nips nips2011 nips2011-165 knowledge-graph by maker-knowledge-mining

165 nips-2011-Matrix Completion for Multi-label Image Classification

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Author: Ricardo S. Cabral, Fernando Torre, Joao P. Costeira, Alexandre Bernardino

Abstract: Recently, image categorization has been an active research topic due to the urgent need to retrieve and browse digital images via semantic keywords. This paper formulates image categorization as a multi-label classiﬁcation problem using recent advances in matrix completion. Under this setting, classiﬁcation of testing data is posed as a problem of completing unknown label entries on a data matrix that concatenates training and testing features with training labels. We propose two convex algorithms for matrix completion based on a Rank Minimization criterion speciﬁcally tailored to visual data, and prove its convergence properties. A major advantage of our approach w.r.t. standard discriminative classiﬁcation methods for image categorization is its robustness to outliers, background noise and partial occlusions both in the feature and label space. Experimental validation on several datasets shows how our method outperforms state-of-the-art algorithms, while effectively capturing semantic concepts of classes. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 pt Abstract Recently, image categorization has been an active research topic due to the urgent need to retrieve and browse digital images via semantic keywords. [sent-9, score-0.375]

2 This paper formulates image categorization as a multi-label classiﬁcation problem using recent advances in matrix completion. [sent-10, score-0.29]

3 Under this setting, classiﬁcation of testing data is posed as a problem of completing unknown label entries on a data matrix that concatenates training and testing features with training labels. [sent-11, score-0.261]

4 We propose two convex algorithms for matrix completion based on a Rank Minimization criterion speciﬁcally tailored to visual data, and prove its convergence properties. [sent-12, score-0.363]

5 standard discriminative classiﬁcation methods for image categorization is its robustness to outliers, background noise and partial occlusions both in the feature and label space. [sent-16, score-0.408]

6 Experimental validation on several datasets shows how our method outperforms state-of-the-art algorithms, while effectively capturing semantic concepts of classes. [sent-17, score-0.069]

7 1 Introduction With the ever-growing amount of digital image data in multimedia databases, there is a great need for algorithms that can provide effective semantic indexing. [sent-18, score-0.22]

8 Categorizing digital images using keywords, however, is the quintessential example of a challenging classiﬁcation problem. [sent-19, score-0.087]

9 Several aspects contribute to the difﬁculty of the image categorization problem, including the large variability in appearance, illumination and pose of different objects. [sent-20, score-0.219]

10 Over the last decade, progress in the image classiﬁcation problem has been achieved by using more powerful classiﬁers and building or learning better image representations. [sent-22, score-0.228]

11 On one hand, standard discriminative approaches such as Support Vector Machines or Boosting have been extended to the multi-label case [28, 14] and incorporated under frameworks such as Multiple Instance Learning [31, 33, 32, 20, 27] and Multi-task Learning [26]. [sent-23, score-0.078]

12 However, a major limitation of discriminative approaches is the lack of robustness to outliers and missing data. [sent-24, score-0.108]

13 Recall most discriminative approaches project the data directly onto linear or non-linear spaces, thus lacking a noise model for it. [sent-25, score-0.235]

14 To address this issue, we propose formulating the image classiﬁcation problem under a matrix completion framework, that has been fueled by recent advances in Rank Minimization [7, 18]. [sent-26, score-0.4]

15 On the other hand, traversal to the use of more powerful classiﬁers, better image representations, such as SIFT [17] or GIST [21] have boosted recognition and categorization performance. [sent-28, score-0.219]

16 Our algorithms make use of the fact that in this model the histogram of an entire image contains information of all of its subparts. [sent-30, score-0.182]

17 By modeling the error in the histogram, our matrix completion algorithm is able to capture semantically discriminative portions of the image, thus obviating the need for training with precise localization, as required by previous methods [31, 33, 32, 20, 27]. [sent-31, score-0.423]

18 1 Our main contributions are twofold: (1) We propose two new Rank Minimization algorithms, MCPos and MC-Simplex, motivated by the image categorization problem. [sent-32, score-0.219]

19 2 Previous Work This section reviews related work in the area of image categorization and the problem of Matrix Completion using a Rank Minimization criterion, optimized with Nuclear Norm methods. [sent-36, score-0.219]

20 Image semantic understanding is now typically formulated as a multi-label problem. [sent-39, score-0.069]

21 In this setting, each image may be simultaneously categorized into more than one of a set of predeﬁned categories. [sent-40, score-0.114]

22 Additionally, Multiple Instance Learning (MIL) approaches can be used to explicitly model the relations between labels and speciﬁc regions of the image, as initially proposed by Maron et al. [sent-43, score-0.066]

23 This framework allows for the localization and classiﬁcation tasks to beneﬁt from each other, thus reducing noise in the corresponding feature space and making the learned semantic models more accurate [31, 33, 32, 20, 27, 26]. [sent-45, score-0.191]

24 [31] exploit asymmetric loss functions to balance false positives and negatives. [sent-51, score-0.067]

25 This is usually obtained by pre-segmenting images to a small ﬁxed number of parts or applied in settings where detectors perform well, such as the problem of associating faces to captioned names [4]. [sent-53, score-0.08]

