nips nips2011 nips2011-141 knowledge-graph by maker-knowledge-mining

141 nips-2011-Large-Scale Category Structure Aware Image Categorization

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Author: Bin Zhao, Fei Li, Eric P. Xing

Abstract: Most previous research on image categorization has focused on medium-scale data sets, while large-scale image categorization with millions of images from thousands of categories remains a challenge. With the emergence of structured large-scale dataset such as the ImageNet, rich information about the conceptual relationships between images, such as a tree hierarchy among various image categories, become available. As human cognition of complex visual world beneﬁts from underlying semantic relationships between object classes, we believe a machine learning system can and should leverage such information as well for better performance. In this paper, we employ such semantic relatedness among image categories for large-scale image categorization. Speciﬁcally, a category hierarchy is utilized to properly deﬁne loss function and select common set of features for related categories. An efﬁcient optimization method based on proximal approximation and accelerated parallel gradient method is introduced. Experimental results on a subset of ImageNet containing 1.2 million images from 1000 categories demonstrate the effectiveness and promise of our proposed approach. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Most previous research on image categorization has focused on medium-scale data sets, while large-scale image categorization with millions of images from thousands of categories remains a challenge. [sent-8, score-1.109]

2 With the emergence of structured large-scale dataset such as the ImageNet, rich information about the conceptual relationships between images, such as a tree hierarchy among various image categories, become available. [sent-9, score-0.469]

3 As human cognition of complex visual world beneﬁts from underlying semantic relationships between object classes, we believe a machine learning system can and should leverage such information as well for better performance. [sent-10, score-0.45]

4 In this paper, we employ such semantic relatedness among image categories for large-scale image categorization. [sent-11, score-1.128]

5 Speciﬁcally, a category hierarchy is utilized to properly deﬁne loss function and select common set of features for related categories. [sent-12, score-0.327]

6 An efﬁcient optimization method based on proximal approximation and accelerated parallel gradient method is introduced. [sent-13, score-0.327]

7 2 million images from 1000 categories demonstrate the effectiveness and promise of our proposed approach. [sent-15, score-0.369]

8 1 Introduction Image categorization / object recognition has been one of the most important research problems in the computer vision community. [sent-16, score-0.25]

9 (A human being has little problem recognizing tens of thousands of visual categories, even with very little “training” data. [sent-18, score-0.12]

10 LabelMe [31] provides 30k labeled and segmented images, covering around 200 image categories. [sent-20, score-0.213]

11 Moreover, the newly released ImageNet [12] data set goes a big step further, in that it further increases the number of classes to over 15000, and has more than 1000 images for each class on average. [sent-21, score-0.138]

12 Similarly, TinyImage [36] contains 80 million 32 × 32 low resolution images, with each image loosely labeled with one of 75,062 English nouns. [sent-22, score-0.268]

13 Clearly, these are no longer artiﬁcial visual categorization problems created for machine learning, but instead more like a human-level cognition problem for real world object recognition with a much bigger set of objects. [sent-23, score-0.359]

14 1 (a) (b) (c) Figure 1: (a) Image category hierarchy in ImageNet; (b) Overlapping group structure; (3) Semantic relatedness measure between image categories. [sent-35, score-0.775]

15 The hierarchical semantic structure stemmed from the WordNet over image categories in the ImageNet data makes it distinctive from other existing large-scale dataset, and it reassembles how human cognitive system stores visual knowledge. [sent-36, score-0.817]

16 Figure 1(a) shows an example such as a tree hierarchy, where leaf nodes are individual categories, and each internal node denotes the cluster of categories corresponding to the leaf nodes in the subtree rooted at the given node. [sent-37, score-0.567]

17 As human cognition of complex visual world beneﬁts from underlying semantic relationships between object classes, we believe a machine learning system can and should leverage such information as well for better performance. [sent-38, score-0.45]

18 It should be noted that our proposed method is applicable to any tree structure for image category, such as the category structure learned to capture visual appearance similarities between image classes [32, 17, 13]. [sent-40, score-0.771]

19 To the best of our knowledge, our attempt in this paper represents an initial foray to systematically utilizing information residing in concept hierarchy, for multi-way classiﬁcation on super large-scale image data sets. [sent-41, score-0.336]

20 More precisely, our approach utilizes the concept hierarchy in two aspects: loss function and feature selection. [sent-42, score-0.186]

