nips nips2010 nips2010-156 knowledge-graph by maker-knowledge-mining

156 nips-2010-Learning to combine foveal glimpses with a third-order Boltzmann machine

Source: pdf

Author: Hugo Larochelle, Geoffrey E. Hinton

Abstract: We describe a model based on a Boltzmann machine with third-order connections that can learn how to accumulate information about a shape over several ﬁxations. The model uses a retina that only has enough high resolution pixels to cover a small area of the image, so it must decide on a sequence of ﬁxations and it must combine the “glimpse” at each ﬁxation with the location of the ﬁxation before integrating the information with information from other glimpses of the same object. We evaluate this model on a synthetic dataset and two image classiﬁcation datasets, showing that it can perform at least as well as a model trained on whole images. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Learning to combine foveal glimpses with a third-order Boltzmann machine Hugo Larochelle and Geoffrey Hinton Department of Computer Science, University of Toronto 6 King’s College Rd, Toronto, ON, Canada, M5S 3G4 {larocheh,hinton}@cs. [sent-1, score-0.253]

2 We evaluate this model on a synthetic dataset and two image classiﬁcation datasets, showing that it can perform at least as well as a model trained on whole images. [sent-5, score-0.118]

3 Human vision, by contrast, uses a retina in which the resolution falls off rapidly with eccentricity and it relies on intelligent, top-down strategies for sequentially ﬁxating parts of the optic array that are relevant for the task at hand. [sent-7, score-0.16]

4 If a system with billions of neurons at its disposal has adopted this strategy, the use of a variable resolution retina and a sequence of intelligently selected ﬁxation points is likely to be even more advantageous for simulated visual systems that have to make do with a few thousand “neurons”. [sent-13, score-0.136]

5 In this paper we explore the computational issues that arise when the ﬁxation point strategy is incorporated in a Boltzmann machine and demonstrate a small system that can make good use of a variable resolution retina containing very few pixels. [sent-14, score-0.102]

6 There are two main computational issues: • What-where combination: How can eye positions be combined with the features extracted from the retinal input (glimpses) to allow evidence for a shape to be accumulated across a sequence of ﬁxations? [sent-15, score-0.286]

7 • Where to look next: Given the results of the current and previous ﬁxations, where should the system look next to optimize its object recognition performance? [sent-16, score-0.106]

8 The center dot marks the pixel at position (i, j) (pixels are drawn as dotted squares). [sent-18, score-0.131]

9 B: examples of glimpses computed by the retinal transformation, at different positions (visualized through reconstructions). [sent-19, score-0.439]

10 To tackle these issues, we rely on a special type of restricted Boltzmann machine (RBM) with thirdorder connections between visible units (the glimpses), hidden units (the accumulated features) and position-dependent units which gate the connections between the visible and hidden units. [sent-21, score-0.413]

11 We describe approaches for training this model to jointly learn and accumulate useful features from the image and control where these features should be extracted, and evaluate it on a synthetic dataset and two image classiﬁcation datasets. [sent-22, score-0.254]

12 2 Vision as a sequential process with retinal ﬁxations Throughout this work, we will assume the following problem framework. [sent-23, score-0.111]

13 We are given a training set of image and label pairs {(It , lt )}N and the task is to predict the value of lt (e. [sent-24, score-0.138]

14 To achieve this, we require that information about an image I (removing the superscript t for simplicity) must be acquired sequentially by ﬁxating (or querying) the image at a series of K positions [(i1 , j1 ), . [sent-32, score-0.175]

15 Given a position (ik , jk ), which identiﬁes a pixel I(ik , jk ) in the image, information in the neighborhood of that pixel is extracted through what we refer to as a retinal transformation r(I, (ik , jk )). [sent-36, score-1.111]

16 copies the value of the pixels) from the image only in the neighborhood of pixel I(ik , jk ). [sent-39, score-0.354]

17 At the periphery of the retina, lower-resolution information is extracted by averaging the values of pixels falling in small hexagonal regions of the image. [sent-40, score-0.14]

18 The hexagons are arranged into a spiral, with the size of the hexagons increasing with the distance from the center (ik , jk ) of the ﬁxation1 . [sent-41, score-0.341]

19 All of the high-resolution and low-resolution information is then concatenated into a single vector given as output by r(I, (ik , jk )). [sent-42, score-0.271]