26 A major breakthrough by [7] states the minimization of the rank function, under broad conditions, can be achieved using the minimizer obtained with the Nuclear Norm (sum of singular values). [sent-59, score-0.191]

27 In the last few years, incremental matrix completion methods have also been proposed [1, 2, 5]. [sent-62, score-0.286]

28 2 3 Multi-label classiﬁcation using Matrix Completion In a supervised setting, a classiﬁer learns a mapping1 W : X → Y between the space of features X and the space of labels Y, from Ntr tuples of known features and labels. [sent-64, score-0.139]

29 Linear classiﬁers deﬁne (xj , yj ) ∈ RF × RK , where F is the feature dimension and K the number of classes, and minimize the loss l between the output space and the projection of the input space, as Ntr minimize W,b l yj , [W b] j=1 xj 1 , (1) with parameters W ∈ RK×F , b ∈ RK . [sent-65, score-0.138]

30 For this purpose, they concatenate all labels and features into matrices Ytst ∈ RK×Ntst , Ytr ∈ RK×Ntr , Xtst ∈ RF ×Ntst , Xtr ∈ RF ×Ntr . [sent-68, score-0.069]

31 If the linear model holds, then the matrix   Ytr Ytst Z0 =  Xtr Xtst  , (2) 1 should be rank deﬁcient. [sent-69, score-0.156]

32 The classiﬁcation process consists in ﬁlling the unknown entries in Ytst such that the Nuclear Norm of Z0 , the convex envelope of rank [7], is minimized. [sent-70, score-0.177]

33 Since in practice we may have errors and partial knowledge in the training labels and in the feature space, let us deﬁne ΩX and ΩY as the set of known feature and label entries and zero out unknown entries in Z0 . [sent-71, score-0.278]

34 Additionally, let the data matrix Z be deﬁned as a sum of Z0 with an error term E, as     Ytr Ytst EY tr 0 ZY (3) Z = ZX =  Xtr Xtst  +  EXtr EXtst  = Z0 + E, Z1 1 0 where ZY , ZX , Z1 respectively stand for the label, feature and last rows of Z. [sent-72, score-0.175]

35 Then, classiﬁcation can be posed as an optimization problem that ﬁnds the best label assignment Ytst and error matrix E such that the rank of Z is minimized. [sent-73, score-0.233]

36 The resulting optimization problem, MC-1 [11], is minimize Ytst ,EXtr ,EY tr ,EXtst µ Z ∗ + 1 |ΩX | cx (zij , z0ij ) + ij∈ΩX λ |ΩY | cy (zij , z0ij ) ij∈ΩY Z = Z0 + E subject to (4) Z1 = 1 . [sent-74, score-0.299]

37 The parameters λ, µ are positive trade-off weights between better feature adaptation and label error correction. [sent-77, score-0.111]

38 We note this problem is equivalent to minimize Z subject to µ Z ∗ + 1 |ΩX | cx (zij , z0ij ) + ij∈ΩX λ |ΩY | cy (zij , z0ij ) ij∈ΩY (5) Z1 = 1 which can be solved using a Fixed Point Continuation method [18], described in Sec. [sent-78, score-0.259]

39 ||A||∗ designates the Nuclear Norm (sum of singular values) of A. [sent-92, score-0.083]

40 1k ∈ Rk×1 is a vector of ones, 0k×n ∈ Rk×n is a matrix of zeros and Ik ∈ Rk×k denotes the identity matrix (dimensions are omitted when trivially inferred). [sent-94, score-0.142]

41 3 4 Matrix completion for multi-label classiﬁcation of visual data In this section, we present the main contributions of this paper: the application of Matrix Completion to the multi-label image classiﬁcation problem and its convergence proof. [sent-95, score-0.364]

42 In the bag of (visual) words (BoW) model [24], visual data is encoded by the distribution of features among entries from a codebook. [sent-96, score-0.085]

43 The codebook is typically created by clustering local feature representations such as SIFT [17] or GIST [21]. [sent-97, score-0.067]

44 In this setting, the formulation MC-1 (5) is inadequate because it introduces negative values to the histograms in ZX . [sent-98, score-0.075]

45 To address this issue, we replace the penalties used so they reﬂect the nature of data: we replace the Least-Squares penalty in cx (·) by Pearson’s χ2 distance, that takes into account the asymmetry in histogram data, as F F χ2 (zj , z0j ) = χ2 (zij , z0ij ) = i i=1 i=1 (zij − z0ij )2 . [sent-99, score-0.159]

46 Additionally, we note that the Log label error in cy (·), albeit asymmetric, incurs in unnecessary penalization of entries belonging to the same class as the original entry (see Fig. [sent-101, score-0.295]

47 Therefore, we generalize this loss to progressively resemble smooth version of the Hinge loss, speciﬁed by the parameter γ as cy (zij , z0ij ) = 1 log(1 + exp (−γz0ij zij )). [sent-103, score-0.499]

48 8 1 z Figure 1: Comparison of Generalized Log loss with Log loss (γ = 1). [sent-115, score-0.076]

49 1 Fixed Point continuation (FPC) for MC-1 Albeit convex, the Nuclear Norm operator makes (5), (7), (8) not smooth. [sent-117, score-0.093]