21 First, the loss function used in our formulation weighs differentially for different misclassiﬁcation outcomes: misclassifying an image to a category that is close to its true identity should receive less penalty than misclassifying it to a totally unrelated one. [sent-43, score-0.643]

22 Second, in an image classiﬁcation problem with thousands of categories, it is not realistic to assume that all of the classes share the same set of relevant features. [sent-44, score-0.369]

23 That is to say, a subset of highly related categories may share a common set of relevant features, whereas weakly related categories are less likely to be affected by the same features. [sent-45, score-0.534]

24 Consequently, the image categorization problem is formulated as augmented logistic regression with overlapping-group-lasso regularization. [sent-46, score-0.552]

25 To solve this optimization problem, we introduce the Accelerated Parallel ProximaL gradiEnT (APPLET) method, which tackles the non-smoothness of overlapping-group-lasso penalty via proximal gradient [20, 9], and the huge number of training samples by Map-Reduce parallel computing [10]. [sent-48, score-0.414]

26 Therefore, the contributions made in this paper are: (1) We incorporate the semantic relationships between object classes, into an augmented multi-class logistic regression formulation, regularized by the overlapping-group-lasso penalty. [sent-49, score-0.54]

27 (2) We propose a proximal gradient based method for solving the resulting non-smooth optimization problem, where the super large scale of the problem is tackled by map-reduce parallel computation. [sent-51, score-0.325]

28 Section 5 demonstrates the effectiveness of the proposed algorithm using millions of training images from 1000 categories, followed by conclusions in Section 6. [sent-56, score-0.134]

29 1 Category Structure Aware Image Categorization Motivation ImageNet organizes the different classes of images in a densely populated semantic hierarchy. [sent-58, score-0.355]

30 Speciﬁcally, image categories in ImageNet are interlinked by several types of relations, with the 2 “IS-A” relation being the most comprehensive and useful [11], resulting in a tree hierarchy over image categories. [sent-59, score-0.887]

31 For example, the ’husky’ category follows a path in the tree composed of ’working dog’, ’dog’, ’canine’, etc. [sent-60, score-0.249]

32 The distance between two nodes in the tree depicts the difference between the two corresponding image categories. [sent-61, score-0.306]

33 Consequently, in the category hierarchy in ImageNet, each internal node near the bottom of the tree shows that the image categories of its subtree are highly correlated, whereas the internal node near the root represents relatively weaker correlations among the categories in its subtree. [sent-62, score-1.256]

34 The class hierarchy provides a measure of relatedness between image classes. [sent-63, score-0.599]

35 Misclassifying an image to a category that is close to its true identity should receive less penalty than misclassifying it to a totally unrelated one. [sent-64, score-0.532]

36 Instead of using 0-1 loss as in conventional image categorization, which treats image categories equally and independently, our approach utilizes a loss function that is aware of the category hierarchy. [sent-66, score-0.928]

37 Moreover, highly related image categories are more likely to share common visual patterns. [sent-67, score-0.567]

38 For example, in Figure 1(a), husky and shepherd share similar object shape and texture. [sent-68, score-0.154]

39 Consequently, recognition of these related categories are more likely to be affected by the same features. [sent-69, score-0.283]

40 Multi-class logistic regression deﬁnes a weight vector wk for each class k ∈ {1, . [sent-76, score-0.233]

41 ,k} P (y|x, W), with the conditional likelihood computed as T exp(wyi xi ) P (yi |xi , W) = ∑ (1) T k exp(wk xi ) ∗ ∗ The optimal weight vectors W∗ = [w1 , . [sent-82, score-0.118]

42 1 Augmented Soft-Max Loss Function Using the tree hierarchy on image categories, we could calculate a semantic relatedness (a. [sent-88, score-0.874]

43 similarity) matrix S ∈ RK×K over all categories, where Sij measures the semantic relatedness of class i and j. [sent-91, score-0.451]

44 Using this modiﬁed softmax loss function, the image categorization problem could be formulated as [ ( ) ( )] N ∑ ∑∑ ∑ T T min log Sk,r exp(wr xi ) − log Syi ,r exp(wr xi ) + λΩ(W) (5) W i=1 r r k 3 2. [sent-95, score-0.505]

45 2 Semantic Relatedness Matrix To compute semantic relatedness matrix S in the above formulation, we ﬁrst deﬁne a metric measuring the semantic distance between image categories. [sent-97, score-0.881]

