nips nips2010 nips2010-99 knowledge-graph by maker-knowledge-mining

99 nips-2010-Gated Softmax Classification

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Author: Roland Memisevic, Christopher Zach, Marc Pollefeys, Geoffrey E. Hinton

Abstract: We describe a ”log-bilinear” model that computes class probabilities by combining an input vector multiplicatively with a vector of binary latent variables. Even though the latent variables can take on exponentially many possible combinations of values, we can efﬁciently compute the exact probability of each class by marginalizing over the latent variables. This makes it possible to get the exact gradient of the log likelihood. The bilinear score-functions are deﬁned using a three-dimensional weight tensor, and we show that factorizing this tensor allows the model to encode invariances inherent in a task by learning a dictionary of invariant basis functions. Experiments on a set of benchmark problems show that this fully probabilistic model can achieve classiﬁcation performance that is competitive with (kernel) SVMs, backpropagation, and deep belief nets. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We describe a ”log-bilinear” model that computes class probabilities by combining an input vector multiplicatively with a vector of binary latent variables. [sent-10, score-0.163]

2 Even though the latent variables can take on exponentially many possible combinations of values, we can efﬁciently compute the exact probability of each class by marginalizing over the latent variables. [sent-11, score-0.137]

3 The bilinear score-functions are deﬁned using a three-dimensional weight tensor, and we show that factorizing this tensor allows the model to encode invariances inherent in a task by learning a dictionary of invariant basis functions. [sent-13, score-0.631]

4 Experiments on a set of benchmark problems show that this fully probabilistic model can achieve classiﬁcation performance that is competitive with (kernel) SVMs, backpropagation, and deep belief nets. [sent-14, score-0.327]

5 1 Introduction Consider the problem of recognizing an image that contains a single hand-written digit that has been approximately normalized but may have been written in one of a number of different styles. [sent-15, score-0.105]

6 Features extracted from the image often provide much better evidence for a combination of a class and a style than they do for the class alone. [sent-16, score-0.384]

7 For example, a diagonal stroke might be highly compatible with an italic 1 or a non-italic 7. [sent-17, score-0.353]

8 A short piece of horizontal stroke at the top right may be compatible with a very italic 3 or a 5 with a disconnected top. [sent-18, score-0.492]

9 A fat piece of vertical stroke at the bottom of the image near the center may be compatible with a 1 written with a very thick pen or a narrow 8 written with a moderately thick pen so that the bottom loop has merged. [sent-19, score-0.737]

10 This would force the way in which local features were interpreted to be globally consistent about style features such as stroke thickness or ”italicness”. [sent-21, score-0.53]

11 But what if the values of the style features are missing from the training data? [sent-22, score-0.247]

12 We describe a way of learning a large set of binary style features from training data that are only labeled with the class. [sent-23, score-0.247]

13 Our ”gated softmax” model allows the 2K possible combinations of the K learned style features to be integrated out. [sent-24, score-0.333]

14 This makes it easy to compute the posterior probability of a class label on test data and easy to get the exact gradient of the log probability of the correct label on training data. [sent-25, score-0.104]

15 [1] describes a bilinear sparse coding model that, similar to our model, can be trained discriminatively to predict classes. [sent-28, score-0.214]

16 Furthermore, the model parameters, unlike in our case, are not factorized, and as a result the model cannot extract features which are shared among classes. [sent-30, score-0.198]

17 Our model is similar to the top layer of the deep network discussed in [2], again, without factorization and feature sharing. [sent-32, score-0.228]

18 Our model can be viewed also as a “degenerate” special case of the image transformation model described in [3], which replaces the output-image in that model with a “one-hot” encoded class label. [sent-34, score-0.275]

19 We describe the basic model, how it relates to logistic regression, and how to perform learning and inference in the following section. [sent-36, score-0.145]

20 R One of the most common, and certainly oldest, approaches to solving this task is logistic regression, which is based on a log-linear relationship between inputs and labels (see, for example, [4]). [sent-40, score-0.325]

21 In particular, using a set of linear, class-speciﬁc score functions t sy (x) = wy x (1) we can obtain probabilities over classes by exponentiating and normalizing: p(y|x) = t exp(wy x) t y exp(wy x) (2) Classiﬁcation decisons for test-cases xtest are given by arg max p(y|xtest ). [sent-41, score-0.576]

