nips nips2010 nips2010-266 knowledge-graph by maker-knowledge-mining

266 nips-2010-The Maximal Causes of Natural Scenes are Edge Filters

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Author: Jose Puertas, Joerg Bornschein, Joerg Luecke

Abstract: We study the application of a strongly non-linear generative model to image patches. As in standard approaches such as Sparse Coding or Independent Component Analysis, the model assumes a sparse prior with independent hidden variables. However, in the place where standard approaches use the sum to combine basis functions we use the maximum. To derive tractable approximations for parameter estimation we apply a novel approach based on variational Expectation Maximization. The derived learning algorithm can be applied to large-scale problems with hundreds of observed and hidden variables. Furthermore, we can infer all model parameters including observation noise and the degree of sparseness. In applications to image patches we ﬁnd that Gabor-like basis functions are obtained. Gabor-like functions are thus not a feature exclusive to approaches assuming linear superposition. Quantitatively, the inferred basis functions show a large diversity of shapes with many strongly elongated and many circular symmetric functions. The distribution of basis function shapes reﬂects properties of simple cell receptive ﬁelds that are not reproduced by standard linear approaches. In the study of natural image statistics, the implications of using different superposition assumptions have so far not been investigated systematically because models with strong non-linearities have been found analytically and computationally challenging. The presented algorithm represents the ﬁrst large-scale application of such an approach. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 However, in the place where standard approaches use the sum to combine basis functions we use the maximum. [sent-9, score-0.371]

2 In applications to image patches we ﬁnd that Gabor-like basis functions are obtained. [sent-13, score-0.66]

3 Quantitatively, the inferred basis functions show a large diversity of shapes with many strongly elongated and many circular symmetric functions. [sent-15, score-0.919]

4 The distribution of basis function shapes reﬂects properties of simple cell receptive ﬁelds that are not reproduced by standard linear approaches. [sent-16, score-0.489]

5 In the study of natural image statistics, the implications of using different superposition assumptions have so far not been investigated systematically because models with strong non-linearities have been found analytically and computationally challenging. [sent-17, score-0.375]

6 1 Introduction If Sparse Coding (SC, [1]) or Independent Component Analysis (ICA; [2, 3]) are applied to image patches, basis functions are inferred that closely resemble Gabor wavelet functions. [sent-19, score-0.605]

7 Because of the similarity of these functions to simple-cell receptive ﬁelds in primary visual cortex, SC and ICA became the standard models to explain simple-cell responses, and they are the primary choice in modelling the local statistics of natural images. [sent-20, score-0.422]

8 Furthermore, different forms of independent sparse priors have been investigated by many modelers [8, 9, 10], while other approaches have gone a step further and studied a relaxation of the assumption of independence between hidden variables [11, 12, 13]. [sent-28, score-0.213]

9 ∗ authors contributed equally 1 An assumption that has, in the context of image statistics, been investigated relatively little is the assumption of linear superpositions of basis functions. [sent-29, score-0.459]

10 For many types of data, linear superposition can be motivated by the actual combination rule of the data components (e. [sent-31, score-0.249]

11 For other types of data, including visual data, linear superposition can represent a severe approximation, however. [sent-34, score-0.234]

12 In this paper we study the application of a probabilistic generative model with strongly non-linear superposition to natural image patches. [sent-50, score-0.386]

13 The basic model has ﬁrst been suggested in [19] where tractable learning algorithms for parameter optimization where inferred for the case of a superposition based on a point-wise maximum. [sent-51, score-0.268]

14 It enables large-scale applications to image patches and, thus, allows for studying the inferred basis functions as it is commonly done for linear approaches. [sent-58, score-0.769]

15 H denotes the number of hidden variables sh , and W ∈ RD×H . [sent-64, score-0.39]

16 , maxh {W � } for vectors W � ∈ RD×1 : 1h p(� | �, Θ) y s Dh = h � N (� ; max{sh Wh }, σ 2 1) y h � � N (� ; h sh Wh , σ 2 1) y (non-linear superposition) (3) (linear superposition) (4) � � where N (� ; µ, Σ) denotes the multi-variate Gaussian distribution (note that h sh Wh = W � ). [sent-74, score-0.551]