20 An illustration of this retinal transformation is given in Figure 1. [sent-43, score-0.175]

21 As a shorthand, we will use xk to refer to the glimpse given by the output of the retinal transformation r(I, (ik , jk )). [sent-44, score-0.558]

22 3 A multi-ﬁxation model We now describe a system that can predict l from a few glimpses x1 , . [sent-45, score-0.253]

23 We know that this problem is solvable: [1] demonstrated that people can “see” a shape by combining information from multiple glimpses through a hole that is much smaller than the whole shape. [sent-49, score-0.273]

24 1 Restricted Boltzmann Machine for classiﬁcation RBMs are undirected generative models which model the distribution of a visible vector v of units using a hidden vector of binary units h. [sent-59, score-0.225]

25 C l∗ =1 Another useful property of this model is that all hidden units can be marginalized over analytically in order to exactly compute exp(dl + j softplus(cj + Ujl + Wj· x)) p(y = el |x) = (5) C ∗ ∗ l∗ =1 exp(dl + j softplus(cj + Ujl + Wj· x)) where softplus(a) = log(1 + exp(a)). [sent-67, score-0.129]

26 Instead, we could redeﬁne the energy function of Equation 1 as follows: K E(y, x1:K , h) = −h W(ik ,jk ) xk − b xk − c h − d y − h Uy (6) k=1 (ik ,jk ) where the connection matrix W now depends on the position of the ﬁxation2 . [sent-75, score-0.194]

27 Such connections are called high-order (here third order) because they can be seen as connecting the hidden 2 To be strictly correct in our notation, we should add the position coordinates (i1 , j1 ), . [sent-76, score-0.131]

28 3 units, input units and implicit position units (one for each possible value of positions (ik , jk )). [sent-84, score-0.552]

29 Conditioned on the position units (which are assumed to be given), this model is still an RBM satisfying the traditional conditional independence properties between the hidden and visible units. [sent-85, score-0.217]

30 For a given m × m grid of possible ﬁxation positions, all W(ik ,jk ) matrices contain m2 HR parameters where H is the number of hidden units and R is the size of the retinal transformation. [sent-86, score-0.216]

31 To reduce that number, we parametrize or factorize the W(ik ,jk ) matrices as follows W(ik ,jk ) = P diag(z(ik , jk )) F (7) where F is R × D, P is D × H, z(ik , jk ) is a (learned) vector associated to position (ik , jk ) and diag(a) is a matrix whose diagonal is the vector a. [sent-87, score-0.889]

32 Hence, W(ik ,jk ) is now an outer product of the D lower-dimensional bases in F (“ﬁlters”) and P (“pooling”), gated by a position speciﬁc vector z(ik , jk ). [sent-88, score-0.347]

33 Instead of learning a separate matrix W(ik ,jk ) for each possible position, we now only need to learn a separate vector z(ik , jk ) for each position. [sent-89, score-0.271]

34 Intuitively, the vector z(ik , jk ) controls which rows of F and columns of P are used to accumulate the glimpse at position (ik , jk ) into the hidden layer of the RBM. [sent-90, score-0.829]

35 We emphasize that z(ik , jk ) is not stochastic but is a deterministic function of position (ik , jk ), trained by backpropagation of gradients from the multi-ﬁxation RBM learning cost. [sent-92, score-0.667]

36 In practice, we force the components of z(ik , jk ) to be in [0, 1]3 . [sent-93, score-0.271]

37 1 Learning the what-where combination For now, let’s assume that we are given the sequence of glimpses xt fed to the multi-ﬁxation RBM 1:K for each image It . [sent-98, score-0.355]

38 As suggested by [9], we can train the RBM to minimize the following hybrid cost over each input xt and label lt : 1:K Hybrid cost: Chybrid = − log p(yt |xt ) − α log p(yt , xt ) 1:K 1:K (8) t where y = elt . [sent-99, score-0.246]

39 The RBM is then trained by doing stochastic or mini-batch gradient descent on the hybrid cost. [sent-103, score-0.138]

40 Putting more emphasis on the discriminative term ensures that more capacity is allocated to predicting the label values than to predicting each pixel value, which is important because there are many more pixels than labels. [sent-105, score-0.119]

