surely to the population measure: ρn → ρ in the sense of the weak topology [34]. Since weak convergence and convergence ˆ in Wp plus convergence of p-th moments are equivalent in Pp (M), this means that, in the Wp sense, the empirical measure ρn converges to ρ, as n → ∞. A fundamental question is therefore how fast ˆ the rate of convergence of ρn → ρ is. ˆ 2 2.1.1 Convergence in expectation The rate of convergence of ρn → ρ in expectation has been widely studied in the past, resultˆ ing in upper bounds of order EW2 (ρ, ρn ) = O(n−1/(d+2) ) [19, 8], and lower bounds of order ˆ EW2 (ρ, ρn ) = Ω(n−1/d ) [29] (both assuming that the absolutely continuous part of ρ is ρA = 0, ˆ with possibly better rates otherwise). More recently, an upper bound of order EWp (ρ, ρn ) = O(n−1/d ) has been proposed [2] by proving ˆ a bound for the Optimal Bipartite Matching (OBM) problem [1], and relating this problem to the expected distance EWp (ρ, ρn ). In particular, given two independent samples Xn , Yn , the OBM ˆ problem is that of ﬁnding a permutation σ that minimizes the matching cost n−1 xi −yσ(i) p [24, p ˆ ˆ ˆ 30]. It is not hard to show that the optimal matching cost is Wp (ˆXn , ρYn ) , where ρXn , ρYn are ρ the empirical measures associated to Xn , Yn . By Jensen’s inequality, the triangle inequality, and (a + b)p ≤ 2p−1 (ap + bp ), it holds EWp (ρ, ρn )p ≤ EWp (ˆXn , ρYn )p ≤ 2p−1 EWp (ρ, ρn )p , ˆ ρ ˆ ˆ and therefore a bound of order O(n−p/d ) for the OBM problem [2] implies a bound EWp (ρ, ρn ) = ˆ O(n−1/d ). The matching lower bound is only known for a special case: ρA constant over a bounded set of non-null measure [2] (e.g. ρA uniform.) Similar results, with matching lower bounds are found for W1 in [11]. 2.1.2 Convergence in probability Results for convergence in probability, one of the main results of this work, appear to be considerably harder to obtain. One fruitful avenue of analysis has been the use of so-called transportation, or Talagrand inequalities Tp , which can be used to prove concentration inequalities on Wp [20]. In particular, we say that ρ satisﬁes a Tp (C) inequality with C > 0 iff Wp (ρ, µ)2 ≤ CH(µ|ρ), ∀µ ∈ Pp (M), where H(·|·) is the relative entropy [20]. As shown in [6, 5], it is possible to obtain probabilistic upper bounds on Wp (ρ, ρn ), with p = 1, 2, if ρ is known to satisfy a Tp inequality ˆ of the same order, thereby reducing the problem of bounding Wp (ρ, ρn ) to that of obtaining a Tp ˆ inequality. Note that, by Jensen’s inequality, and as expected from the behavior of Wp , the inequality T2 is stronger than T1 [20]. While it has been shown that ρ satisﬁes a T1 inequality iff it has a ﬁnite square-exponential moment 2 (E[eα x ] ﬁnite for some α > 0) [4, 7], no such general conditions have been found for T2 . As an example, consider that, if M is compact with diameter D then, by theorem 6.15 of [35], and the celebrated Csisz´ r-Kullback-Pinsker inequality [27], for all ρ, µ ∈ Pp (M), it is a Wp (ρ, µ)2p ≤ (2D)2p ρ − µ where · does not. TV 2 TV ≤ 22p−1 D2p H(µ|ρ), is the total variation norm. Clearly, this implies a Tp=1 inequality, but for p ≥ 2 it The T2 inequality has been shown by Talagrand to be satisﬁed by the Gaussian distribution [31], and then slightly more generally by strictly log-concave measures (see [20, p. 123], and [3].) However, as noted in [6], “contrary to the T1 case, there is no hope to obtain T2 inequalities from just integrability or decay estimates.” Structure of this paper. In this work we obtain bounds in probability (learning rates) for the problem of learning a probability measure in the sense of W2 . We begin by establishing (lower) bounds for the convergence of empirical to population measures, which serve to set up the problem and introduce the connection between quantization and measure learning (sec. 3.) We then describe how existing unsupervised learning algorithms that compute a set (k-means, k-ﬂats, PCA,. . . ) can be easily extended to produce a measure (sec. 4.) Due to its simplicity and widespread use, we focus here on k-means. Since the two measure estimates that we consider are the empirical measure, and the measure induced by k-means, we next set out to prove upper bounds on their convergence to the data-generating measure (sec. 5.) We arrive at these bounds by means of intermediate measures, which are related to the problem of optimal quantization. The bounds apply in a very broad setting (unlike existing bounds based on transportation inequalities, they are not restricted to log-concave measures [20, 3].) 