17 B Two generated patches constructed from two Gabor basis functions with approximately orthogonal wave vectors. [sent-76, score-0.578]

18 In the upper-right the basis functions were combined using linear superposition. [sent-77, score-0.402]

19 D Cross-sections through basis functions (along maximum amplitude direction). [sent-82, score-0.371]

20 This is the case for diagonally overlapping basis functions (Fig. [sent-90, score-0.371]

21 1B) and, at closer inspection, also for overlapping collinear basis functions (Fig. [sent-91, score-0.422]

22 Strong interferences as with linear combinations can not be expected from combinations of image components. [sent-93, score-0.239]

23 4D), it could thus be argued that the maximum combination is closer to the actual combination rule of image causes. [sent-95, score-0.22]

24 For s the learning algorithm, K n in (6) is chosen to contain hidden states � with at most γ active causes � sh ≤ γ. [sent-114, score-0.417]

25 To determine the H � hidden variables for I we use those variables h with the H � largest y values of a selection function Sh (� (n) ) which is given by: eﬀ � eﬀ y y Sh (� (n) ) = π N (� (n) ; Wh , σ 2 1) , with an effective weight Wdh = max{yd , Wdh } . [sent-118, score-0.204]

27 N cut is hereby the �∈K n p(�, � s expected number of data points that have been generated by states with less or equal to γ non-zero entries: γ � ��H � � � cut π γ (1 − π)H−γ . [sent-135, score-0.205]

28 If such a correction is taken into account, we obtain the update rule: H � A(π) π 1 � sh , (13) s �|� |�qn with |� | = s π new = B(π) |M| n∈M h=1 γ γ � �H� ��H� γ� H−γ� π (1 − π) and B(π) = γ� π γ� (1 − π)H−γ� . [sent-145, score-0.294]

29 Crucial for the scalability to large-scale problems is hereby the preselection of H � < H hidden variables using the selection function in Eqn. [sent-159, score-0.252]

30 patches basis functions Figure 2: Illustration of patch preprocessing and basis function visualization. [sent-162, score-0.976]

31 The left-hand-side shows data points obtained from gray-value patches after DoG ﬁltering. [sent-163, score-0.244]

32 These patches are transformed to non-negative data by Eqn. [sent-164, score-0.207]

33 The algorithm maximizes the data likelihood under the MCA model (1) and (2), and infers basis functions (second from the right). [sent-166, score-0.371]

34 For visualization, the basis functions are displayed after their parts have been recombined again. [sent-167, score-0.371]

35 More formally, we use a DoG ﬁlter to generate patches � with D = 26 × 26 y ˜ pixels. [sent-173, score-0.207]

36 Such a patch is then converted to a patch of size D = 2 D by assigning: yd = [˜d ]+ and yD+d = [−˜d ]+ y y (15) (for d = 1, . [sent-174, score-0.278]

37 As inferred basis functions of images most commonly resemble Gabor wavelets, we used Gabor functions for the generation of artiﬁcial data. [sent-183, score-0.677]

38 The Gabor basis functions were combined according to the MCA generative model (1) and (2). [sent-184, score-0.487]

39 The parameters were chosen to lie in the same range as the parameters inferred in preliminary runs of the algorithm on natural image patches. [sent-190, score-0.222]

40 s We then selected the |�| corresponding Gabor functions and used channel-splitting (15) to convert s them into basis functions with only non-negative parts. [sent-192, score-0.465]

41 To form an artiﬁcial patch, these basis functions were combined using the point-wise maximum according to (2). [sent-193, score-0.402]

42 We generated N = 150 000 patches as data points in this way (Fig. [sent-194, score-0.244]

43 We generated the data with a larger number of basis functions to better match the continuous � distribution of the real generating components of images. [sent-197, score-0.43]

44 The basis functions Wh were initialized 5 A B Figure 3: A Artiﬁcial patches generated by combining artiﬁcial Gabors using a point-wise maximum. [sent-198, score-0.578]