50 Since the natural reformulation of a Nuclear Norm minimization is a Semideﬁnite Program, existing off-the-shelf interior point methods are not applicable due to the large dimension of Z. [sent-118, score-0.095]

51 The FPC method [18], in particular, is comprised by a series of gradient updates h(·) = I(·) − τ g(·) with step size τ and gradient g(·) given by the error penalizations cx (·) and cy (·). [sent-120, score-0.44]

52 These steps are alternated with a shrinkage operator Sν (·) = max (0, · − ν), applied to the singular values of the resulting matrix, so the rank is minimized. [sent-121, score-0.125]

53 In this paper, we prove the convergence of FPC to the constrained problem class by using the fact that projections onto Convex sets are also non-expansive; thus, the composition of gradient, shrinkage and projection steps is also a contraction. [sent-125, score-0.185]

54 Error > do Gradient Descent: A = h(A) = Z − τ g(Z) Shrink: A = UΣV , Z = USτ µ (Σ)V Project onto feasible set: Z1 = 1 end while end for Output: Complete Matrix Z Lemma 1 Let pC (·) be a projection operator onto any given convex set C. [sent-129, score-0.276]

55 These values are λγ easily obtained by noting the gradient of the Log loss function is Lipschitz continuous with L = 0. [sent-148, score-0.077]

56 Key to the feasibility of (7) and (8) within this algorithmic framework, however, is an efﬁcient way to project Z onto the newly deﬁned constraint sets. [sent-150, score-0.157]

57 While for MC-Pos (7) projecting a vector onto the Non-Negative Orthant is done in closed form by truncating negative components to zero, efﬁciently performing the projection onto the Probability Simplex in MC-Simplex (8) is not straightforward. [sent-151, score-0.234]

58 We note, however, this is a projection onto a convex subset of an 1 ball [9]. [sent-152, score-0.192]

59 Therefore, we can explore the dual of the projection problem and use a sorting procedure to implement this projection in closed form, as described in Alg. [sent-153, score-0.132]

60 Algorithm 2 Projection of a vector onto probability Simplex Input: Vector v ∈ RF to be projected Sort v into µ : µ1 ≥ µ2 ≥ . [sent-158, score-0.084]

61 Error > do Gradient Descent: A = Z − τ g(Z) Shrink: A = UΣV Shrink: Z = USτ µ (Σ)V Project ZX onto P (Alg. [sent-164, score-0.084]

62 We compare our results with MC-1 (5) and standard discriminative and MIL approaches [30, 20, 27, 26, 33, 32] on three datasets: CMU-Face , MSRC and 15 Scene. [sent-167, score-0.078]

63 The continuation steps require a decreasing sequence of µ, which we chose as µk = 0. [sent-169, score-0.093]

64 CMU-Face dataset This dataset consists in 624 images of 20 subject faces with several expressions and poses, under two conditions: wearing sunglasses and not. [sent-174, score-0.122]

65 As in [20], our training set is built using images of the ﬁrst 8 subjects (126 images with glasses and 128 without), leaving the remainder for testing (370, equally split among the classes). [sent-176, score-0.129]

66 We describe each image by extracting 10000 SIFT features [17] at random scales and positions and quantizing them onto a 1000 visual codebook, obtained by performing hierarchical k-means clustering on 100000 features randomly selected from the training set. [sent-177, score-0.297]

67 For this dataset, note that subjects were captured in a very similar environment, so the most discriminative part is the eye region. [sent-178, score-0.078]

68 Since the face position varies, they propose using a Multiple Instance Learning framework (MIL-SegSVM), that localizes the most discriminative region in each image while learning a classiﬁer to split both classes. [sent-181, score-0.192]

69 For the SVM, we either trained with the entire image information (SVM-Img) or with only the features extracted from the relevant, manually labeled, region of the eyes. [sent-183, score-0.149]

70 We ﬁll Z with the label vector and the BoW histograms of each entire image and leave the test set labels Ytst as unknown entries. [sent-185, score-0.27]

71 For the MCSimplex case, we preprocess Z by 1 -normalizing each histogram in ZX . [sent-186, score-0.068]

72 This is done to avoid the Simplex projection picking a single bin and zeroing out the others, due to scale disparities in the bin counts. [sent-187, score-0.222]

73 These indicate both the fully supervised and the MIL approaches are more robust to the variability introduced by background noise, when compared to what is obtained when training without localization information (SVM-Img). [sent-189, score-0.152]

74 By using Matrix Completion, in turn, we are able to surpass these classiﬁcation scores by solving a single convex minimization, since our error term E removes noise introduced by non-discriminative parts of the image. [sent-191, score-0.072]

75 We search for the box having the normalized histogram most closely resemblant to the corrected version in ZX according to the χ2 distance, and get the results shown in Fig. [sent-193, score-0.129]

76 These show how the corrected histograms capture the semantic concept being trained. [sent-195, score-0.205]

77 Moreover, MC-1 does not allow to pursue further localization of the class representative since it introduces erroneous negative numbers in the histograms (Fig. [sent-197, score-0.196]

78 bin count Figure 2: Histograms corrected by MC-Pos (7) preserve semantic meaning. [sent-199, score-0.208]