3 3 Learning probability measures, optimal transport and quantization We address the problem of learning a probability measure ρ when the only observations we have at our disposal are n i.i.d. samples Xn = (x1 , . . . , xn ). We begin by establishing some notation and useful intermediate results. Given a closed set S ⊆ X , let {Vq : q ∈ S} be a Borel Voronoi partition of X composed of sets Vq closest to each q ∈ S, that is, such that each Vq ⊆ {x ∈ X : x − q = minr∈S x − r } is measurable (see for instance [15].) Consider the projection function πS : X → S mapping each x ∈ Vq to q. By virtue of {Vq }q∈S being a Borel Voronoi partition, the map πS is measurable [15], and it is d (x, πS (x)) = minq∈S x − q for all x ∈ X . For any ρ ∈ Pp (M), let πS ρ be the pushforward, or image measure of ρ under the mapping πS , −1 which is deﬁned to be (πS ρ)(A) := ρ(πS (A)) for all Borel measurable sets A. From its deﬁnition, it is clear that πS ρ is supported on S. We now establish a connection between the expected distance to a set S, and the distance between ρ and the set’s induced pushforward measure. Notice that, for discrete sets S, the expected Lp distance to S is exactly the expected quantization error Ep,ρ (S) := Ex∼ρ d(x, S)p = Ex∼ρ x − πS (x) p incurred when encoding points x drawn from ρ by their closest point πS (x) in S [14]. This close connection between optimal quantization and Wasserstein distance has been pointed out in the past in the statistics [28], optimal quantization [14, p. 33], and approximation theory [16] literatures. The following two lemmas are key tools in the reminder of the paper. The ﬁrst highlights the close link between quantization and optimal transport. Lemma 3.1. For closed S ⊆ X , ρ ∈ Pp (M), 1 ≤ p < ∞, it holds Ex∼ρ d(x, S)p = Wp (ρ, πS ρ)p . Note that the key element in the above lemma is that the two measures in the expression Wp (ρ, πS ρ) must match. When there is a mismatch, the distance can only increase. That is, Wp (ρ, πS µ) ≥ Wp (ρ, πS ρ) for all µ ∈ Pp (M). In fact, the following lemma shows that, among all the measures with support in S, πS ρ is closest to ρ. Lemma 3.2. For closed S ⊆ X , and all µ ∈ Pp (M) with supp(µ) ⊆ S, 1 ≤ p < ∞, it holds Wp (ρ, µ) ≥ Wp (ρ, πS ρ). When combined, lemmas 3.1 and 3.2 indicate that the behavior of the measure learning problem is limited by the performance of the optimal quantization problem. For instance, Wp (ρ, ρn ) can only ˆ be, in the best-case, as low as the optimal quantization cost with codebook of size n. The following section makes this claim precise. 3.1 Lower bounds Consider the situation depicted in ﬁg. 1, in which a sample X4 = {x1 , x2 , x3 , x4 } is drawn from a distribution ρ which we assume here to be absolutely continuous on its support. As shown, the projection map πX4 sends points x to their closest point in X4 . The resulting Voronoi decomposition of supp(ρ) is drawn in shades of blue. By lemma 5.2 of [9], the pairwise intersections of Voronoi regions have null ambient measure, and since ρ is absolutely continuous, the pushforward measure 4 can be written in this case as πX4 ρ = j=1 ρ(Vxj )δxj , where Vxj is the Voronoi region of xj . Note that, even for ﬁnite sets S, this particular decomposition is not always possible if the {Vq }q∈S form a Borel Voronoi tiling, instead of a Borel Voronoi partition. If, for instance, ρ has an atom falling on two Voronoi regions in a tiling, then both regions would count the atom as theirs, and double-counting would imply q ρ(Vq ) > 1. The technicalities required to correctly deﬁne a Borel Voronoi partition are such that, in general, it is simpler to write πS ρ, even though (if S is discrete) this measure can clearly be written as a sum of deltas with appropriate masses. By lemma 3.1, the distance Wp (ρ, πX4 ρ)p is the (expected) quantization cost of ρ when using X4 as codebook. Clearly, this cost can never be lower than the optimal quantization cost of size 4. This reasoning leads to the following lower bound between empirical and population measures. 