41 We can also take advantage of the following obvious fact: If the sequence xt is associated with a 1:K particular target label yt , then so are all the subsequences xt where k < K. [sent-108, score-0.17]

42 While being more expensive than the hybrid cost, the hybrid-sequential cost could yield better generalization performance by better exploiting the training data. [sent-113, score-0.131]

43 2 Learning where to look Now that we have a model for processing the glimpses resulting from ﬁxating at different positions, we need to deﬁne a model which will determine where those ﬁxations should be made on the m × m grid of possible positions. [sent-117, score-0.278]

44 After k − 1 ﬁxations, this model should take as input some vector sk containing information about the glimpses accumulated so far (e. [sent-118, score-0.359]

45 the current activation probabilities of the multi-ﬁxation RBM hidden layer), and output a score f (sk , (ik , jk )) for each possible ﬁxation position (ik , jk ). [sent-120, score-0.658]

46 Ideally, the ﬁxation position with highest score under the controller should be the one which maximizes the chance of correctly classifying the input image. [sent-123, score-0.253]

47 For instance, a good controller could be such that t f (sk , (ik , jk )) ∝ log p(yt |xt 1:k−1 , xk = r(I, (ik , jk ))) (10) t i. [sent-124, score-0.778]

48 its output is proportional to the log-probability the RBM will assign to the true target y of the image It once it has ﬁxated at position (ik , jk ) and incorporated the information in that glimpse. [sent-126, score-0.397]

49 In other words, we would like the controller to assign high scores to ﬁxation positions which are more likely to provide the RBM with the necessary information to make a correct prediction of yt . [sent-127, score-0.299]

50 A simple training cost for the controller could then be to reduce the absolute difference between its prediction f (sk , (ik , jk )) and the observed value of log p(yt |xt 1:k−1 , xk = r(I, (ik , jk ))) for the sequences of glimpses generated while training the multi-ﬁxation RBM. [sent-128, score-1.116]

51 At test time however, for each k, the position that is the most likely under the controller is chosen4 . [sent-130, score-0.253]

52 In our experiments, we used a linear model for f (sk , (ik , jk )), with separate weights for each possible value of (ik , jk ). [sent-131, score-0.542]

53 f (sk , (ik , jk )) only depends on the values of sk and (ik , jk ) (though one could consider training a separate controller for each k). [sent-134, score-0.811]

54 As for the value taken by sk , we set it to k−1 k−1 W(ik∗ ,jk∗ ) xk∗ sigm c + = sigm c + k∗ =1 P diag(z(ik∗ , jk∗ )) F xk∗ (12) k∗ =1 which can be seen as an estimate of the probability vector for each hidden unit of the RBM to be 1, given the previous glimpses x1:k−1 . [sent-137, score-0.483]

55 For the special case k = 1, s1 is computed based on a ﬁxation at the center of the image but all the information in this initial glimpse is then “forgotten”, i. [sent-138, score-0.144]

56 We also concatenate to sk a binary vector of size m2 (one component for each possible ﬁxation position), where a component is 1 if the associated position has been ﬁxated. [sent-141, score-0.146]

57 Finally, in order to ensure that a ﬁxation position is never sampled twice, we impose that pcontroller ((ik , jk )|xt 1:k−1 ) = 0 for all positions previously sampled. [sent-142, score-0.458]

58 3 Putting it all together Figure 2 summarizes how the multi-ﬁxation RBM and the controller are jointly trained, for either the hybrid cost or the hybrid-sequential cost. [sent-144, score-0.305]

59 Details on gradient computations for both costs are 4 While it might not be optimal, this greedy search for the best sequence of ﬁxation positions is simple and worked well in practice. [sent-145, score-0.111]

60 To our knowledge, this is the ﬁrst implemented system for combining glimpses that jointly trains a recognition component (the RBM) with an attentional component (the ﬁxation controller). [sent-147, score-0.319]

61 5 Related work A vast array of work has been dedicated to modelling the visual search behavior of humans [11, 12, 13, 14], typically through the computation of saliency maps [15, 16]. [sent-148, score-0.117]

62 Surprisingly little work has been done on how best to combine multiple glimpses in a recognition system. [sent-150, score-0.284]

63 SIFT features have been proposed either as a preﬁlter for reducing the number of possible ﬁxation positions [17] or as a way of preprocessing the raw glimpses [13]. [sent-151, score-0.365]

