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143 nips-2010-Learning Convolutional Feature Hierarchies for Visual Recognition

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Author: Koray Kavukcuoglu, Pierre Sermanet, Y-lan Boureau, Karol Gregor, Michael Mathieu, Yann L. Cun

Abstract: We propose an unsupervised method for learning multi-stage hierarchies of sparse convolutional features. While sparse coding has become an increasingly popular method for learning visual features, it is most often trained at the patch level. Applying the resulting ﬁlters convolutionally results in highly redundant codes because overlapping patches are encoded in isolation. By training convolutionally over large image windows, our method reduces the redudancy between feature vectors at neighboring locations and improves the efﬁciency of the overall representation. In addition to a linear decoder that reconstructs the image from sparse features, our method trains an efﬁcient feed-forward encoder that predicts quasisparse features from the input. While patch-based training rarely produces anything but oriented edge detectors, we show that convolutional training produces highly diverse ﬁlters, including center-surround ﬁlters, corner detectors, cross detectors, and oriented grating detectors. We show that using these ﬁlters in multistage convolutional network architecture improves performance on a number of visual recognition and detection tasks. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 fr Abstract We propose an unsupervised method for learning multi-stage hierarchies of sparse convolutional features. [sent-5, score-0.775]

2 While sparse coding has become an increasingly popular method for learning visual features, it is most often trained at the patch level. [sent-6, score-0.519]

3 By training convolutionally over large image windows, our method reduces the redudancy between feature vectors at neighboring locations and improves the efﬁciency of the overall representation. [sent-8, score-0.262]

4 In addition to a linear decoder that reconstructs the image from sparse features, our method trains an efﬁcient feed-forward encoder that predicts quasisparse features from the input. [sent-9, score-0.677]

5 While patch-based training rarely produces anything but oriented edge detectors, we show that convolutional training produces highly diverse ﬁlters, including center-surround ﬁlters, corner detectors, cross detectors, and oriented grating detectors. [sent-10, score-0.706]

6 We show that using these ﬁlters in multistage convolutional network architecture improves performance on a number of visual recognition and detection tasks. [sent-11, score-0.76]

7 1 Introduction Over the last few years, a growing amount of research on visual recognition has focused on learning low-level and mid-level features using unsupervised learning, supervised learning, or a combination of the two. [sent-12, score-0.348]

8 The present paper introduces a new class of techniques for learning features extracted though convolutional ﬁlter banks. [sent-15, score-0.577]

9 The techniques are applicable to Convolutional Networks and their variants, which use multiple stages of trainable convolutional ﬁlter banks, interspersed with non-linear operations, and spatial feature pooling operations [1, 2]. [sent-16, score-0.738]

10 While ConvNets have traditionally been trained in supervised mode, a number of recent systems have proposed to use unsupervised learning to pretrain the ﬁlters, followed by supervised ﬁne-tuning. [sent-17, score-0.374]

11 Some authors have used convolutional forms of Restricted Boltzmann Machines (RBM) trained with contrastive divergence [3], but many of them have relied on sparse coding and sparse modeling [4, 5, 6]. [sent-18, score-1.033]

12 In sparse coding, a sparse feature vector z is computed so as to best reconstruct the input x through a linear operation with a learned dictionary matrix D. [sent-19, score-0.603]

13 Right: A dictionary with 128 elements, learned with convolutional sparse coding model. [sent-21, score-1.076]

14 The dictionary learned with the convolutional model spans the orientation space much more uniformly. [sent-22, score-0.771]

15 In addition it can be seen that the diversity of ﬁlters obtained by convolutional sparse model is much richer compared to patch based one. [sent-23, score-0.742]

16 The dictionary is obtained by minimizing the energy 1 wrt D: minz,D L(x, z, D) averaged over a training set of input samples. [sent-24, score-0.344]

17 In most applications of sparse coding to image analysis [7, 8], the system is trained on single image patches whose dimensions match those of the ﬁlters. [sent-27, score-0.62]

18 This procedure completely ignores the fact that the ﬁlters are eventually going to be used in a convolutional fashion. [sent-29, score-0.518]

19 Learning will produce a dictionary of ﬁlters that are essentially shifted versions of each other over the patch, so as to reconstruct each patch in isolation. [sent-30, score-0.351]

20 To address the second problem, we follow the idea of [4, 5], and use a trainable, feed-forward, nonlinear encoder module to produce a fast approximation of the sparse code. [sent-34, score-0.591]

21 The contribution of this paper is to address both issues simultaneously, thus allowing convolutional approaches to sparse coding to scale up, and opening the road to real-time applications. [sent-38, score-0.823]

22 2 Algorithms and Method In this section, we analyze the beneﬁts of convolutional sparse coding for object recognition systems, and propose convolutional extensions to the coordinate descent sparse coding (CoD) [9] algorithm and the dictionary learning procedure. [sent-39, score-2.13]

23 Any existing sparse coding algorithm can then be used. [sent-42, score-0.305]

24 Unfortunately, this method incurs a cost, since the size of the dictionary then depends on the size of the input signal. [sent-43, score-0.253]

25 In this work, we use the coordinate descent sparse coding algorithm [9] as a starting point and generalize it using convolution operations. [sent-45, score-0.491]