22 Training amounts to 1 adapting the vectors wy by maximizing the average conditional log-probability N α log p(y α |xα ) α α N for a set {(x , y )}α=1 of training cases. [sent-42, score-0.375]

23 In the case of two or more labels, logistic regression is also referred to as the “multinomial logit model” or the “maximum entropy model” [5]. [sent-44, score-0.203]

24 Important properties of logistic regression are that (a) the training objective is convex, so there are no local optima, and (b) the model is probabilistic, hence it comes with well-calibrated estimates of uncertainty in the classiﬁcation decision (ref. [sent-48, score-0.376]

25 2 A log-bilinear model Logistic regression makes the assumption that classes can be separated in the input space with hyperplanes (up to noise). [sent-53, score-0.157]

26 We take the stance that we do not know what form the separation manifold takes on, and instead introduce a set of probabilistic hidden variables which cooperate to model the decision surface jointly. [sent-57, score-0.18]

27 To obtain classiﬁcation decisions at test-time and for training the model, we then need to marginalize over these hidden variables. [sent-58, score-0.133]

28 2 h h hk hk f xi xi f f y x y x (a) (b) Figure 1: (a) A log-bilinear model: Binary hidden variables hk can blend in log-linear dependencies that connect input features xi with labels y. [sent-59, score-0.67]

29 More speciﬁcally, we consider the following variation of logistic regression: We introduce a vector h of binary latent variables (h1 , . [sent-61, score-0.145]

30 1) with a bilinear score of x and h: sy (x, h) = ht Wy x. [sent-65, score-0.359]

31 (3) The bilinear score combines, quadratically, all pairs of input components xi with hidden variables hk . [sent-66, score-0.481]

32 The score for each class is thus a quadratic product, parameterized by a class-speciﬁc matrix Wy . [sent-67, score-0.154]

33 This is in contrast to the inner product, parameterized by class-speciﬁc vectors wy , for logistic regression. [sent-68, score-0.478]

34 t y h exp(h Wy x) p(y, h|x) = (4) In contrast to logistic regression, we obtain a distribution over both the hidden variables h and labels y. [sent-70, score-0.291]

35 (5) h∈{0,1}K As with logistic regression, we thus get a distribution over labels y, conditioned on inputs x. [sent-72, score-0.287]

36 3, we obtain the logistic regression score (up to normalization), since the argument in the “exp()” collapses to the class-speciﬁc row-vector ht Wy . [sent-76, score-0.405]

37 5 is therefore exactly one logistic classiﬁer, showing that the model is equivalent to a mixture of 2K logistic regressors with shared weights. [sent-78, score-0.386]

38 In the following, we let W denote the three-way tensor of parameters (by “stacking” the matrices Wy ). [sent-80, score-0.102]

39 As we shall show, it is the conditioning on the inputs x that renders this model useful. [sent-86, score-0.137]

40 Note that in our case, the distribution that the model outputs is a distribution over a countable (and, in particular, fairly small2 ) number of possible values, so we can compute the constant Ω = y k (1 + exp( i Wyik xi )), that normalizes the left-hand side in Eqs. [sent-89, score-0.098]

41 3 Sharing features among classes The score (or “activation”) that class label y receives from each of the 2K terms in Eq. [sent-93, score-0.255]

42 In most natural problems, inputs from different classes share the same domain, and therefore show similar characteristics. [sent-99, score-0.136]

43 Note that parameter sharing would not be possible in models like logistic regression or SVMs, which are based on linear score functions. [sent-102, score-0.331]

44 This factorization of a three-way parameter tensor was previously used by [3] to reduce the number of parameters in an unsupervised model of images. [sent-104, score-0.152]

45 Plugging the factorized form for the weight tensor into the deﬁnition of the probability (Eq. [sent-105, score-0.102]

46 4) and re-arranging terms yields y h hk Wkf Wyf x xi Wif p(y, h|x) ∝ exp f i (8) k This shows that, after factorizing, we obtain a classiﬁcation decision by ﬁrst projecting the input vector x (and the vector of hidden variables h) onto F basis functions, or ﬁlters. [sent-106, score-0.371]