64 [18] used a ﬁxed and hand-tuned saliency map to sample small patches in images of hand-written characters and trained a recursive neural network from sequences of such patches. [sent-152, score-0.113]

65 For instance, [19] use a saliency map based on ﬁlters previously trained on natural images for the where to look component, and the what-where combination component for recognition is a nearest neighbor density estimator. [sent-156, score-0.169]

66 The second is on a synthetic dataset and is meant to analyze the controller learning algorithm and its interaction with the multi-ﬁxation RBM. [sent-165, score-0.212]

67 Finally, results on a facial expression recognition problem are presented. [sent-166, score-0.103]

68 2 separately from the controller model, we trained a multi-ﬁxation RBM5 on a ﬁxed set of 4 ﬁxations (i. [sent-169, score-0.226]

69 Those ﬁxations were centered around the pixels at positions {(9, 9), (9, 19), (19, 9), (19, 19)} (MNIST images are of size 28 × 28) and their order was chosen at random for every parameter update of the RBM. [sent-172, score-0.16]

70 The retinal transformation had a high-resolution fovea covering 38 pixels and 60 hexagonal low-resolution regions in the periphery (see Figure 2 for an illustration). [sent-173, score-0.345]

71 The results are given in Figure 2, with comparisons with an RBF kernel SVM classiﬁer and a single hidden layer neural network initialized using unsupervised training of an RBM on the training set (those two baselines were trained on the full MNIST images). [sent-175, score-0.168]

72 The multi-ﬁxation RBM yields performance comparable to the baselines despite only having four glimpses, and the hybrid-sequential cost function works better than the non-sequential, hybrid cost. [sent-176, score-0.109]

73 2 Experiment 2: evaluation of the controller In this second experiment, we designed a synthetic problem where the optimal ﬁxation policy is known, to validate the proposed training algorithm for the controller. [sent-178, score-0.218]

74 The task is to identify whether 5 The RBM used H = 500 hidden units and was trained with a constant learning rate of 0. [sent-179, score-0.154]

75 The learned position vectors z(ik , jk ) were of size D = 250. [sent-181, score-0.347]

76 We report results when using either the hybrid cost of Equation 8 or the hybrid-sequential cost of Equation 9, with α = 0. [sent-183, score-0.15]

77 B: illustration of glimpses and results for experiment on MNIST. [sent-195, score-0.292]

78 At the center of the image is one of 8 visual symbols, indicating the location of the bar. [sent-197, score-0.106]

79 Since, as described earlier, the input s1 of the controller contains information about the center of the image, only one ﬁxation decision by the controller sufﬁces to solve this problem. [sent-203, score-0.376]

80 A multi-ﬁxation RBM was trained jointly with a controller on this problem6 , with only K = 1 ﬁxation. [sent-204, score-0.245]

81 When trained according to the hybrid cost of Equation 8 (α = 1), the model was able to solve this problem perfectly without errors, i. [sent-205, score-0.158]

82 the controller always proposes to ﬁxate at the region containing the white bar and the multi-ﬁxation RBM always correctly recognizes the orientation of the bar. [sent-207, score-0.177]

83 This is because the purely discriminative RBM never learns meaningful features for the non-discriminative visual symbol at the center, which are essential for the controller to be able to predict the position of the white bar. [sent-211, score-0.344]

84 3 Experiment 3: facial expression recognition experiment Finally, we applied the multi-ﬁxation RBM with its controller to a problem of facial expression recognition. [sent-213, score-0.372]

85 A multi-ﬁxation RBM learned jointly with a controller was trained on this problem7 , with K = 6 ﬁxations. [sent-219, score-0.245]

86 Possible ﬁxation positions were layed out every 10 pixels on a 7 × 7 grid, with the top-left 6 Hyper-parameters: H = 500, D = 250. [sent-220, score-0.137]

87 The controller had the choice of 9 possible ﬁxation positions, each covering either one of the eight regions where bars can be found or the middle region where the visual symbol is. [sent-223, score-0.229]

88 The retinal transformation was such that information from only one of those regions is transferred. [sent-224, score-0.156]

89 The RBM was trained with the hybrid cost of Equation 8 with α = 0. [sent-228, score-0.158]

90 001 (the hybrid cost was preferred mainly because it is faster). [sent-229, score-0.109]