26 Two important issues arise when learning convolutional dictionaries: 1. [sent-46, score-0.518]

27 Since the loss is not jointly convex in D and z, but is convex in each variable when the other one is kept ﬁxed, sparse dictionaries are usually learned by an approach similar to block coordinate descent, which alternatively minimizes over z and D (e. [sent-50, score-0.241]

28 One can use either batch [7] (by accumulating derivatives over many samples) or online updates [8, 6, 5] (updating the dictionary after each sample). [sent-53, score-0.285]

29 In this work, we use a stochastic online procedure for updating the dictionary elements. [sent-54, score-0.283]

30 The updates to the dictionary elements, calculated from equation 2, are sensitive to the boundary effects introduced by the convolution operator. [sent-55, score-0.331]

31 Algorithm 1 Convolutional extension to coordinate descent sparse coding[9]. [sent-59, score-0.234]

32 repeat z = hα (β) ¯ (k, p, q) = arg maxi,m,n |zimn − zimn | (k : dictionary index, (p. [sent-63, score-0.292]

33 q) : location index) ¯ bi = βkpq β = β + (zkpq − zkpq ) × align(S(:, k, :, :), (p, q)) ¯ zkpq = zkpq , βkpq = bi ¯ until change in z is below a threshold end function The second important point in training convolutional dictionaries is the computation of the S = DT ∗ D operator. [sent-64, score-0.873]

34 For convolutional modeling, the same approach can be followed with some additional care. [sent-66, score-0.561]

35 In patch based sparse coding, each element (i, j) of S equals the dot product of dictionary elements i and j. [sent-67, score-0.52]

36 Since the similarity of a pair of dictionary elements has to be also considered in spatial dimensions, each term is expanded as “full” convolution of two dictionary elements (i, j), producing 2s−1×2s−1 matrix. [sent-68, score-0.711]

37 One can also use the steepest descent algorithm for ﬁnding the solution to convolutional sparse coding given in equation 2, however using this method would be orders of magnitude slower compared to specialized algorithms like CoD [9] and the solution would never contain exact zeros. [sent-71, score-0.931]

38 In algorithm 1 we explain the extension of the coordinate descent algorithm [9] for convolutional inputs. [sent-72, score-0.626]

39 Having formulated convolutional sparse coding, the overall learning procedure is simple stochastic (online) gradient descent over dictionary D: ∀xi ∈ X training set : z ∗ = arg min L(xi , z, D), D ← D − η z ∂L(xi , z ∗ , D) ∂D (4) The columns of D are normalized after each iteration. [sent-73, score-1.011]

40 A convolutional dictionary with 128 elements which was trained on images from Berkeley dataset [13] is shown in ﬁgure 1. [sent-74, score-0.982]

41 It is clear that for both encoder functions, hessian update improves the convergence signiﬁcantly. [sent-80, score-0.462]

42 Right: 128 convolutional ﬁlters (W ) learned in the encoder using smooth shrinkage function. [sent-81, score-1.011]

43 In this work, we extend their encoder module training to convolutional domain and also propose a new encoder function that approximates sparse codes more closely. [sent-85, score-1.595]

44 The encoder used in [14] is a simple feedforward function which can also be seen as a small convolutional neural network: z = g k × ˜ tanh(x ∗ W k ) (k = 1. [sent-86, score-0.947]

45 This function has been shown to produce good features for object recognition [14], however it does not include a shrinkage operator, thus its ability to produce sparse representations is very limited. [sent-89, score-0.372]

46 In order to be able to train a ﬁlter bank that is applied to the input before the shrinkage operator, we propose to use an encoder with a smooth shrinkage operator z = shβ k ,bk (x ∗ W k ) where k = 1. [sent-92, score-0.627]

47 In a ˜ sense, training the encoder module is similar to training a ConvNet. [sent-101, score-0.579]

48 It can be seen that especially for the tanh encoder function, the effect of using second order information on the convergence is signiﬁcant. [sent-104, score-0.503]

49 3 Patch Based vs Convolutional Sparse Modeling Natural images, sounds, and more generally, signals that display translation invariance in any dimension, are better represented using convolutional dictionaries. [sent-106, score-0.518]

50 For example, if k × k image patches are sampled from a set of natural images, an edge at a given orientation may appear at any location, forcing local models to allocate multiple dictionary elements to represent a single underlying orientation. [sent-108, score-0.409]

51 By contrast, a convolutional model only needs to record the oriented structure once, since dictionary elements can be used at all locations. [sent-109, score-0.851]

52 Figure 1 shows atoms from patch-based and convolutional dictionaries comprising the same number of elements. [sent-110, score-0.582]

53 The convolutional dictionary does not waste resources modeling similar ﬁlter structure at multiple locations. [sent-111, score-0.771]

54 In this work, we present two encoder architectures, 1. [sent-113, score-0.429]

55 steepest descent sparse coding with tanh encoding function using g k × tanh(x ∗ W k ), 2. [sent-114, score-0.536]

56 convolutional CoD sparse coding with shrink 4 encoding function using shβ,b (x ∗ W k ). [sent-115, score-0.872]