4 Theorem 3.3. For ρ ∈ Pp (M) with absolutely continuous part ρA = 0, and 1 ≤ p < ∞, it holds Wp (ρ, ρn ) = Ω(n−1/d ) uniformly over ρn , where the constants depend on d and ρA only. ˆ ˆ Proof: Let Vn,p (ρ) := inf S⊂M,|S|=n Ex∼ρ d(x, S)p be the optimal quantization cost of ρ of order p with n centers. Since ρA = 0, and since ρ has a ﬁnite (p + δ)-th order moment, for some δ > 0 (since it is supported on the unit ball), then it is Vn,p (ρ) = Θ(n−p/d ), with constants depending on d and ρA (see [14, p. 78] and [16].) Since supp(ˆn ) = Xn , it follows that ρ Wp (ρ, ρn )p ˆ ≥ lemma 3.2 Wp (ρ, πXn ρ)p = lemma 3.1 Ex∼ρ d(x, Xn )p ≥ Vn,p (ρ) = Θ(n−p/d ) Note that the bound of theorem 3.3 holds for ρn derived from any sample Xn , and is therefore ˆ stronger than the existing lower bounds on the convergence rates of EWp (ρ, ρn ) → 0. In particular, ˆ it trivially induces the known lower bound Ω(n−1/d ) on the rate of convergence in expectation. 4 Unsupervised learning algorithms for learning a probability measure As described in [21], several of the most widely used unsupervised learning algorithms can be ˆ interpreted to take as input a sample Xn and output a set Sk , where k is typically a free parameter of the algorithm, such as the number of means in k-means1 , the dimension of afﬁne spaces in PCA, n ˆ etc. Performance is measured by the empirical quantity n−1 i=1 d(xi , Sk )2 , which is minimized among all sets in some class (e.g. sets of size k, afﬁne spaces of dimension k,. . . ) This formulation is general enough to encompass k-means and PCA, but also k-ﬂats, non-negative matrix factorization, and sparse coding (see [21] and references therein.) Using the discussion of Sec. 3, we can establish a clear connection between unsupervised learning and the problem of learning probability measures with respect to W2 . Consider as a running example the k-means problem, though the argument is general. Given an input Xn , the k-means problem is ˆ ˆ to ﬁnd a set |Sk | = k minimizing its average distance from points in Xn . By associating to Sk the pushforward measure πSk ρn , we ﬁnd that ˆ ˆ 1 n n ˆ ˆ d(xi , Sk )2 = Ex∼ρn d(x, Sk )2 ˆ i=1 = lemma 3.1 W2 (ˆn , πSk ρn )2 . ρ ˆ ˆ (2) Since k-means minimizes equation 2, it also ﬁnds the measure that is closest to ρn , among those ˆ with support of size k. This connection between k-means and W2 measure approximation was, to the best of the authors’ knowledge, ﬁrst suggested by Pollard [28] though, as mentioned earlier, the argument carries over to many other unsupervised learning algorithms. Unsupervised measure learning algorithms. We brieﬂy clarify the steps involved in using an existing unsupervised learning algorithm for probability measure learning. Let Uk be a parametrized algorithm (e.g. k-means) that takes a sample Xn and outputs a set Uk (Xn ). The measure learning algorithm Ak : Mn → Pp (M) corresponding to Uk is deﬁned as follows: ˆ 1. Ak takes a sample Xn and outputs the measure πSk ρn , supported on Sk = Uk (Xn ); ˆ ˆ 2. since ρn is discrete, then so must πSk ρn be, and thus Ak (Xn ) = ˆ ˆ ˆ 1 n n ˆ i=1 δπSk (xi ) ; 3. in practice, we can simply store an n-vector πSk (x1 ), . . . , πSk (xn ) , from which Ak (Xn ) ˆ ˆ can be reconstructed by placing atoms of mass 1/n at each point. In the case that Uk is the k-means algorithm, only k points and k masses need to be stored. Note that any algorithm A that attempts to output a measure A (Xn ) close to ρn can be cast in the ˆ above framework. Indeed, if S is the support of A (Xn ) then, by lemma 3.2, πS ρn is the measure ˆ closest to ρn with support in S . This effectively reduces the problem of learning a measure to that of ˆ 1 In a slight abuse of notation, we refer to the k-means algorithm here as an ideal algorithm that solves the k-means problem, even though in practice an