91 The vectors 7 Experiment 2: synthetic dataset Positive examples Negative examples Experiment 3: facial expression recognition dataset Examples Results 0. [sent-231, score-0.122]

92 B: examples and results for the facial expression recognition dataset. [sent-239, score-0.103]

93 The retinal transformation covered around 2000 pixels and didn’t use a periphery8 (all pixels were from the fovea). [sent-241, score-0.28]

94 Moreover, glimpses were passed through a “preprocessing” hidden layer of size 250, initialized by unsupervised training of an RBM with Gaussian visible units (but without target units) on glimpses from the 7 × 7 grid. [sent-242, score-0.704]

95 During training of the multiﬁxation RBM, the discriminative part of its gradient was also passed through the preprocessing hidden layer for ﬁne-tuning of its parameters. [sent-243, score-0.164]

96 Finally, we also computed the performance of a linear SVM classiﬁer trained on the concatenation of the hidden units from a unique RBM with Gaussian visible units applied at all 7 × 7 positions (the same RBM used for initializing the preprocessing layer of the multi-ﬁxation RBM was used). [sent-257, score-0.387]

97 Most computer vision work on object recognition ignores this fact and can be viewed as modelling tachistoscopic recognition of very small objects that lie entirely within the fovea. [sent-262, score-0.123]

98 We believe that the intelligent choice of ﬁxation points and the integration of multiple glimpses will be essential for making biologically inspired vision systems work well on large images. [sent-266, score-0.273]

99 8 This simpliﬁcation of the retinal transformation makes it more convenient to estimate the percentage of high-resolution pixels used by the multi-ﬁxation RBM and contrast it with the SVM trained on the full image. [sent-276, score-0.267]

100 Q-learning of sequential attention for visual object recognition from informative local descriptors. [sent-359, score-0.109]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('xation', 0.622), ('rbm', 0.383), ('ik', 0.302), ('jk', 0.271), ('glimpses', 0.253), ('controller', 0.177), ('xations', 0.169), ('retinal', 0.111), ('retina', 0.078), ('position', 0.076), ('positions', 0.075), ('chybrid', 0.072), ('glimpse', 0.072), ('sk', 0.07), ('hybrid', 0.068), ('units', 0.065), ('xating', 0.064), ('accumulate', 0.064), ('pixels', 0.062), ('sigm', 0.06), ('xk', 0.059), ('geoffrey', 0.059), ('boltzmann', 0.057), ('facial', 0.055), ('xt', 0.052), ('image', 0.05), ('trained', 0.049), ('fovea', 0.049), ('yt', 0.047), ('transformation', 0.045), ('periphery', 0.042), ('cost', 0.041), ('saliency', 0.041), ('hidden', 0.04), ('hugo', 0.039), ('accumulated', 0.036), ('hexagonal', 0.036), ('pcontroller', 0.036), ('seq', 0.036), ('softplus', 0.036), ('visible', 0.036), ('layer', 0.035), ('visual', 0.034), ('pixel', 0.033), ('lt', 0.033), ('larochelle', 0.033), ('optic', 0.032), ('recognition', 0.031), ('dl', 0.03), ('eye', 0.029), ('svm', 0.027), ('array', 0.026), ('object', 0.025), ('look', 0.025), ('yoshua', 0.025), ('discriminative', 0.024), ('el', 0.024), ('toronto', 0.024), ('christof', 0.024), ('hexagons', 0.024), ('susskind', 0.024), ('ujl', 0.024), ('resolution', 0.024), ('mnist', 0.024), ('images', 0.023), ('training', 0.022), ('center', 0.022), ('preprocessing', 0.022), ('josh', 0.021), ('itti', 0.021), ('gradient', 0.021), ('vision', 0.02), ('shape', 0.02), ('experiment', 0.02), ('synthetic', 0.019), ('hasselmo', 0.019), ('touretzky', 0.019), ('subsequences', 0.019), ('uy', 0.019), ('attention', 0.019), ('generative', 0.019), ('contrastive', 0.019), ('illustration', 0.019), ('jointly', 0.019), ('laurent', 0.018), ('roland', 0.018), ('mozer', 0.018), ('memisevic', 0.018), ('symbol', 0.018), ('classi', 0.018), ('cj', 0.017), ('gregory', 0.017), ('expression', 0.017), ('attentional', 0.016), ('meant', 0.016), ('modelling', 0.016), ('connections', 0.015), ('costs', 0.015), ('features', 0.015), ('pseudocode', 0.015)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 1.0000002 156 nips-2010-Learning to combine foveal glimpses with a third-order Boltzmann machine