45 B Inferred basis functions if the MCA learning algorithm is applied. [sent-199, score-0.371]

46 C Comparison between the shapes of generating (green) and inferred (blue) Gabors. [sent-200, score-0.278]

47 The brighter the blue data points the larger the error between the basis function and the matched Gabor (also for Fig. [sent-201, score-0.349]

48 C ny nx by setting them to the average over all the preprocessed input patches plus a small Gaussian white noise (≈ 0. [sent-203, score-0.486]

49 5% of the average basis function value) was added during the ﬁrst 20 iterations, was linearly decreased to zero between iterations 20 and 40, and kept at zero for the last 20 iterations. [sent-211, score-0.314]

50 3B displays some of the typical basis functions that were recovered in a run of the algorithm on artiﬁcial patches. [sent-216, score-0.429]

51 When we matched the obtained basis functions with Gabor functions (compare, e. [sent-218, score-0.5]

52 We thus plotted the values parameterizing the Gabor shapes in an nx /ny -plot. [sent-221, score-0.27]

53 Although some few recovered basis functions lie a relatively distant from the generating distribution, it is in general recovered well. [sent-225, score-0.546]

54 This is presumably due to the smaller number of basis function in the model H < H gen . [sent-228, score-0.341]

55 These outliers are usually basis functions that represent more than one Gabor or small Gabor parts. [sent-232, score-0.371]

56 Importantly, however, we found that the large majority of inferred Gabors consistently recovered the generating Gabor functions in nx /ny -plots. [sent-233, score-0.48]

57 In particular, when we changed the angle of the generating distribution in the nx /ny -plots (e. [sent-234, score-0.219]

58 The dataset used in the experiment on natural images was prepared ˜ by sampling N = 200 000 patches of D = 26 × 26 pixels from the van Hateren image database [28] (while constraining random selection to patches of images without man-made structures). [sent-239, score-0.615]

59 We preprocessed the patches as described above using a DoG ﬁlter1 with a ratio of 3 : 1 between positive and negative parts (see, e. [sent-240, score-0.285]

60 The inferred basis functions we found to resembled Gabor-like functions at different locations, and with different orientations and frequencies. [sent-249, score-0.574]

61 Additionally, we obtained many globular basis functions with no or very little orientation preferences. [sent-250, score-0.456]

62 A Random selection of 125 basis functions of the H=400 inferred. [sent-257, score-0.403]

63 B Selection of most globular functions and C most elongated functions. [sent-258, score-0.291]

64 D Selection of preprocessed patches extracted from natural images. [sent-259, score-0.316]

65 E Selection of data points generated according to the model using the inferred basis functions and sparseness level (but no noise). [sent-260, score-0.602]

66 4D,E were chosen to demonstrate the high similarity between preprocessed natural patches (in D) and generated ones (in E). [sent-263, score-0.316]

67 To highlight the diversity of obtained basis functions, Figs. [sent-264, score-0.357]

68 Instead of matching the basis functions directly, we ﬁrst computed estimates of their corresponding receptive ﬁelds (RFs). [sent-270, score-0.473]

69 These estimates were obtained by convoluting the basis functions with the same DoG ﬁlter as used for preprocessing (see, e. [sent-271, score-0.418]

70 C Distribution measured in vivo [33] (red triangles) and corresponding distribution of MCA basis functions (blue). [sent-282, score-0.419]

71 0◦ nx nx After matching the (convoluted) ﬁelds with Gabor functions, we found a relatively homogeneous distribution of the ﬁelds’ orientations as it is commonly observed (Fig. [sent-283, score-0.32]

72 The broad distribution in nx /ny -space hereby reﬂects the high diversity of basis functions obtained by our algorithm (see Fig. [sent-288, score-0.691]

73 However, the MCA basis functions do quantitatively not match the measurements exactly (see Fig. [sent-291, score-0.402]

74 5C): the MCA distribution contains a higher percentage of strongly elongated basis functions, and many MCA functions are shifted slightly to the right relative to the measurements. [sent-292, score-0.512]