79 96 100 50 0 0 200 400 600 800 1000 600 800 1000 bin count bin 4 2 0 2 4 0 200 400 bin Figure 3: Erroneous histogram correction performed by MC-1 (5). [sent-207, score-0.302]

80 The MSRC dataset consists of 591 real world images distributed among 21 classes, with an average of 3 classes present per image. [sent-211, score-0.116]

81 We mimic the setup of [27] and use as features histograms of Textons [23] concatenated with histograms of colors in the L+U+V space. [sent-212, score-0.185]

82 In this dataset, we compare our formulations to MC-1 and several state-of-the-art approaches for categorization using Multiple-Label Multiple Instance Learning: Multiple Set Kernel MIL SVM (MSK-MIL) by Vijayanarasimhan et al. [sent-215, score-0.137]

83 Again, the fact that feature errors are corrected allows us to achieve good results while training with the entire image. [sent-222, score-0.124]

84 15 Scene dataset Finally, we test the performance of our algorithm for scene classiﬁcation. [sent-225, score-0.101]

85 The 15 scene dataset is a multi-label dataset with 4485 images. [sent-227, score-0.137]

86 According to the feature study in [30], we use GIST [21], the non-histogram feature achieving best results on this dataset. [sent-228, score-0.068]

87 94 Conclusions We presented two new convex methods for performing semi-supervised multi-label classiﬁcation of histogram data, with proven convergence properties. [sent-254, score-0.145]

88 Moreover, since histograms of full images contain the information for parts contained therein, the error embedded in our formulation is able to capture intra class variability arising from different backgrounds. [sent-256, score-0.155]

89 Experiments show that our methods perform comparably to state-of-the-art MIL methods in several image datasets, surpassing them in several cases, without the need for precise localization of objects in the training set. [sent-257, score-0.231]

90 Acknowledgements: Support for this research was provided by the Portuguese Foundation for Science and Technology through the Carnegie Mellon Portugal program under the project FCT/CMU/P11. [sent-258, score-0.073]

91 Fast incremental method for matrix completion: an application to trajectory correction. [sent-296, score-0.071]

92 Efﬁcient projections onto the l1-ball for learning in high dimensions. [sent-323, score-0.084]

93 A rank minimization heuristic with application to minimum order system approximation. [sent-330, score-0.151]

94 Transduction with matrix completion: Three birds with one stone. [sent-339, score-0.071]

95 Fixed point and bregman iterative methods for matrix rank minimization. [sent-384, score-0.156]

96 Weakly supervised discriminative localization and classiﬁcation: a joint learning process. [sent-397, score-0.201]

97 An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. [sent-429, score-0.212]

98 Towards weakly supervised semantic segmentation by means of multiple instance and multitask learning. [sent-434, score-0.139]

99 Region-based image annotation using asymmetrical support vector machine-based multiple-instance learning. [sent-469, score-0.114]