Author: Hugo Larochelle, Geoffrey E. Hinton

2 0.13574882 95 nips-2010-Feature Transitions with Saccadic Search: Size, Color, and Orientation Are Not Alike

Author: Stella X. Yu

Abstract: Size, color, and orientation have long been considered elementary features whose attributes are extracted in parallel and available to guide the deployment of attention. If each is processed in the same fashion with simply a different set of local detectors, one would expect similar search behaviours on localizing an equivalent ﬂickering change among identically laid out disks. We analyze feature transitions associated with saccadic search and ﬁnd out that size, color, and orientation are not alike in dynamic attribute processing over time. The Markovian feature transition is attractive for size, repulsive for color, and largely reversible for orientation. 1

3 0.11645656 99 nips-2010-Gated Softmax Classification

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

4 0.11139044 206 nips-2010-Phone Recognition with the Mean-Covariance Restricted Boltzmann Machine

Author: George Dahl, Marc'aurelio Ranzato, Abdel-rahman Mohamed, Geoffrey E. Hinton

Abstract: Straightforward application of Deep Belief Nets (DBNs) to acoustic modeling produces a rich distributed representation of speech data that is useful for recognition and yields impressive results on the speaker-independent TIMIT phone recognition task. However, the ﬁrst-layer Gaussian-Bernoulli Restricted Boltzmann Machine (GRBM) has an important limitation, shared with mixtures of diagonalcovariance Gaussians: GRBMs treat different components of the acoustic input vector as conditionally independent given the hidden state. The mean-covariance restricted Boltzmann machine (mcRBM), ﬁrst introduced for modeling natural images, is a much more representationally efﬁcient and powerful way of modeling the covariance structure of speech data. Every conﬁguration of the precision units of the mcRBM speciﬁes a different precision matrix for the conditional distribution over the acoustic space. In this work, we use the mcRBM to learn features of speech data that serve as input into a standard DBN. The mcRBM features combined with DBNs allow us to achieve a phone error rate of 20.5%, which is superior to all published results on speaker-independent TIMIT to date. 1