57 The time required for training the ﬁrst system is much higher than for the second system due to steepest descent sparse coding. [sent-116, score-0.401]

58 4 Multi-stage architecture Our convolutional encoder can be used to replace patch-based sparse coding modules used in multistage object recognition architectures such as the one proposed in our previous work [14]. [sent-119, score-1.481]

59 Building on our previous ﬁndings, for each stage, the encoder is followed by and absolute value rectiﬁcation, contrast normalization and average subsampling. [sent-120, score-0.502]

60 The last operation, average pooling is simply a spatial pooling operation that is applied on each feature map independently. [sent-125, score-0.301]

61 In the next sections, we present experiments showing that using convolutionally trained encoders in this architecture lead to better object recognition performance. [sent-129, score-0.401]

62 Steepest descent sparse coding with tanh encoder: SDtanh . [sent-132, score-0.436]

63 Coordinate descent sparse coding with shrink encoder: CDshrink . [sent-134, score-0.362]

64 In the following, we give details of the unsupervised training and supervised recognition experiments. [sent-135, score-0.314]

65 Architecture : We use the unsupervised trained encoders in a multi-stage system identical to the one proposed in [14]. [sent-147, score-0.309]

66 At ﬁrst layer 64 features are extracted from the input image, followed by a second layers that produces 256 features. [sent-148, score-0.272]

67 Second layer features are connected to ﬁst layer features through a sparse connection table to break the symmetry and to decrease the number of parameters. [sent-149, score-0.518]

68 Unsupervised Training : The input to unsupervised training consists of contrast normalized grayscale images [20] obtained from the Berkeley segmentation dataset [13]. [sent-150, score-0.272]

69 First Layer: We have trained both systems using 64 dictionary elements. [sent-152, score-0.337]

70 Each dictionary item is a 9 × 9 convolution kernel. [sent-153, score-0.331]

71 The resulting system to be solved is a 64 times overcomplete sparse coding problem. [sent-154, score-0.36]

72 Second Layer: Using the 64 feature maps output from the ﬁrst layer encoder on Berkeley images, we train a second layer convolutional sparse coding. [sent-158, score-1.475]

73 Thus, we aim to learn 4096 convolutional kernels at the second layer. [sent-160, score-0.518]

74 To the best of our knowledge, none of the previous convolutional RBM [3] and sparse coding [6] methods have learned such a large number of dictionary elements. [sent-161, score-1.076]

75 Left: Encoder kernels that correspond to the dictionary elements. [sent-166, score-0.253]

76 Right: 128 dictionary elements, each row shows 16 dictionary elements, connecting to a single second layer feature map. [sent-167, score-0.707]

77 2% Table 1: Comparing SDtanh encoder to CDshrink encoder on Caltech 101 dataset using a single stage architecture. [sent-181, score-0.972]

78 One Stage System: We train 64 convolutional unsupervised features using both SDtanh and CDshrink methods. [sent-184, score-0.738]

79 We use the encoder function obtained from this training followed by absolute value rectiﬁcation, contrast normalization and average pooling. [sent-185, score-0.559]

80 The output of ﬁrst layer is then 64 × 26 × 26 and fed into a logistic regression classiﬁer and Lazebnik’s PMK-SVM classiﬁer [21] (that is, the spatial pyramid pipeline is used, using our features to replace the SIFT features). [sent-188, score-0.341]

81 Two Stage System: We train 4096 convolutional ﬁlters with SDtanh method using 64 input feature maps from ﬁrst stage to produce 256 feature maps. [sent-189, score-0.792]

82 We only use multinomial logistic regression classiﬁer after the second layer feature extraction stage. [sent-192, score-0.248]

83 We denote unsupervised trained one stage systems with U , two stage unsupervised trained systems with U U and “+ ” represents supervised training is performed afterwards. [sent-193, score-0.709]

84 2% reported in [14], we see that convolutional training results in signiﬁcant improvement. [sent-213, score-0.575]

85 5%) −1 10 0 10 1 10 false positives per image false positives per image Figure 4: Results on the INRIA dataset with per-image metric. [sent-248, score-0.372]

86 7% with a spatial pyramid on top of our single-layer U system (with 256 codewords jointly encoding 2 × 2 neighborhoods of our features by hard quantization, then max pooling in each cell of the pyramid, with a linear SVM, as proposed by authors in [22]). [sent-253, score-0.358]

87 Our experiments have shown that sparse features achieve superior recognition performance compared to features obtained using a dictionary trained by a patch-based procedure as shown in Table 2. [sent-254, score-0.649]

88 It is interesting to note that the improvement is larger when using feature extractors trained in a purely unsupervised way, than when unsupervised training is followed by a supervised training phase (57. [sent-255, score-0.654]

89 Recalling that the supervised tuning is a convolutional procedure, this last training step might have the additional beneﬁt of decreasing the redundancy between patch-based dictionary elements. [sent-258, score-0.886]

90 On the other hand, this contribution would be minor for dictionaries which have already been trained convolutionally in the unsupervised stage. [sent-259, score-0.357]

91 A fully connected linear layer with 2 output scores (for pedestrian and background) was used as the classiﬁer. [sent-268, score-0.284]