47 4 where ay = log 1 + exp k y x h xi Wif )Wkf Wyf ( f . [sent-114, score-0.118]

48 (10) i x Note that in this model, learning of features (the F basis functions W·f ) is tied in with learning of the classiﬁer itself. [sent-115, score-0.102]

49 In contrast to neural networks and deep learners ([11], [12]), the model does not try to learn a feature hierarchy. [sent-116, score-0.228]

50 Instead, learned features are combined multiplicatively with hidden variables and the results added up to provide the inputs to the class-units. [sent-117, score-0.32]

51 In terms of neural networks nomenclature, the factored model can best be thought of as a single-hidden-layer network. [sent-118, score-0.181]

52 4 Interpretation An illustration of the graphical model is shown in Figure 1 (non-factored model on the left, factored model on the right). [sent-121, score-0.281]

53 Each hidden variable hk that is “on” contributes a slice W·k· of the parameter tensor to a blend k hk W·k· of at most K matrices. [sent-122, score-0.531]

54 A single blend is simply a linear logistic classiﬁer. [sent-124, score-0.215]

55 An alternative view is that each output unit y accumulates evidence for or against its class by projecting the input onto K basis functions (the rows of Wy in Eq. [sent-125, score-0.2]

56 Each instantiation of h constitutes one way of combining a subset of basis function responses that are considered to be consistent into a single piece of evidence. [sent-127, score-0.105]

57 The model can accomodate each if necessary by making appropriate use of the hidden variables. [sent-131, score-0.141]

58 The factored model, where basis function responses are computed jointly for all classes and then weighted differently for each class, can be thought of as accumulating evidence accordingly in the “spatial frequency domain”. [sent-132, score-0.274]

59 2) takes the form: ¯ ∂ log p(y|x) = δyy − p(¯|x) σ y ¯ ∂Wyik ¯ xi Wyik hk xi with σ(a) = 1 + exp(−a) −1 . [sent-143, score-0.134]

60 (11) i To compute gradients of the factored model (Section 2. [sent-144, score-0.226]

61 As with logistic regression, we can thus perform gradient based optimization of the model likelihood for training. [sent-149, score-0.195]

62 However, in contrast to logistic regression, the model’s objective function is non-linear, so it can contain local optima. [sent-151, score-0.226]

63 Like logistic regression, and in contrast to SVMs, the model computes probabilities and thus provides well-calibrated estimates of uncertainty in its decisions. [sent-153, score-0.195]

64 6 Optimization The log-probability is non-linear and can contain local optima w. [sent-155, score-0.16]

65 W , so some care has to be taken to obtain good local optima during training. [sent-158, score-0.16]

66 In general we found that simply deploying a generalpurpose conjugate gradient solver on random parameter initializations does not reliably yield good local optima (even though it can provide good solutions in some cases). [sent-159, score-0.218]

67 More speciﬁcally, the distribution p(x|y) of inputs given labels is a factored Restricted Boltzmann machine, that can be optimized using contrastive divergence [3]. [sent-163, score-0.273]

68 We found that performing a few iterations of class-conditional optimization as an initialization reliably yields good local optima of the model‘s objective function. [sent-164, score-0.202]

69 This pre-training is reminiscent of training deep networks, which also rely on a pre-training phase. [sent-166, score-0.22]

70 It is possible to perform a different kind of annealing by adding the class-speciﬁc and the model’s actual objective function, and slowly reducing the class-speciﬁc inﬂuence using some weighting scheme. [sent-168, score-0.099]

71 We used both the simple and the annealed optimization in some of our experiments, but did not ﬁnd clear evidence that annealing leads to better local optima. [sent-169, score-0.14]

72 We found that, given an initialization near a local optimum of the objective function, conjugate gradients can signiﬁcantly outperform stochastic gradient descent in terms of the speed at which one can optimize both the model’s own objective function and the cost on validation data. [sent-170, score-0.168]

73 In practice, one can add a regularization (or “weight-decay”) penalty −λ W 2 to the objective function, as is common for logistic regression and other classiﬁers, where λ is chosen by crossvalidation. [sent-171, score-0.245]

74 The benchmark consists of a set of classiﬁcation problems, that are difﬁcult, because they contain many subtle, and highly complicated, dependencies of classes on inputs. [sent-173, score-0.109]