5 0.10405187 103 nips-2010-Generating more realistic images using gated MRF's

Author: Marc'aurelio Ranzato, Volodymyr Mnih, Geoffrey E. Hinton

Abstract: Probabilistic models of natural images are usually evaluated by measuring performance on rather indirect tasks, such as denoising and inpainting. A more direct way to evaluate a generative model is to draw samples from it and to check whether statistical properties of the samples match the statistics of natural images. This method is seldom used with high-resolution images, because current models produce samples that are very different from natural images, as assessed by even simple visual inspection. We investigate the reasons for this failure and we show that by augmenting existing models so that there are two sets of latent variables, one set modelling pixel intensities and the other set modelling image-speciﬁc pixel covariances, we are able to generate high-resolution images that look much more realistic than before. The overall model can be interpreted as a gated MRF where both pair-wise dependencies and mean intensities of pixels are modulated by the states of latent variables. Finally, we conﬁrm that if we disallow weight-sharing between receptive ﬁelds that overlap each other, the gated MRF learns more efﬁcient internal representations, as demonstrated in several recognition tasks. 1 Introduction and Prior Work The study of the statistical properties of natural images has a long history and has inﬂuenced many ﬁelds, from image processing to computational neuroscience [1]. In this work we focus on probabilistic models of natural images. These models are useful for extracting representations [2, 3, 4] that can be used for discriminative tasks and they can also provide adaptive priors [5, 6, 7] that can be used in applications like denoising and inpainting. Our main focus, however, will be on improving the quality of the generative model, rather than exploring its possible applications. Markov Random Fields (MRF’s) provide a very general framework for modelling natural images. In an MRF, an image is assigned a probability which is a normalized product of potential functions, with each function typically being deﬁned over a subset of the observed variables. In this work we consider a very versatile class of MRF’s in which potential functions are deﬁned over both pixels and latent variables, thus allowing the states of the latent variables to modulate or gate the effective interactions between the pixels. This type of MRF, that we dub gated MRF, was proposed as an image model by Geman and Geman [8]. Welling et al. [9] showed how an MRF in this family1 could be learned for small image patches and their work was extended to high-resolution images by Roth and Black [6] who also demonstrated its success in some practical applications [7]. Besides their practical use, these models were speciﬁcally designed to match the statistical properties of natural images, and therefore, it seems natural to evaluate them in those terms. Indeed, several authors [10, 7] have proposed that these models should be evaluated by generating images and 1 Product of Student’s t models (without pooling) may not appear to have latent variables but each potential can be viewed as an inﬁnite mixture of zero-mean Gaussians where the inverse variance of the Gaussian is the latent variable. 1 checking whether the samples match the statistical properties observed in natural images. It is, therefore, very troublesome that none of the existing models can generate good samples, especially for high-resolution images (see for instance ﬁg. 2 in [7] which is one of the best models of highresolution images reported in the literature so far). In fact, as our experiments demonstrate the generated samples from these models are more similar to random images than to natural images! When MRF’s with gated interactions are applied to small image patches, they actually seem to work moderately well, as demonstrated by several authors [11, 12, 13]. The generated patches have some coherent and elongated structure and, like natural image patches, they are predominantly very smooth with sudden outbreaks of strong structure. This is unsurprising because these models have a built-in assumption that images are very smooth with occasional strong violations of smoothness [8, 14, 15]. However, the extension of these patch-based models to high-resolution images by replicating ﬁlters across the image has proven to be difﬁcult. The receptive ﬁelds that are learned no longer resemble Gabor wavelets but look random [6, 16] and the generated images lack any of the long range structure that is so typical of natural images [7]. The success of these methods in applications such as denoising is a poor measure of the quality of the generative model that has been learned: Setting the parameters to random values works almost as well for eliminating independent Gaussian noise [17], because this can be done quite well by just using a penalty for high-frequency variation. In this work, we show that the generative quality of these models can be drastically improved by jointly modelling both pixel mean intensities and pixel covariances. This can be achieved by using two sets of latent variables, one that gates pair-wise interactions between pixels and another one that sets the mean intensities of pixels, as we already proposed in some earlier work [4]. Here, we show that this modelling choice is crucial to make the gated MRF work well on high-resolution images. Finally, we show that the most widely used method of sharing weights in MRF’s for high-resolution images is overly constrained. 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. Tang and C. Eliasmith. Deep networks for robust visual recognition. In ICML, 2010. [20] M. Ranzato, A. Krizhevsky, and G.E. Hinton. Factored 3-way restricted boltzmann machines for modeling natural images. In AISTATS, 2010. [21] N. Heess, C.K.I. Williams, and G.E. Hinton. Learning generative texture models with extended ﬁelds-ofexperts. In BMCV, 2009. [22] A. Krizhevsky. Learning multiple layers of features from tiny images, 2009. MSc Thesis, Dept. of Comp. Science, Univ. of Toronto. [23] A. Waibel, T. Hanazawa, G. Hinton, K. Shikano, and K. Lang. Phoneme recognition using time-delay neural networks. IEEE Acoustics Speech and Signal Proc., 37:328–339, 1989. [24] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. [25] T. Tieleman and G.E. Hinton. Using fast weights to improve persistent contrastive divergence. In ICML, 2009. [26] R.M. Neal. Bayesian learning for neural networks. Springer-Verlag, 1996. [27] T. Tieleman. Training restricted boltzmann machines using approximations to the likelihood gradient. In ICML, 2008. [28] http://www.cs.berkeley.edu/projects/vision/grouping/segbench/. [29] M. Welling, M. Rosen-Zvi, and G.E. Hinton. Exponential family harmoniums with an application to information retrieval. In NIPS, 2005. [30] http://yann.lecun.com/exdb/mnist/. [31] H. Lee, R. Grosse, R. Ranganath, and A. Y. Ng. Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. In Proc. ICML, 2009. [32] G.E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14:1771–1800, 2002. [33] A. Torralba, R. Fergus, and W.T. Freeman. 80 million tiny images: a large dataset for non-parametric object and scene recognition. PAMI, 30:1958–1970, 2008. 9