92 We have trained our system with and without unsupervised initialization, followed by ﬁne-tuning of the entire architecture in supervised manner. [sent-270, score-0.448]

93 After one pass of unsupervised and/or supervised training, several bootstrapping passes were performed to augment the negative set with the 10 most offending samples on each full negative image and the bigger/smaller scaled positives. [sent-272, score-0.368]

94 The value of 1 FPPI is meaningful for pedestrian detection because in real world applications, it is desirable to limit the number of false alarms. [sent-318, score-0.235]

95 However, an unsupervised method can model large variations in pedestrian pose, scale and clutter with much better success. [sent-323, score-0.278]

96 Our aim in this work was to show that an adaptable feature extraction system that learns its parameters from available data can perform comparably to best systems for pedestrian detection. [sent-330, score-0.313]

97 Two different methods were presented for convolutional sparse coding, it was shown that convolutional training of feature extractors reduces the redundancy among ﬁlters compared with those obtained from patch based models. [sent-333, score-1.41]

98 Additionally, we have introduced two different convolutional encoder functions for performing efﬁcient feature extraction which is crucial for using sparse coding in real world applications. [sent-334, score-1.363]

99 We have applied the proposed sparse modeling systems using a successful multi-stage architecture on object recognition and pedestrian detection problems and performed comparably to similar systems. [sent-335, score-0.509]

100 In the pedestrian detection task, we have presented the advantage of using unsupervised learning for feature extraction. [sent-336, score-0.378]