75 It was initially introduced to evaluate the performance of deep neural networks. [sent-174, score-0.178]

76 We do not make use of any random re-starts or other additional ways to ﬁnd good local optima of the objective function. [sent-182, score-0.202]

77 For the class-speciﬁc initializations, we use a class-speciﬁc RBM with binary observables on the datasets “rectangles”, “mnist-rot”, “convex” and “mnist”, because they contain essentially binary inputs (or a heavily-skewed histogram), and Gaussian observables on the others. [sent-183, score-0.181]

78 We also tried “hybrid” approaches on some data-sets where we optimize a sum of the RBM and the model objective function, and decrease the inﬂuence of the RBM as training progresses. [sent-185, score-0.134]

79 1 Learning task-dependent invariances The “rectangles” task requires the classiﬁcation of rectangle images into the classes horizontal vs. [sent-187, score-0.3]

80 Figure 2 (left) shows random sets of 50 rows of the matrix Wy learned by the unfactored model (class horizontal on the top, class vertical on 4 5 An implementation of the model is available at http://learning. [sent-189, score-0.459]

81 com/exdb/mnist/ 6 Figure 2: Left: Class-speciﬁc ﬁlters learned from the rectangle task – top: ﬁlters in support of the label horizontal, bottom: ﬁlters in support of the class label vertical. [sent-194, score-0.139]

82 The plot shows that the hidden units, like “Hough-cells”, make it possible to accumulate evidence for the different classes, by essentially counting horizontal and vertical strokes in the images. [sent-199, score-0.357]

83 An example of ﬁlters learned by the factored model is shown in Figure 2 (right). [sent-203, score-0.22]

84 The task is classiﬁcation of rotated digits in this example. [sent-204, score-0.137]

85 In this task, learning invariances with respect to rotation is crucial for achieving good classiﬁcation performance. [sent-206, score-0.129]

86 The ﬁlters resemble basis functions learned by an image transformation model trained to rotate image patches described in [3]. [sent-209, score-0.305]

87 (a) (b) (c) (d) Figure 3: Example images from four of the “deep learning” benchmark tasks: (a) Rectangles (2class): Distinguish horizontal from vertical rectangles; (b) Rotated digits (10-class): Determine the class of the digit; (c) Convex vs. [sent-212, score-0.33]

88 non-convex (2-class): Determine if the image shows a convex or non-convex shape; (d) Rectangles with images (2-class): Like (a), but rectangles are rendered using natural images. [sent-213, score-0.21]

89 For comparison, we also show the error rates achieved with the unfactored model (Section 2. [sent-224, score-0.144]

90 2), which also performs fairly well as compared to deep networks and SVMs, but is signiﬁcantly weaker in most cases than the factored model. [sent-225, score-0.357]

91 4 Discussion/Future work Several extensions of deep learning methods, including deep kernel methods, have been suggested recently (see, for example, [13], [14]), giving similar performance to the networks that we compare to here. [sent-299, score-0.356]

92 Instead, our model gets its power from the fact that inputs, hidden variables and labels interact in three-way cliques. [sent-301, score-0.196]

93 Factored three-way interactions make it possible to learn task-speciﬁc features and to learn transformational invariances inherent in the task at hand. [sent-302, score-0.27]

94 Interestingly, the ﬁlters that the model learns (see previous Section; Figure 2) resemble those learned be recent models of image transformations (see, for example, [3]). [sent-305, score-0.247]

95 In fact, learning of invariances in general is typically addressed in the context of learning transformations. [sent-306, score-0.129]

96 In contrast to a model of transformations, it is the classiﬁcation task that deﬁnes the invariances here, and the model learns the invariant representations from that task only. [sent-308, score-0.352]

97 Combining the explicit examples of transformations provided by video sequences with the implicit information about transformational invariances provided by labels is a promising future direction. [sent-309, score-0.29]

98 Another direction is the extension to structured prediction problems, for example, by deploying the model as clique potential in a CRF. [sent-315, score-0.108]

99 Learning to represent spatial transformations with factored higher-order Boltzmann machines. [sent-326, score-0.186]

100 An empirical evaluation of deep architectures on problems with many factors of variation. [sent-353, score-0.178]

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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. 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