100 Image tag reﬁnement towards low-rank, content-tag prior and error sparsity. [sent-493, score-0.073]

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Abstract: Variability in single neuron models is typically implemented either by a stochastic Leaky-Integrate-and-Fire model or by a model of the Generalized Linear Model (GLM) family. We use analytical and numerical methods to relate state-of-theart models from both schools of thought. First we ﬁnd the analytical expressions relating the subthreshold voltage from the Adaptive Exponential Integrate-andFire model (AdEx) to the Spike-Response Model with escape noise (SRM as an example of a GLM). Then we calculate numerically the link-function that provides the ﬁring probability given a deterministic membrane potential. We ﬁnd a mathematical expression for this link-function and test the ability of the GLM to predict the ﬁring probability of a neuron receiving complex stimulation. 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The original SRM could account for subthreshold resonance, refractory effects and spike-frequency adaptation [11]. Mathematically similar models were developed independently in the study of the visual system [12] where spike-frequency adaptation has also been modeled [13]. Recently, this approach has retained increased attention since the probabilistic framework can be linked with the Bayesian theory of neural systems [14] and because Bayesian inference can be applied to the population of neurons [15]. In this paper, we investigate the similarity and differences between the state-of-the-art GLM and the stochastic AdEx. The motivation behind this work is to relate the traditional threshold neuron models to Bayesian theory. Our results extend the work of Plesser and Gerstner (2000) [16] since we include the non-linearity for spike initiation and spike-frequency adaptation. We also provide relationships between the parameters of the AdEx and the equivalent GLM. 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Finally, we explore the relations between the statistic of the noise and the sharpness of the non-linearity for spike initiation with the parameters of the SRM. 2 Presentation of the Models In this section we present the general formula for the stochastic AdEx model (Sect. 2.1) and the SRM (Sect 2.2). 2.1 The Stochastic Adaptive Exponential Integrate-and-Fire Model The voltage dynamics of the stochastic AdEx is given by: V −Θ ˙ τm V = El − V + ∆T exp − Rw + RI + R (1) ∆T τw w = a(V − El ) − w ˙ (2) where τm is the membrane time constant, El the reverse potential, R the membrane resistance, Θ is the threshold, ∆T is the shape factor and I(t) the input current which is chosen to be an Ornstein−Θ Uhlenbeck process with correlation time-constant of 5 ms. The exponential term ∆T exp( V∆T ) is a non-linear function responsible for the emission of spikes and is a diffusive white noise with standard deviation σ (i.e. ∼ N (0, σ)). Note that the diffusive white-noise does not imply white noise ﬂuctuations of the voltage V (t), the probability distribution of V (t) will depend on ∆T and Θ. The second variable, w, describes the subthreshold as well as the spike-triggered adaptation both ˆ parametrized by the coupling strength a and the time constant τw . Each time tj the voltage goes to inﬁnity, we assumed that a spike is emitted. Then the voltage is reset to a ﬁxed value Vr and w is increased by a constant value b. 2.2 The Generalized Linear Model In the SRM, The voltage V (t) is given by the convolution of the injected current I(t) with the membrane ﬁlter κ(t) plus the additional kernel η(t) that acts after each spikes (here we split the 2 spike-triggered kernel in two η(t) = ηv (t) + ηw (t) for reasons that will become clear later): V (t) = ˆ ˆ ηv (t − tj ) + ηw (t − tj ) El + [κ ∗ I](t) + (3) ˆ {tj } ˆ Then at each time tj a spike is emitted which results in a change of voltage described by η(t) = ηv (t) + ηw (t). Given the deterministic voltage, (Eq. 3) a spike is emitted according to the ﬁring intensity λ(V ): λ(t) = f (V (t)) (4) where f (·) is an arbitrary function called the link-function. Then the ﬁring behavior of the SRM depends on the choice of the link-function and its parameters. The most common link-function used to model single neuron activities are the linear-rectiﬁer and the exponential function. 3 Mapping In order to map the stochastic AdEx to the SRM we follow a two-step procedure. First we derive the ﬁlter κ(t) and the kernels ηv (t) and ηw (t) analytically as a function of AdEx parameters. Second, we derive the link-function of the SRM from the stochastic spike emission of the AdEx. Figure 1: Mapping of the subthreshold dynamics of an AdEx to an equivalent SRM. A. Membrane ﬁlter κ(t) for three different sets of parameters of the AdEx leading to over-damped, critically damped and under-damped cases (upper, middle and lower panel, respectively). B. Spike-Triggered η(t) (black), ηv (t) (light gray) and ηw (gray) for the three cases. C. Example of voltage trace produced when an AdEx is stimulated with a step of colored noise (black). The corresponding voltage from a SRM stimulated with the same current and where we forced the spikes to match those of the AdEx (red). D. Error in the subthreshold voltage (VAdEx − VGLM ) as a function of the mean voltage of the AdEx, for the three different cases: over-, critically and under-damped (light gray, gray and black, respectively) with ∆T = 1 mV. Red line represents the voltage threshold Θ. E. Root Mean Square Error (RMSE) ratio for the three cases with ∆T = 1 mV. The RMSE ratio is the RMSE between the deterministic VSRM and the stochastic VAdEx divided by the RMSE between repetitions of the stochastic AdEx voltage. The error bar shows a single standard deviation as the RMSE ratio is averaged accross multiple value of σ. 3.1 