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Earlier work considered homogeneous MRF’s in which each potential is replicated at all image locations. This has the subtle effect of making learning very difﬁcult because of strong correlations at nearby sites. Following Gregor and LeCun [18] and also Tang and Eliasmith [19], we keep the number of parameters under control by using local potentials, but unlike Roth and Black [6] we only share weights between potentials that do not overlap. 2 Augmenting Gated MRF’s with Mean Hidden Units A Product of Student’s t (PoT) model [15] is a gated MRF deﬁned on small image patches that can be viewed as modelling image-speciﬁc, pair-wise relationships between pixel values by using the states of its latent variables. It is very good at representing the fact that two-pixel have very similar intensities and no good at all at modelling what these intensities are. Failure to model the mean also leads to impoverished modelling of the covariances when the input images have nonzero mean intensity. The covariance RBM (cRBM) [20] is another model that shares the same limitation since it only differs from PoT in the distribution of its latent variables: The posterior over the latent variables is a product of Bernoulli distributions instead of Gamma distributions as in PoT. We explain the fundamental limitation of these models by using a simple toy example: Modelling two-pixel images using a cRBM with only one binary hidden unit, see ﬁg. 1. This cRBM assumes that the conditional distribution over the input is a zero-mean Gaussian with a covariance that is determined by the state of the latent variable. Since the latent variable is binary, the cRBM can be viewed as a mixture of two zero-mean full covariance Gaussians. The latent variable uses the pairwise relationship between pixels to decide which of the two covariance matrices should be used to model each image. When the input data is pre-proessed by making each image have zero mean intensity (the empirical histogram is shown in the ﬁrst row and ﬁrst column), most images lie near the origin because most of the times nearby pixels are strongly correlated. Less frequently we encounter edge images that exhibit strong anti-correlation between the pixels, as shown by the long tails along the anti-diagonal line. A cRBM could model this data by using two Gaussians (ﬁrst row and second column): one that is spherical and tight at the origin for smooth images and another one that has a covariance elongated along the anti-diagonal for structured images. If, however, the whole set of images is normalized by subtracting from every pixel the mean value of all pixels over all images (second row and ﬁrst column), the cRBM fails at modelling structured images (second row and second column). It can ﬁt a Gaussian to the smooth images by discovering 2 Figure 1: In the ﬁrst row, each image is zero mean. In the second row, the whole set of data points is centered but each image can have non-zero mean. The ﬁrst column shows 8x8 images picked at random from natural images. The images in the second column are generated by a model that does not account for mean intensity. The images in the third column are generated by a model that has both “mean” and “covariance” hidden units. The contours in the ﬁrst column show the negative log of the empirical distribution of (tiny) natural two-pixel images (x-axis being the ﬁrst pixel and the y-axis the second pixel). The plots in the other columns are toy examples showing how each model could represent the empirical distribution using a mixture of Gaussians with components that have one of two possible covariances (corresponding to the state of a binary “covariance” latent variable). Models that can change the means of the Gaussians (mPoT and mcRBM) can represent better structured images (edge images lie along the anti-diagonal and are ﬁtted by the Gaussians shown in red) while the other models (PoT and cRBM) fail, overall when each image can have non-zero mean. the direction of strong correlation along the main diagonal, but it is very likely to fail to discover the direction of anti-correlation, which is crucial to represent discontinuities, because structured images with different mean intensity appear to be evenly spread over the whole input space. If the model has another set of latent variables that can change the means of the Gaussian distributions in the mixture (as explained more formally below and yielding the mPoT and mcRBM models), then the model can represent both changes of mean intensity and the correlational structure of pixels (see last column). The mean latent variables effectively subtract off the relevant mean from each data-point, letting the covariance latent variable capture the covariance structure of the data. As before, the covariance latent variable needs only to select between two covariance matrices. In fact, experiments on real 8x8 image patches conﬁrm these conjectures. Fig. 1 shows samples drawn from PoT and mPoT. mPoT (and similarly mcRBM [4]) is not only better at modelling zero mean images but it can also represent images that have non zero mean intensity well. We now describe mPoT, referring the reader to [4] for a detailed description of mcRBM. In PoT [9] the energy function is: E PoT (x, hc ) = i 1 [hc (1 + (Ci T x)2 ) + (1 − γ) log hc ] i i 2 (1) where x is a vectorized image patch, hc is a vector of Gamma “covariance” latent variables, C is a ﬁlter bank matrix and γ is a scalar parameter. The joint probability over input pixels and latent variables is proportional to exp(−E PoT (x, hc )). Therefore, the conditional distribution over the input pixels is a zero-mean Gaussian with covariance equal to: Σc = (Cdiag(hc )C T )−1 . (2) In order to make the mean of the conditional distribution non-zero, we deﬁne mPoT as the normalized product of the above zero-mean Gaussian that models the covariance and a spherical covariance Gaussian that models the mean. The overall energy function becomes: E mPoT (x, hc , hm ) = E PoT (x, hc ) + E m (x, hm ) 3 (3) Figure 2: Illustration of different choices of weight-sharing scheme for a RBM. Links converging to one latent variable are ﬁlters. Filters with the same color share the same parameters. Kinds of weight-sharing scheme: A) Global, B) Local, C) TConv and D) Conv. E) TConv applied to an image. Cells correspond to neighborhoods to which ﬁlters are applied. Cells with the same color share the same parameters. F) 256 ﬁlters learned by a Gaussian RBM with TConv weight-sharing scheme on high-resolution natural images. Each ﬁlter has size 16x16 pixels and it is applied every 16 pixels in both the horizontal and vertical directions. Filters in position (i, j) and (1, 1) are applied to neighborhoods that are (i, j) pixels away form each other. Best viewed in color. where hm is another set of latent variables that are assumed to be Bernoulli distributed (but other distributions could be used). The new energy term is: E m (x, hm ) = 1 T x x− 2 hm Wj T x j (4) j yielding the following conditional distribution over the input pixels: p(x|hc , hm ) = N (Σ(W hm ), Σ), Σ = (Σc + I)−1 (5) with Σc deﬁned in eq. 2. As desired, the conditional distribution has non-zero mean2 . Patch-based models like PoT have been extended to high-resolution images by using spatially localized ﬁlters [6]. While we can subtract off the mean intensity from independent image patches to successfully train PoT, we cannot do that on a high-resolution image because overlapping patches might have different mean. Unfortunately, replicating potentials over the image ignoring variations of mean intensity has been the leading strategy to date [6]3 . This is the major reason why generation of high-resolution images is so poor. Sec. 4 shows that generation can be drastically improved by explicitly accounting for variations of mean intensity, as performed by mPoT and mcRBM. 