46 A simple way to compute semantic distance in a structure such as the one provided by ImageNet is to utilize the paths connecting the two corresponding nodes to the root node. [sent-98, score-0.297]

47 We construct the semantic relatedness matrix S = exp(−κ(1− D)), where κ is a constant controlling the decay factor of semantic relatedness with respect to semantic distance. [sent-100, score-1.119]

48 Figure 1(c) shows the semantic relatedness matrix computed with κ = 5. [sent-101, score-0.451]

49 3 Tree-Guided Sparse Feature Coding In ImageNet, image categories are grouped at multiple granularity as a tree hierarchy. [sent-103, score-0.522]

50 1, the image categories in each internal node are likely to be inﬂuenced by a common set of features. [sent-105, score-0.542]

51 Speciﬁcally, given the tree hierarchy T = (V, E) over image categories, each node v ∈ V of tree T is associated with group Gv , composed of all leaf nodes in the subtree rooted at v, as illustrated in Figure 1(b). [sent-107, score-0.668]

52 , J} associated with categories in Gv , and each group Gv is associated with weight γv that reﬂects the strength of correlation within the group. [sent-115, score-0.286]

53 Specifically, we ﬁrst reformulate the overlapping-group-lasso penalty Ω(W) into a max problem over auxiliary variables using dual norm, and then introduce its smooth lower bound [20, 9]. [sent-122, score-0.216]

54 Instead of optimizing the original non-smooth penalty, we run the accelerated gradient descent method [27] under a Map-Reduce framework [10] to optimize the smooth lower bound. [sent-123, score-0.22]

55 For each input j and group gi associated with wjgi , we introduce a vector of auxiliary variables αjgi ∈ R|gi | . [sent-130, score-0.205]

56 Since the dual norm of L2 norm is also an L2 norm, we can reformulate ||wjgi ||2 as ||wjgi ||2 = ∑ max||αjgi ||2 ≤1 αT i wjgi . [sent-131, score-0.163]

57 Following [9], the overlappinggroup-lasso penalty in (8) can be equivalently reformulated as ∑∑ Ω(W) = γi max αT i wjgi = max⟨CWT , A⟩ (10) jg j i ||αjgi ||2 ≤1 A∈O ∑ where i = 1, . [sent-151, score-0.282]

58 Moreover, the matrix C is deﬁned with rows indexed by (s, gi ) such that s ∈ gi and i ∈ {1, . [sent-158, score-0.12]

59 Speciﬁcally, our smooth approximation function is deﬁned as: fµ (W) = max⟨CWT , A⟩ − µd(A) A∈O (11) where µ is the positive smoothness parameter and d(A) is an arbitrary smooth strongly-convex function deﬁned on O. [sent-167, score-0.124]

60 According to Theorem 1, fµ (W) is a smooth function for any µ > 0, with a simple form of gradient and can be viewed as a smooth approximation of f0 (W) with the maximum gap of µD. [sent-172, score-0.201]

61 2 Accelerated Parallel Gradient Method Given the smooth approximation of Ω(W) in (11) and the corresponding gradient presented in Theorem 1, we could apply gradient descent method to solve the problem. [sent-174, score-0.216]

62 wk could be calculated as follows [ ∑ ] N T T Syi ,k exp(wk xi ) ∂g(W) ∑ q Sk,q exp(wk xi ) = xi ∑ ∑ −∑ (14) T T ∂wk r q Sr,q exp(wr xi ) r Syi ,r exp(wr xi ) i=1 5 Therefore, the gradient of g(W) w. [sent-179, score-0.46]

63 ∂w1 ∂wK ˜ According to Theorem 1, the gradient of f (W) is given by ˜ ∇f (W) = ∇g(W) + λA∗T C (15) ˜ Although f (W) is a smooth function of W, it is represented as a summation over all training sam˜ ples. [sent-186, score-0.17]

64 Due to the huge number of samples in the training set, we adopt a Map-Reduce parallel framework [10] to compute ∇g(W) as shown in Eq. [sent-188, score-0.136]

65 In this paper, we adopt this accelerated gradient method , and the whole algorithm is shown in Algorithm 1. [sent-192, score-0.158]

66 Recent approaches transfer information across classes by regularizing the parameters of the classiﬁers across classes [37, 28, 15, 33, 34, 2, 26, 30]. [sent-201, score-0.182]