Subthreshold voltage dynamics We start by assuming that the non-linearity for spike initiation does not affect the mean subthreshold voltage of the stochastic AdEx (see Figure 1 D). This assumption is motivated by the small ∆T 3 observed in in-vitro recordings (from 0.5 to 2 mV [8, 9]) which suggest that the subthreshold dynamics are mainly linear except very close to Θ. Also, we expect that the non-linear link-function will capture some of the dynamics due to the non-linearity for spike initiation. Thus it is possible to rewrite the deterministic subthreshold part of the AdEx (Eq. 1-2 without and without ∆T exp((V − Θ)/∆T )) using matrices: ˙ x = Ax (5) with x = V w and A = − τ1 m a τw − gl1m τ − τ1 w (6) In this form, the dynamics of the deterministic AdEx voltage is a damped oscillator with a driving force. Depending on the eigenvalues of A the system could be over-damped, critically damped or under-damped. The ﬁlter κ(t) of the GLM is given by the impulse response of the system of coupled differential equations of the AdEx, described by Eq. 5 and 6. In other words, one has to derive the response of the system when stimulating with a Dirac-delta function. The type of damping gives three different qualitative shapes of the kernel κ(t), which are summarized in Table 3.1 and Figure 1 A. Since the three different ﬁlters also affect the nature of the stochastic voltage ﬂuctuations, we will keep the distinction between over-damped, critically damped and under-damped scenarios throughout the paper. This means that our approach is valid for at least 3 types of diffusive voltage-noise (i.e. the white noise in Eq. 1 ﬁltered by 3 different membrane ﬁlters κ(t)). To complete the description of the deterministic voltage, we need an expression for the spiketriggered kernels. The voltage reset at each spike brings a spike-triggered jump in voltage of magˆ nitude ∆ = Vr − V (t). This perturbation is superposed to the current ﬂuctuations due to I(t) and can be mediated by a Delta-diract pulse of current. Thus we can write the voltage reset kernel by: ηv (t) = ∆ ∆ [δ ∗ κ] (t) = κ(t) κ(0) κ(0) (7) where δ(t) is the Dirac-delta function. The shape of this kernel depends on κ(t) and can be computed from Table 3.1 (see Figure 1 B). Finally, the AdEx mediates spike-frequency adaptation by the jump of the second variables w. From Eq. 2 we can see that this produces a current wspike (t) = b exp (−t/τw ) that can cumulate over subsequent spikes. The effect of this current on voltage is then given by the convolution of wspike (t) with the membrane ﬁlter κ(t). Thus in the SRM framework the spike-frequency adaptation is taken into account by: ηw (t) = [wspike ∗ κ](t) (8) Again the precise form of ηw (t) depends on κ(t) and can be computed from Table 3.1 (see Figure 1 B). At this point, we would like to verify our assumption that the non-linearity for spike emission can be neglected. Fig. 1 C and D shows that the error between the voltage from Eq. 3 and the voltage from the stochastic AdEx is generally small. Moreover, we see that the main contribution to the voltage prediction error is due to the mismatch close to the spikes. However the non-linearity for spike initiation may change the probability distribution of the voltage ﬂuctuations, which in turn inﬂuences the probability of spiking. This will inﬂuence the choice of the link-function, as we will see in the next section. 3.2 Spike Generation Using κ(t), ηv (t) and ηw (t), we must relate the spiking probability of the stochastic AdEx as a function of its deterministic voltage. According to [2] the probability of spiking in time bin dt given the deterministic voltage V (t) is given by: p(V ) = prob{spike in [t, t + dt]} = 1 − exp (−f (V (t))dt) (9) where f (·) gives the ﬁring intensity as a function of the deterministic V (t) (Eq. 3). Thus to extract the link-function f we have to compute the probability of spiking given V (t) for our SRM. To do so we apply the method proposed by Jolivet et al. (2004) [18], where the probability of spiking is simply given by the distribution of the deterministic voltage estimated at the spike times divided by the distribution of the SRM voltage when there is no spike (see ﬁgure 2 A). One can numerically compute these two quantities for our models using N repetitions of the same stimulus. 4 Table 1: Analytical expressions for the membrane ﬁlter κ(t) in terms of the parameters of the AdEx for over-, critically-, and under-damped cases. Membrane Filter: κ(t) over-damped if: (τm + τw )2 > 4τm τw (gl +a) gl κ(t) = k1 eλ1 t + k2 eλ2 t λ1 = 1 2τm τw (−(τm + τw ) + critically-damped if: (τm + τw )2 = 4τm τw (gl +a) gl κ(t) = (αt + β)eλt λ= under-damped if: (τm + τw )2 < 4τm τw (gl +a) gl κ(t) = (k1 cos (ωt) + k2 sin (ωt)) eλt −(τm +τw ) 2τm τw λ= −(τm +τw ) 2τm τw (τm + τw )2 − 4 τm τw (gl + a) gl λ2 = 1 2τm τw (−(τm + τw ) − α= τm −τw 2Cτm τw ω= τw −τm 2τm τw 2 − a g l τm τw (τm + τw )2 − 4 τm τw (gl + a) gl k1 = −(1+(τm λ2 )) Cτm (λ1 −λ2 ) k2 = 1+(τm λ1 ) Cτm (λ1 −λ2 ) β= 1 C k1 = k2 = 1 C −(1+τm λ) Cωτm The standard deviation σ of the noise and the parameter ∆T of the AdEx non-linearity may affect the shape of the link-function. We thus extract p(V ) for different σ and ∆T (Fig. 2 B). Then using visual heuristics and previous knowledge about the potential analytical expression of the link-funtion, we try to ﬁnd a simple analytical function that captures p(V ) for a large range of combinations of σ and ∆T . We observed that the log(− log(p)) is close to linear in most studied conditions Fig. 2 B suggesting the following two distributions of p(V ): V − VT (10) p(V ) = 1 − exp − exp ∆V V − VT p(V ) = exp − exp − (11) ∆V Once we have p(V ), we can use Eq. 4 to obtain the equivalent SRM link-function, which leads to: −1 f (V ) = log (1 − p(V )) (12) dt Then the two potential link-functions of the SRM can be derived from Eq. 10 and Eq. 11 (respectively): V − VT f (V ) = λ0 exp (13) ∆V V − VT (14) f (V ) = −λ0 log 1 − exp − exp − ∆V 1 with λ0 = dt , VT the threshold of the SRM and ∆V the sharpness of the link-function (i.e. the parameters that governs the degree of the stochasticity). Note that the exact value of λ0 has no importance since it is redundant with VT . Eq. 13 is the standard exponential link-function, but we call Eq. 14 the log-exp-exp link-function. 