3 Weight-Sharing Schemes By integrating out the latent variables, we can write the density function of any gated MRF as a normalized product of potential functions (for mPoT refer to eq. 6). In this section we investigate different ways of constraining the parameters of the potentials of a generic MRF. Global: The obvious way to extend a patch-based model like PoT to high-resolution images is to deﬁne potentials over the whole image; we call this scheme global. This is not practical because 1) the number of parameters grows about quadratically with the size of the image making training too slow, 2) we do not need to model interactions between very distant pairs of pixels since their dependence is negligible, and 3) we would not be able to use the model on images of different size. Conv: The most popular way to handle big images is to deﬁne potentials on small subsets of variables (e.g., neighborhoods of size 5x5 pixels) and to replicate these potentials across space while 2 The need to model the means was clearly recognized in [21] but they used conjunctive latent features that simultaneously represented a contribution to the “precision matrix” in a speciﬁc direction and the mean along that same direction. 3 The success of PoT-like models in Bayesian denoising is not surprising since the noisy image effectively replaces the reconstruction term from the mean hidden units (see eq. 5), providing a set of noisy mean intensities that are cleaned up by the patterns of correlation enforced by the covariance latent variables. 4 sharing their parameters at each image location [23, 24, 6]. This yields a convolutional weightsharing scheme, also called homogeneous ﬁeld in the statistics literature. This choice is justiﬁed by the stationarity of natural images. This weight-sharing scheme is extremely concise in terms of number of parameters, but also rather inefﬁcient in terms of latent representation. First, if there are N ﬁlters at each location and these ﬁlters are stepped by one pixel then the internal representation is about N times overcomplete. The internal representation has not only high computational cost, but it is also highly redundant. Since the input is mostly smooth and the parameters are the same across space, the latent variables are strongly correlated as well. This inefﬁciency turns out to be particularly harmful for a model like PoT causing the learned ﬁlters to become “random” looking (see ﬁg 3-iii). A simple intuition follows from the equivalence between PoT and square ICA [15]. If the ﬁlter matrix C of eq. 1 is square and invertible, we can marginalize out the latent variables and write: p(y) = i S(yi ), where yi = Ci T x and S is a Student’s t distribution. In other words, there is an underlying assumption that ﬁlter outputs are independent. However, if the ﬁlters of matrix C are shifted and overlapping versions of each other, this clearly cannot be true. Training PoT with the Conv weight-sharing scheme forces the model to ﬁnd ﬁlters that make ﬁlter outputs as independent as possible, which explains the very high-frequency patterns that are usually discovered [6]. Local: The Global and Conv weight-sharing schemes are at the two extremes of a spectrum of possibilities. For instance, we can deﬁne potentials on a small subset of input variables but, unlike Conv, each potential can have its own set of parameters, as shown in ﬁg. 2-B. This is called local, or inhomogeneous ﬁeld. Compared to Conv the number of parameters increases only slightly but the number of latent variables required and their redundancy is greatly reduced. In fact, the model learns different receptive ﬁelds at different locations as a better strategy for representing the input, overall when the number of potentials is limited (see also ﬁg. 2-F). TConv: Local would not allow the model to be trained and tested on images of different resolution, and it might seem wasteful not to exploit the translation invariant property of images. We therefore advocate the use of a weight-sharing scheme that we call tiled-convolutional (TConv) shown in ﬁg. 2-C and E [18]. Each ﬁlter tiles the image without overlaps with copies of itself (i.e. the stride equals the ﬁlter diameter). This reduces spatial redundancy of latent variables and allows the input images to have arbitrary size. At the same time, different ﬁlters do overlap with each other in order to avoid tiling artifacts. Fig. 2-F shows ﬁlters that were (jointly) learned by a Restricted Boltzmann Machine (RBM) [29] with Gaussian input variables using the TConv weight-sharing scheme. 4 Experiments We train gated MRF’s with and without mean hidden units using different weight-sharing schemes. The training procedure is very similar in all cases. We perform approximate maximum likelihood by using Fast Persistence Contrastive Divergence (FPCD) [25] and we draw samples by using Hybrid Monte Carlo (HMC) [26]. Since all latent variables can be exactly marginalized out we can use HMC on the free energy (negative logarithm of the marginal distribution over the input pixels). For mPoT this is: F mPoT (x) = − log(p(x))+const. = k,i 1 1 γ log(1+ (Cik T xk )2 )+ xT x− 2 2 T log(1+exp(Wjk xk )) (6) k,j where the index k runs over spatial locations and xk is the k-th image patch. FPCD keeps samples, called negative particles, that it uses to represent the model distribution. These particles are all updated after each weight update. For each mini-batch of data-points a) we compute the derivative of the free energy w.r.t. the training samples, b) we update the negative particles by running HMC for one HMC step consisting of 20 leapfrog steps. We start at the previous set of negative particles and use as parameters the sum of the regular parameters and a small perturbation vector, c) we compute the derivative of the free energy at the negative particles, and d) we update the regular parameters by using the difference of gradients between step a) and c) while the perturbation vector is updated using the gradient from c) only. The