approximation algorithm may be used. 5 ﬁnding a set, and is akin to how the fact that every optimal quantizer is a nearest-neighbor quantizer (see [15], [12, p. 350], and [14, p. 37–38]) reduces the problem of ﬁnding an optimal quantizer to that of ﬁnding an optimal quantizing set. Clearly, the minimum of equation 2 over sets of size k (the output of k-means) is monotonically ˆ ˆ non-increasing with k. In particular, since Sn = Xn and πSn ρn = ρn , it is Ex∼ρn d(x, Sn )2 = ˆ ˆ ˆ ˆ 2 W2 (ˆn , πSn ρn ) = 0. That is, we can always make the learned measure arbitrarily close to ρn ρ ˆ ˆ ˆ by increasing k. However, as pointed out in Sec. 2, the problem of measure learning is concerned with minimizing the 2-Wasserstein distance W2 (ρ, πSk ρn ) to the data-generating measure. The ˆ ˆ actual performance of k-means is thus not necessarily guaranteed to behave in the same way as the empirical one, and the question of characterizing its behavior as a function of k and n naturally arises. ˆ Finally, we note that, while it is Ex∼ρn d(x, Sk )2 = W2 (ˆn , πSk ρn )2 (the empirical performances ρ ˆ ˆ ˆ are the same in the optimal quantization, and measure learning problem formulations), the actual performances satisfy ˆ Ex∼ρ d(x, Sk )2 = W2 (ρ, π ˆ ρ)2 ≤ W2 (ρ, π ˆ ρn )2 , 1 ≤ k ≤ n. ˆ lemma 3.1 Sk lemma 3.2 Sk Consequently, with the identiﬁcation between sets S and measures πS ρn , the measure learning ˆ problem is, in general, harder than the set-approximation problem (for example, if M = Rd and ρ is absolutely continuous over a set of non-null volume, it is not hard to show that the inequality is ˆ almost surely strict: Ex∼ρ d(x, Sk )2 < W2 (ρ, πSk ρn )2 for 1 < k < n.) ˆ ˆ In the remainder, we characterize the performance of k-means on the measure learning problem, for varying k, n. Although other unsupervised learning algorithms could have been chosen as basis for our analysis, k-means is one of the oldest and most widely used, and the one for which the deep connection between optimal quantization and measure approximation is most clearly manifested. Note that, by setting k = n, our analysis includes the problem of characterizing the behavior of the distance W2 (ρ, ρn ) between empirical and population measures which, as indicated in Sec. 2.1, ˆ is a fundamental question in statistics (i.e. the speed of convergence of empirical to population measures.) 5 Learning rates In order to analyze the performance of k-means as a measure learning algorithm, and the convergence of empirical to population measures, we propose the decomposition shown in ﬁg. 2. The diagram includes all the measures considered in the paper, and shows the two decompositions used to prove upper bounds. The upper arrow (green), illustrates the decomposition used to bound the distance W2 (ρ, ρn ). This decomposition uses the measures πSk ρ and πSk ρn as intermediates to arrive ˆ ˆ at ρn , where Sk is a k-point optimal quantizer of ρ, that is, a set Sk minimizing Ex∼ρ d(x, S)2 over ˆ all sets of size |S| = k. The lower arrow (blue) corresponds to the decomposition of W2 (ρ, πSk ρn ) ˆ ˆ (the performance of k-means), whereas the labelled black arrows correspond to individual terms in the bounds. We begin with the (slightly) simpler of the two results. 5.1 Convergence rates for the empirical to population measures Let Sk be the optimal k-point quantizer of ρ of order two [14, p. 31]. By the triangle inequality and the identity (a + b + c)2 ≤ 3(a2 + b2 + c2 ), it follows that W2 (ρ, ρn )2 ≤ 3 W2 (ρ, πSk ρ)2 + W2 (πSk ρ, πSk ρn )2 + W2 (πSk ρn , ρn )2 . ˆ ˆ ˆ ˆ (3) This is the decomposition depicted in the upper arrow of ﬁg. 2. By lemma 3.1, the ﬁrst term in the sum of equation 3 is the optimal k-point quantization error of ρ over a d-manifold M which, using recent techniques from [16] (see also [17, p. 491]), is shown in the proof of theorem 5.1 (part a) to be of order Θ(k −2/d ). The remaining terms, b) and c), are slightly more technical and are bounded in the proof of theorem 5.1. Since equation 3 holds for all 1 ≤ k ≤ n, the best bound on W2 (ρ, ρn ) can be obtained by optimizˆ ing the right-hand side over all possible values of k, resulting in the following probabilistic bound for the rate of convergence of the empirical to population measures. 