3.3 Prediction The next point is to evaluate the ﬁt quality of each link-function. To do this, we ﬁrst estimate the parameters VT and ∆V of the GLM link-function that maximize the likelihood of observing a spike 5 Figure 2: SRM link-function. A. Histogram of the SRM voltage at the AdEx ﬁring times (red) and at non-ﬁring times (gray). The ratio of the two distributions gives p(V ) (Eq. 9, dashed lines). Inset, zoom to see the voltage histogram evaluated at the ﬁring time (red). B. log(− log(p)) as a function of the SRM voltage for three different noise levels σ = 0.07, 0.14, 0.18 nA (pale gray, gray, black dots, respectively) and ∆T = 1 mV. The line is a linear ﬁt corresponding to the log-exp-exp linkfunction and the dashed line corresponds to a ﬁt with the exponential link-function. C. Same data and labeling scheme as B, but plotting f (V ) according to Eq. 12. The lines are produced with Eq. 14 with parameters ﬁtted as described in B. and the dashed lines are produced with Eq. 13. Inset, same plot but on a semi-log(y) axis. train generated with an AdEx. Second we look at the predictive power of the resulting SRM in terms of Peri-Stimulus Time Histogram (PSTH). In other words we ask how close the spike trains generated with a GLM are from the spike train generated with a stochastic AdEx when both models are stimulated with the same input current. For any GLM with link-function f (V ) ≡ f (t|I, θ) and parameters θ regulating the shape of κ(t), ˆ ηv (t) and ηw (t), the Negative Log-Likelihood (NLL) of observing a spike-train {t} is given by:   NLL = − log(f (t|I, θ)) − f (t|I, θ) (15) t ˆ t It has been shown that the negative log-likelihood is convex in the parameters if f is convex and logconcave [19]. It is easy to show that a linear-rectiﬁer link-function, the exponential link-function and the log-exp-exp link-function all satisfy these conditions. This allows efﬁcient estimation of ˆ ˆ the optimal parameters VT and ∆V using a simple gradient descent. One can thus estimate from a single AdEx spike train the optimal parameters of a given link-function, which is more efﬁcient than the method used in Sect. 3.2. The minimal NLL resulting from the gradient descent gives an estimation of the ﬁt quality. A better estimate of the ﬁt quality is given by the distance between the PSTHs in response to stimuli not used for parameter ﬁtting . Let ν1 (t) be the PSTH of the AdEx, and ν2 (t) be the PSTH of the ﬁtted SRM, 6 Figure 3: PSTH prediction. A. Injected current. B. Voltage traces produced by an AdEx (black) and the equivalent SRM (red), when stimulated with the current in A. C. Raster plot for 20 realizations of AdEx (black tick marks) and equivalent SRM (red tick marks). D. PSTH of the AdEx (black) and the SRM (red) obtained by averaging 10,000 repetitions. E. Optimal log-likelihood for the three cases of the AdEx, using three different link-functions, a linear-rectiﬁer (light gray), an exponential link-function (gray) and the link-function deﬁned by Eq. 14 (dark gray), these values are obtained by averaging over 40 different combinations σ and ∆T (see Fig. 4). Error bars are one standard deviation, the stars denote a signiﬁcant difference, two-sample t-test with α = 0.01. F. same as E. but for Md (Eq. 16). then we use Md ∈ [0, 1] as a measure of match: Md = 2 2 (ν1 (t) − ν2 (t)) dt ν1 (t)2 dt + ν2 (t)2 dt (16) Md = 1 means that it is impossible to differentiate the SRM from the AdEx in terms of their PSTHs, whereas a Md of 0 means that the two PSTHs are completely different. Thus Md is a normalized similarity measure between two PSTHs. In practice, Md is estimated from the smoothed (boxcar average of 1 ms half-width) averaged spike train of 1 000 repetitions for each models. We use both the NLL and Md to quantify the ﬁt quality for each of the three damping cases and each of the three link-functions. Figure 3 shows the match between the stochastic AdEx used as a reference and the derived GLM when both are stimulated with the same input current (Fig. 3 A). The resulting voltage traces are almost identical (Fig. 3 B) and both models predict almost the same spike trains and so the same PSTHs (Fig. 3 C and D). More quantitalively, we see on Fig. 3 E and F, that the linear-rectiﬁer ﬁts signiﬁcantly worse than both the exponential and log-exp-exp link-functions, both in terms of NLL and of Md . The exponential link-function performs as well as the log-exp-exp link-function, with a spike train similarity measure Md being almost 1 for both. Finally the likelihood-based method described above gives us the opportunity to look at the relationship between the AdEx parameters σ and ∆T that governs its spike emission and the parameters VT and ∆V of the link-function (Fig. 4). We observe that an increase of the noise level produces a ﬂatter link-function (greater ∆V ) while an increase in ∆T also produces an increase in ∆V and VT (note that Fig. 4 shows ∆V and VT for the exponential link-function only, but equivalent results are obtained with the log-exp-exp link-function). 