perturbation is also strongly decayed to zero and is subject to a larger learning rate. The aim is to encourage the negative particles to explore the space more quickly by slightly and temporarily raising the energy at their current position. Note that the use of FPCD as opposed to other estimation methods (like Persistent Contrastive Divergence [27]) turns out to be crucial to achieve good mixing of the sampler even after training. We train on mini-batches of 32 samples using gray-scale images of approximate size 160x160 pixels randomly cropped from the Berkeley segmentation dataset [28]. We perform 160,000 weight updates decreasing the learning by a factor of 4 by the end of training. The initial learning rate is set to 0.1 for the covariance 5 Figure 3: 160x160 samples drawn by A) mPoT-TConv, B) mHPoT-TConv, C) mcRBM-TConv and D) PoTTConv. On the side also i) a subset of 8x8 “covariance” ﬁlters learned by mPoT-TConv (the plot below shows how the whole set of ﬁlters tile a small patch; each bar correspond to a Gabor ﬁt of a ﬁlter and colors identify ﬁlters applied at the same 8x8 location, each group is shifted by 2 pixels down the diagonal and a high-resolution image is tiled by replicating this pattern every 8 pixels horizontally and vertically), ii) a subset of 8x8 “mean” ﬁlters learned by the same mPoT-TConv, iii) ﬁlters learned by PoT-Conv and iv) by PoT-TConv. ﬁlters (matrix C of eq. 1), 0.01 for the mean parameters (matrix W of eq. 4), and 0.001 for the other parameters (γ of eq. 1). During training we condition on the borders and initialize the negative particles at zero in order to avoid artifacts at the border of the image. We learn 8x8 ﬁlters and pre-multiply the covariance ﬁlters by a whitening transform retaining 99% of the variance; we also normalize the norm of the covariance ﬁlters to prevent some of them from decaying to zero during training4 . Whenever we use the TConv weight-sharing scheme the model learns covariance ﬁlters that mostly resemble localized and oriented Gabor functions (see ﬁg. 3-i and iv), while the Conv weight-sharing scheme learns structured but poorly localized high-frequency patterns (see ﬁg. 3-iii) [6]. The TConv models re-use the same 8x8 ﬁlters every 8 pixels and apply a diagonal offset of 2 pixels between neighboring ﬁlters with different weights in order to reduce tiling artifacts. There are 4 sets of ﬁlters, each with 64 ﬁlters for a total of 256 covariance ﬁlters (see bottom plot of ﬁg. 3). Similarly, we have 4 sets of mean ﬁlters, each with 32 ﬁlters. These ﬁlters have usually non-zero mean and exhibit on-center off-surround and off-center on-surround patterns, see ﬁg. 3-ii. In order to draw samples from the learned models, we run HMC for a long time (10,000 iterations, each composed of 20 leap-frog steps). Some samples of size 160x160 pixels are reported in ﬁg. 3 A)D). Without modelling the mean intensity, samples lack structure and do not seem much different from those that would be generated by a simple Gaussian model merely ﬁtting the second order statistics (see ﬁg. 3 in [1] and also ﬁg. 2 in [7]). By contrast, structure, sharp boundaries and some simple texture emerge only from models that have mean latent variables, namely mcRBM, mPoT and mHPoT which differs from mPoT by having a second layer pooling matrix on the squared covariance ﬁlter outputs [11]. A more quantitative comparison is reported in table 1. We ﬁrst compute marginal statistics of ﬁlter responses using the generated images, natural images from the test set, and random images. The statistics are the normalized histogram of individual ﬁlter responses to 24 Gabor ﬁlters (8 orientations and 3 scales). We then calculate the KL divergence between the histograms on random images and generated images and the KL divergence between the histograms on natural images and generated images. The table also reports the average difference of energies between random images and natural images. All results demonstrate that models that account for mean intensity generate images 4 The code used in the experiments can be found at the ﬁrst author’s web-page. 6 MODEL F (R) − F (T ) (104 ) KL(R G) KL(T G) KL(R G) − KL(T PoT - Conv 2.9 0.3 0.6 PoT - TConv 2.8 0.4 1.0 -0.6 mPoT - TConv 5.2 1.0 0.2 0.8 mHPoT - TConv 4.9 1.7 0.8 0.9 mcRBM - TConv 3.5 1.5 1.0 G) -0.3 0.5 Table 1: Comparing MRF’s by measuring: difference of energy (negative log ratio of probabilities) between random images (R) and test natural images (T), the KL divergence between statistics of random images (R) and generated images (G), KL divergence between statistics of test natural images (T) and generated images (G), and difference of these two KL divergences. Statistics are computed using 24 Gabor ﬁlters. that are closer to natural images than to random images, whereas models that do not account for the mean (like the widely used PoT-Conv) produce samples that are actually closer to random images. 4.1 Discriminative Experiments on Weight-Sharing Schemes In future work, we intend to use the features discovered by the generative model for recognition. To understand how the different weight sharing schemes affect recognition performance we have done preliminary tests using the discriminative performance of a simpler model on simpler data. We consider one of the simplest and most versatile models, namely the RBM [29]. Since we also aim to test the Global weight-sharing scheme we are constrained to using fairly low resolution datasets such as the MNIST dataset of handwritten digits [30] and the CIFAR 10 dataset of generic object categories [22]. The MNIST dataset has soft binary images of size 28x28 pixels, while the CIFAR 10 dataset has color images of size 32x32 pixels. CIFAR 10 has 10 classes, 5000 training samples per class and 1000 test samples per class. MNIST also has 10 classes with, on average, 6000 training samples per class and 1000 test samples per class. The energy function of the RBM trained on the CIFAR 10 dataset, modelling input pixels with 3 (R,G,B) Gaussian variables [31], is exactly the one shown in eq. 4; while the RBM trained on MNIST uses logistic units for the pixels and the energy function is again the same as before but without any quadratic term. All models are trained in an unsupervised way to approximately maximize the likelihood in the training set using Contrastive Divergence [32]. They are then used to represent each input image with a feature vector (mean of the posterior over the latent variables) which is fed to a multinomial logistic classiﬁer for discrimination. Models are compared in terms of: 1) recognition accuracy, 2) convergence time and 3) dimensionality of the representation. In general, assuming ﬁlters much smaller than the input image and assuming equal number of latent variables, Conv, TConv and Local models process each sample faster than Global by a factor approximately equal to the ratio between the area of the image and the area of the ﬁlters, which can be