75 If the basis functions are matched with Gabors directly, we actually do not observe the latter effect (see suppl. [sent-293, score-0.406]

76 If simple-cell responses are associated with the posterior probabilities of multiple-cause models, the basis functions should, however, not be compared to measured RFs directly (although it is frequently done in the literature). [sent-297, score-0.371]

77 In both cases we observed qualitatively and quantitatively similar distributions of basis functions. [sent-299, score-0.308]

78 Runs with H = 200 thus also contained many circular symmetric basis functions (see suppl. [sent-300, score-0.479]

79 Based on standard generative models with linear superposition it has recently been argued [32] that such functions are only obtained in a regime with large numbers of hidden variables relative to the input dimensionality (see [34] for an early contribution). [sent-305, score-0.478]

80 The model combines basis functions using a point-wise maximum as an alternative to the linear combination as assumed by Sparse Coding, ICA, and most other approaches. [sent-307, score-0.419]

81 Our results suggest that changing the component combination rule has a strong impact on the distribution of inferred basis functions. [sent-308, score-0.515]

82 While we still obtain Gabor-like functions, we robustly observe a large variety of basis functions. [sent-309, score-0.277]

83 Most notably, we obtain circular symmetric functions as well as many elongated functions that are closely associated with edges traversing the entire patch (compare Figs. [sent-310, score-0.482]

84 The differences in basis function shapes between non-linear and linear approaches are, in this respect, consistent with the different types of interferences between basis functions. [sent-315, score-0.749]

85 The maximum results in basis function combinations with much less pronounced interferences, while the stronger interferences of linear combinations might result in a repulsive effect fostering less elongated ﬁelds (compare Fig. [sent-316, score-0.546]

86 For linear approaches, a large diversity of Gabor shapes (including circular symmetric ﬁelds) could only be obtained in very over-complete settings [34], or speciﬁcally modelled priors with hand-set sparseness levels [10]. [sent-318, score-0.422]

87 Such studies were motivated by a recently observed discrepancy of receptive ﬁelds as predicted by SC or ICA, and receptive ﬁelds as measured in vivo [33]. [sent-319, score-0.252]

88 Compared to these measurements, the MCA basis functions and their approximate receptive ﬁelds show a similar diversity of shapes. [sent-320, score-0.553]

89 MCA functions and measured RFs both show circular symmetric ﬁelds and in both cases there is a tendency towards ﬁelds elongated orthogonal to the wave-vector direction (compare Fig. [sent-321, score-0.314]

90 Possible factors that can inﬂuence the distributions of basis functions, for MCA as well as for other methods, are hereby different types of preprocessing, different prior distributions, and different noise models. [sent-323, score-0.398]

91 Even if the prior type is ﬁxed, differences for the basis functions have been reported for different settings of prior parameters (e. [sent-324, score-0.371]

92 If possible, these parameters should thus be learned along with the basis functions. [sent-327, score-0.277]

93 However, while linear component combinations have extensively been studied in the context of image statistics, the investigation of other combination rules has been limited to relatively small scale applications [17, 16, 35, 19]. [sent-337, score-0.205]

94 As with linear approaches, we found that Gabor-like basis functions are obtained. [sent-339, score-0.371]

95 To recover the generating causes of image patches, a linear combination might, furthermore, not be the best choice. [sent-342, score-0.248]

96 A network that uses few active neurones to code visual input predicts the diverse shapes of cortical receptive ﬁelds. [sent-406, score-0.321]

97 An evaluation of the two-dimensional gabor ﬁlter model of simple receptive ﬁelds in cat striate cortex. [sent-497, score-0.415]

98 Independent component ﬁlters of natural images compared with simple cells in primary visual cortex. [sent-514, score-0.206]

99 Spatial structure and symmetry of simple-cell receptive ﬁelds in macaque primary visual cortex. [sent-549, score-0.21]

100 Sparse coding with an overcomplete basis set: A strategy employed by V1? [sent-560, score-0.313]

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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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