6 x2 x W2 (ρ, ρn ) ˆ supp ρ x1 π{x1 ,x2 ,x3 ,x4 } ρ a) x3 πSk ρ b) πSk ρn ˆ c) d) ρn ˆ πSk ρn ˆ ˆ W2 (ρ, πSk ρn ) ˆ ˆ x4 Figure 1: A sample {x1 , x2 , x3 , x4 } is drawn from a distribution ρ with support in supp ρ. The projection map π{x1 ,x2 ,x3 ,x4 } sends points x to their closest one in the sample. The induced Voronoi tiling is shown in shades of blue. Figure 2: The measures considered in this paper are linked by arrows for which upper bounds for their distance are derived. Bounds for the quantities of interest W2 (ρ, ρn )2 , and W2 (ρ, πSk ρn )2 , ˆ ˆ ˆ are decomposed by following the top and bottom colored arrows. Theorem 5.1. Given ρ ∈ Pp (M) with absolutely continuous part ρA = 0, sufﬁciently large n, and τ > 0, it holds W2 (ρ, ρn ) ≤ C · m(ρA ) · n−1/(2d+4) · τ, ˆ where m(ρA ) := 5.2 M 2 with probability 1 − e−τ . ρA (x)d/(d+2) dλM (x), and C depends only on d. Learning rates of k-means The key element in the proof of theorem 5.1 is that the distance between population and empirical measures can be bounded by choosing an intermediate optimal quantizing measure of an appropriate size k. In the analysis, the best bounds are obtained for k smaller than n. If the output of k-means is close to an optimal quantizer (for instance if sufﬁcient data is available), then we would similarly expect that the best bounds for k-means correspond to a choice of k < n. The decomposition of the bottom (blue) arrow in ﬁgure 2 leads to the following bound in probability. Theorem 5.2. Given ρ ∈ Pp (M) with absolutely continuous part ρA = 0, and τ > 0, then for all sufﬁciently large n, and letting k = C · m(ρA ) · nd/(2d+4) , it holds W2 (ρ, πSk ρn ) ≤ C · m(ρA ) · n−1/(2d+4) · τ, ˆ ˆ where m(ρA ) := M 2 with probability 1 − e−τ . ρA (x)d/(d+2) dλM (x), and C depends only on d. Note that the upper bounds in theorem 5.1 and 5.2 are exactly the same. Although this may appear ˆ surprising, it stems from the following fact. Since S = Sk is a minimizer of W2 (πS ρn , ρn )2 , the ˆ ˆ bound d) of ﬁgure 2 satisﬁes: W2 (πSk ρn , ρn )2 ≤ W2 (πSk ρn , ρn )2 ˆ ˆ ˆ ˆ ˆ and therefore (by the deﬁnition of c), the term d) is of the same order as c). It follows then that adding term d) to the bound only affects the constants, but otherwise leaves it unchanged. Since d) is the term that takes the output measure of k-means to the empirical measure, this implies that the rate of convergence of k-means (for suitably chosen k) cannot be worse than that of ρn → ρ. ˆ Conversely, bounds for ρn → ρ are obtained from best rates of convergence of optimal quantizers, ˆ whose convergence to ρ cannot be slower than that of k-means (since the quantizers that k-means produces are suboptimal.) 7 Since the bounds obtained for the convergence of ρn → ρ are the same as those for k-means with ˆ k of order k = Θ(nd/(2d+4) ), this suggests that estimates of ρ that are as accurate as those derived from an n point-mass measure ρn can be derived from k point-mass measures with k ˆ n. Finally, we note that the introduced bounds are currently limited by the statistical bound sup |W2 (πS ρn , ρn )2 − W2 (πS ρ, ρ)2 | ˆ ˆ |S|=k = sup |Ex∼ρn d(x, S)2 − Ex∼ρ d(x, S)2 | ˆ lemma 3.1 |S|=k (4) (see for instance [21]), for which non-matching lower bounds are known. This means that, if better upper bounds can be obtained for equation 4, then both bounds in theorems 5.1 and 5.2 would automatically improve (would become closer to the lower bound.) References [1] M. Ajtai, J. Komls, and G. Tusndy. On optimal matchings. Combinatorica, 4:259–264, 1984. [2] Franck Barthe and Charles Bordenave. Combinatorial optimization over two random point sets. Technical Report arXiv:1103.2734, Mar 2011. [3] Gordon Blower. The Gaussian isoperimetric inequality and transportation. Positivity, 7:203–224, 2003. [4] S. G. Bobkov and F. G¨ tze. Exponential integrability and transportation cost related to logarithmic o Sobolev inequalities. Journal of Functional Analysis, 163(1):1–28, April 1999. 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