4 Discussion In Sect. 3.3 we have shown that it is possible to predict with almost perfect accuracy the PSTH of a stochastic AdEx model using an appropriate set of parameters in the SRM. Moreover, since 7 Figure 4: Inﬂuence of the AdEx parameters on the parameters of the exponential link-function. A. VT as a function of ∆T and σ. B. ∆V as a function of ∆T and σ. the subthreshold voltage of the AdEx also gives a good match with the deterministic voltage of the SRM, we expect that the AdEx and the SRM will not differ in higher moments of the spike train probability distributions beyond the PSTH. We therefore conclude that diffusive noise models of the type of Eq. 1-2 are equivalent to GLM of the type of Eq. 3-4. Once combined with similar results on other types of stochastic LIF (e.g. correlated noise), we could bridge the gap between the literature on GLM and the literature on diffusive noise models. Another noteworthy observation pertains to the nature of the link-function. The link-function has been hypothesized to be a linear-rectiﬁer, an exponential, a sigmoidal or a Gaussian [16]. We have observed that for the AdEx the link-function follows Eq. 14 that we called the log-exp-exp linkfunction. Although the link-function is log-exp-exp for most of the AdEx parameters, the exponential link-function gives an equivalently good prediction of the PSTH. This can be explained by the fact that the difference between log-exp-exp and exponential link-functions happens mainly at low voltage (i.e. far from the threshold), where the probability of emitting a spike is so low (Figure 2 C, until -50 mv). Therefore, even if the exponential link-function overestimates the ﬁring probability at these low voltages it rarely produces extra spikes. At voltages closer to the threshold, where most of the spikes are emitted, the two link-functions behave almost identically and hence produce the same PSTH. The Gaussian link-function can be seen as lying in-between the exponential link-function and the log-exp-exp link-function in Fig. 2. This means that the work of Plesser and Gerstner (2000) [16] is in agreement with the results presented here. The importance of the time-derivative of the ˙ voltage stressed by Plesser and Gerstner (leading to a two-dimensional link-function f (V, V )) was not studied here to remain consistent with the typical usage of GLM in neural systems [14]. Finally we restricted our study to exponential non-linearity for spike initiation and do not consider other cases such as the Quadratic Integrate-and-ﬁre (QIF, [5]) or other polynomial functional shapes. We overlooked these cases for two reasons. First, there are many evidences that the non-linearity in neurons (estimated from in-vitro recordings of Pyramidal neurons) is well approximated by a single exponential [9]. Second, the exponential non-linearity of the AdEx only affects the subthreshold voltage at high voltage (close to threshold) and thus can be neglected to derive the ﬁlters κ(t) and η(t). Polynomial non-linearities on the other hand affect a larger range of the subthreshold voltage so that it would be difﬁcult to justify the linearization of subthreshold dynamics essential to the method presented here. References [1] R. B. Stein, “Some models of neuronal variability,” Biophys J, vol. 7, no. 1, pp. 37–68, 1967. [2] W. Gerstner and W. Kistler, Spiking neuron models. Cambridge University Press New York, 2002. [3] E. Izhikevich, “Resonate-and-ﬁre neurons,” Neural Networks, vol. 14, no. 883-894, 2001. [4] M. J. E. Richardson, N. Brunel, and V. Hakim, “From subthreshold to ﬁring-rate resonance,” Journal of Neurophysiology, vol. 89, pp. 2538–2554, 2003. 8 [5] E. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, pp. 1569–1572, 2003. [6] S. Mensi, R. Naud, M. Avermann, C. C. H. Petersen, and W. Gerstner, “Parameter extraction and classiﬁcation of three neuron types reveals two different adaptation mechanisms,” Under review. [7] N. Fourcaud-Trocme, D. Hansel, C. V. Vreeswijk, and N. Brunel, “How spike generation mechanisms determine the neuronal response to ﬂuctuating inputs,” Journal of Neuroscience, vol. 23, no. 37, pp. 11 628–11 640, 2003. [8] R. Brette and W. Gerstner, “Adaptive exponential integrate-and-ﬁre model as an effective description of neuronal activity,” Journal of Neurophysiology, vol. 94, pp. 3637–3642, 2005. [9] L. Badel, W. Gerstner, and M. Richardson, “Dependence of the spike-triggered average voltage on membrane response properties,” Neurocomputing, vol. 69, pp. 1062–1065, 2007. [10] P. McCullagh and J. A. Nelder, Generalized linear models, 2nd ed. Chapman & Hall/CRC, 1998, vol. 37. [11] W. Gerstner, J. van Hemmen, and J. Cowan, “What matters in neuronal locking?” Neural computation, vol. 8, pp. 1653–1676, 1996. [12] D. Hubel and T. Wiesel, “Receptive ﬁelds and functional architecture of monkey striate cortex,” Journal of Physiology, vol. 195, pp. 215–243, 1968. [13] J. Pillow, L. Paninski, V. Uzzell, E. Simoncelli, and E. Chichilnisky, “Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model,” Journal of Neuroscience, vol. 25, no. 47, pp. 11 003–11 013, 2005. [14] K. Doya, S. Ishii, A. Pouget, and R. P. N. Rao, Bayesian brain: Probabilistic approaches to neural coding. The MIT Press, 2007. [15] S. Gerwinn, J. H. Macke, M. Seeger, and M. Bethge, “Bayesian inference for spiking neuron models with a sparsity prior,” in Advances in Neural Information Processing Systems, 2007. [16] H. Plesser and W. Gerstner, “Noise in integrate-and-ﬁre neurons: From stochastic input to escape rates,” Neural Computation, vol. 12, pp. 367–384, 2000. [17] J. Schemmel, J. Fieres, and K. Meier, “Wafer-scale integration of analog neural networks,” in Neural Networks, 2008. IJCNN 2008. (IEEE World Congress on Computational Intelligence). IEEE International Joint Conference on, june 2008, pp. 431 –438. [18] R. Jolivet, T. Lewis, and W. Gerstner, “Generalized integrate-and-ﬁre models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy,” Journal of Neurophysiology, vol. 92, pp. 959–976, 2004. [19] L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network: Computation in Neural Systems, vol. 15, pp. 243–262, 2004. 9

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