67 It is unclear how these approaches would perform on super large-scale data sets containing thousands of image categories. [sent-203, score-0.341]

68 Some of these approaches would encounter severe computational bottleneck when scaling up to thousands of classes [16]. [sent-204, score-0.124]

69 However, none of these approaches utilize the semantic relationships deﬁned among image categories in ImageNet, which we argue is a crucial source of information for further improvement in such super large scale classiﬁcation problem. [sent-207, score-0.851]

70 The number of images for each category ranges from 668 to 3047. [sent-210, score-0.204]

71 Our work, on the other hand, looks at this problem from a different angle, focusing on principled 6 methodology that explores the beneﬁt of utilizing class structure in image categorization and proposing a model and related optimization technique to properly incorporate such information. [sent-214, score-0.353]

72 1 Image Features Each image is resized to have a max side length of 300 pixels. [sent-218, score-0.213]

73 We then perform k-means clustering on a random subset of 10 million SIFT descriptors to form a visual vocabulary of 1000 visual words. [sent-221, score-0.23]

74 Finally, a single feature vector is computed for each image using max pooling on a spatial pyramid [22]. [sent-223, score-0.256]

75 Consequently, each image is represented as a vector in a 21,000 dimensional space. [sent-225, score-0.213]

76 Speciﬁcally, for every image, each tested algorithm will produce a list of 5 object categories in the descending order of conﬁdence. [sent-228, score-0.329]

77 012 Figure 2: Left: image classes with highest accuracy. [sent-277, score-0.288]

78 We use the conventional L2 regularized logistic regression [5] as baseline. [sent-282, score-0.145]

79 According to the classiﬁcation results, we could clearly see the advantage of APPLET over conventional logistic regression, especially on the top-5 error rate. [sent-285, score-0.125]

80 It should be noted 7 that hierarchical error is measured by the height of the lowest common ancestor in the hierarchy, and moving up a level can more than double the number of descendants. [sent-289, score-0.141]

81 Moreover, Figure 2 shows the image categories with highest and lowest classiﬁcation accuracy. [sent-294, score-0.507]

82 One key reason for introducing the augmented loss function is to ensure that predicted image class falls not too far from its true class on the semantic hierarchy. [sent-295, score-0.518]

83 As shown in Table 1, a systematic reduction in classiﬁcation error using APPLET shows that acknowledging semantic relationships between image classes enables the system to discriminate at more informative semantic levels. [sent-300, score-0.801]

84 4 We present in Figure 3 how categorization performance scales with λ and κ. [sent-303, score-0.14]

85 According to Figure 3, APPLET achieves lowest categorization error around λ = 0. [sent-304, score-0.216]

86 When κ is too small, a large number of categories are mixed together, resulting in a much higher ﬂat loss. [sent-330, score-0.251]

87 On the other hand, when κ ≥ 50, the semantic relatedness matrix is close to diagonal, resulting in treating all categories independently, and categorization performance becomes similar as LR. [sent-331, score-0.842]

88 6 Conclusions In this paper, we argue the positive effect of incorporating category hierarchy information in super large scale image categorization. [sent-332, score-0.585]

89 2 million training images from 1000 categories demonstrates the effectiveness and promise of our proposed approach. [sent-335, score-0.4]

90 Smoothing proximal gradient method for general structured sparse learning. [sent-396, score-0.177]

91 What does classifying more than 10,000 image categories tell us? [sent-414, score-0.495]

92 Fast and balanced: Efﬁcient label tree learning for large scale object recognition. [sent-431, score-0.136]

93 Learning generative visual models from few training examples: an incremental bayesian approach tested on 101 object categories. [sent-437, score-0.18]

94 Discriminative learning of relaxed hierarchy for large-scale visual recognition. [sent-455, score-0.223]

95 Large-scale image classiﬁcation: fast feature extraction and svm training. [sent-505, score-0.213]

96 Incremental learning of object detectors using a visual shape alphabet. [sent-528, score-0.149]

97 Transfer learning for image classiﬁcation with sparse prototype representations. [sent-540, score-0.213]

98 Labelme: A database and web-based tool for image annotation. [sent-547, score-0.213]

99 Learning to share visual appearance for multiclass object detection. [sent-552, score-0.181]

100 80 million tiny images: A large data set for nonparametric object and scene recognition. [sent-578, score-0.133]

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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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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. 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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. 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