very large in practice. In the ﬁrst set of experiments reported on the left of ﬁg. 4 we study the internal representation in terms of discrimination and dimensionality using the MNIST dataset. For each choice of dimensionality all models are trained using the same number of operations. This is set to the amount necessary to complete one epoch over the training set using the Global model. This experiment shows that: 1) Local outperforms all other weight-sharing schemes for a wide range of dimensionalities, 2) TConv does not perform as well as Local probably because the translation invariant assumption is clearly violated for these relatively small, centered, images, 3) Conv performs well only when the internal representation is very high dimensional (10 times overcomplete) otherwise it severely underﬁts, 4) Global performs well when the representation is compact but its performance degrades rapidly as this increases because it needs more than the allotted training time. The right hand side of ﬁg. 4 shows how the recognition performance evolves as we increase the number of operations (or training time) using models that produce a twice overcomplete internal representation. With only very few ﬁlters Conv still underﬁts and it does not improve its performance by training for longer, but Global does improve and eventually it reaches the performance of Local. If we look at the crossing of the error rate at 2% we can see that Local is about 4 times faster than Global. To summarize, Local provides more compact representations than Conv, is much faster than Global while achieving 7 6 2.4 error rate % 5 error rate % 2.6 Global Local TConv Conv 4 3 2 1 0 2.2 Global Local 2 Conv 1.8 1000 2000 3000 4000 5000 dimensionality 6000 7000 1.6 0 8000 2 4 6 8 # flops (relative to # flops per epoch of Global model) 10 Figure 4: Experiments on MNIST using RBM’s with different weight-sharing schemes. Left: Error rate as a function of the dimensionality of the latent representation. Right: Error rate as a function of the number of operations (normalized to those needed to perform one epoch in the Global model); all models have a twice overcomplete latent representation. similar performance in discrimination. Also, Local can easily scale to larger images while Global cannot. Similar experiments are performed using the CIFAR 10 dataset [22] of natural images. Using the same protocol introduced in earlier work by Krizhevsky [22], the RBM’s are trained in an unsupervised way on a subset of the 80 million tiny images dataset [33] and then “ﬁne-tuned” on the CIFAR 10 dataset by supervised back-propagation of the error through the linear classiﬁer and feature extractor. All models produce an approximately 10,000 dimensional internal representation to make a fair comparison. Models using local ﬁlters learn 16x16 ﬁlters that are stepped every pixel. Again, we do not experiment with the TConv weight-sharing scheme because the image is not large enough to allow enough replicas. Similarly to ﬁg. 3-iii the Conv weight-sharing scheme was very difﬁcult to train and did not produce Gabor-like features. Indeed, careful injection of sparsity and long training time seem necessary [31] for these RBM’s. By contrast, both Local and Global produce Gabor-like ﬁlters similar to those shown in ﬁg. 2 F). The model trained with Conv weight-sharing scheme yields an accuracy equal to 56.6%, while Local and Global yield much better performance, 63.6% and 64.8% [22], respectively. Although Local and Global have similar performance, training with the Local weight-sharing scheme took under an hour while using the Global weight-sharing scheme required more than a day. 5 Conclusions and Future Work This work is motivated by the poor generative quality of currently popular MRF models of natural images. These models generate images that are actually more similar to white noise than to natural images. Our contribution is to recognize that current models can beneﬁt from 1) the addition of a simple model of the mean intensities and from 2) the use of a less constrained weight-sharing scheme. By augmenting these models with an extra set of latent variables that model mean intensity we can generate samples that look much more realistic: they are characterized by smooth regions, sharp boundaries and some simple high frequency texture. We validate our approach by comparing the statistics of ﬁlter outputs on natural images and generated images. In the future, we plan to integrate these MRF’s into deeper hierarchical models and to use their internal representation to perform object recognition in high-resolution images. The hope is to further improve generation by capturing longer range dependencies and to exploit this to better cope with missing values and ambiguous sensory inputs. References [1] E.P. Simoncelli. Statistical modeling of photographic images. Handbook of Image and Video Processing, pages 431–441, 2005. 8 [2] A. Hyvarinen, J. Karhunen, and E. Oja. Independent Component Analysis. John Wiley & Sons, 2001. [3] G.E. Hinton and R. R Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006. [4] M. Ranzato and G.E. Hinton. Modeling pixel means and covariances using factorized third-order boltzmann machines. In CVPR, 2010. [5] M.J. Wainwright and E.P. Simoncelli. Scale mixtures of gaussians and the statistics of natural images. In NIPS, 2000. [6] S. Roth and M.J. Black. Fields of experts: A framework for learning image priors. In CVPR, 2005. [7] U. Schmidt, Q. Gao, and S. Roth. A generative perspective on mrfs in low-level vision. In CVPR, 2010. [8] S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. PAMI, 6:721–741, 1984. [9] M. Welling, G.E. Hinton, and S. Osindero. Learning sparse topographic representations with products of student-t distributions. In NIPS, 2003. [10] S.C. Zhu and D. Mumford. Prior learning and gibbs reaction diffusion. PAMI, pages 1236–1250, 1997. [11] S. Osindero, M. Welling, and G. E. Hinton. Topographic product models applied to natural scene statistics. Neural Comp., 18:344–381, 2006. [12] S. Osindero and G. E. Hinton. Modeling image patches with a directed hierarchy of markov random ﬁelds. In NIPS, 2008. [13] Y. Karklin and M.S. Lewicki. Emergence of complex cell properties by learning to generalize in natural scenes. Nature, 457:83–86, 2009. [14] B. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set: a strategy employed by v1? Vision Research, 37:3311–3325, 1997. [15] Y. W. Teh, M. Welling, S. Osindero, and G. E. Hinton. Energy-based models for sparse overcomplete representations. JMLR, 4:1235–1260, 2003. [16] Y. Weiss and W.T. Freeman. What makes a good model of natural images? In CVPR, 2007. [17] S. Roth and M. J. Black. Fields of experts. Int. Journal of Computer Vision, 82:205–229, 2009. [18] K. Gregor and Y. LeCun. Emergence of complex-like cells in a temporal product network with local receptive ﬁelds. arXiv:1006.0448, 2010. [19] C. 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