6 0.095019527 179 nips-2010-Natural Policy Gradient Methods with Parameter-based Exploration for Control Tasks

7 0.066424839 224 nips-2010-Regularized estimation of image statistics by Score Matching

8 0.062519975 29 nips-2010-An Approximate Inference Approach to Temporal Optimization in Optimal Control

9 0.057429746 143 nips-2010-Learning Convolutional Feature Hierarchies for Visual Recognition

10 0.050974391 133 nips-2010-Kernel Descriptors for Visual Recognition

11 0.050145026 272 nips-2010-Towards Holistic Scene Understanding: Feedback Enabled Cascaded Classification Models

12 0.049256116 209 nips-2010-Pose-Sensitive Embedding by Nonlinear NCA Regression

13 0.048238743 149 nips-2010-Learning To Count Objects in Images

14 0.046415109 86 nips-2010-Exploiting weakly-labeled Web images to improve object classification: a domain adaptation approach

15 0.045937464 271 nips-2010-Tiled convolutional neural networks

16 0.045926407 107 nips-2010-Global seismic monitoring as probabilistic inference

17 0.045092475 241 nips-2010-Size Matters: Metric Visual Search Constraints from Monocular Metadata

18 0.044175461 140 nips-2010-Layer-wise analysis of deep networks with Gaussian kernels

19 0.043773841 70 nips-2010-Efficient Optimization for Discriminative Latent Class Models

20 0.043681633 240 nips-2010-Simultaneous Object Detection and Ranking with Weak Supervision

similar papers computed by lsi model

lsi for this paper:

topicId topicWeight

[(0, 0.126), (1, 0.024), (2, -0.087), (3, -0.07), (4, 0.024), (5, -0.023), (6, -0.001), (7, 0.025), (8, -0.047), (9, 0.039), (10, 0.034), (11, -0.017), (12, 0.066), (13, -0.082), (14, -0.033), (15, 0.023), (16, 0.029), (17, -0.051), (18, -0.009), (19, -0.028), (20, -0.03), (21, 0.136), (22, 0.06), (23, 0.07), (24, 0.002), (25, 0.074), (26, 0.026), (27, 0.009), (28, 0.024), (29, -0.027), (30, -0.038), (31, -0.068), (32, 0.122), (33, 0.022), (34, 0.058), (35, 0.033), (36, 0.025), (37, -0.116), (38, 0.057), (39, 0.05), (40, 0.129), (41, 0.04), (42, 0.038), (43, 0.15), (44, 0.052), (45, 0.007), (46, 0.015), (47, -0.102), (48, 0.023), (49, -0.011)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.92010856 156 nips-2010-Learning to combine foveal glimpses with a third-order Boltzmann machine

Author: Hugo Larochelle, Geoffrey E. Hinton

2 0.64786673 95 nips-2010-Feature Transitions with Saccadic Search: Size, Color, and Orientation Are Not Alike

Author: Stella X. Yu

3 0.5638957 206 nips-2010-Phone Recognition with the Mean-Covariance Restricted Boltzmann Machine

Author: George Dahl, Marc'aurelio Ranzato, Abdel-rahman Mohamed, Geoffrey E. Hinton

4 0.54841298 99 nips-2010-Gated Softmax Classification

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

5 0.52676046 271 nips-2010-Tiled convolutional neural networks

Author: Jiquan Ngiam, Zhenghao Chen, Daniel Chia, Pang W. Koh, Quoc V. Le, Andrew Y. Ng

Abstract: Convolutional neural networks (CNNs) have been successfully applied to many tasks such as digit and object recognition. Using convolutional (tied) weights signiﬁcantly reduces the number of parameters that have to be learned, and also allows translational invariance to be hard-coded into the architecture. In this paper, we consider the problem of learning invariances, rather than relying on hardcoding. We propose tiled convolution neural networks (Tiled CNNs), which use a regular “tiled” pattern of tied weights that does not require that adjacent hidden units share identical weights, but instead requires only that hidden units k steps away from each other to have tied weights. By pooling over neighboring units, this architecture is able to learn complex invariances (such as scale and rotational invariance) beyond translational invariance. Further, it also enjoys much of CNNs’ advantage of having a relatively small number of learned parameters (such as ease of learning and greater scalability). We provide an efﬁcient learning algorithm for Tiled CNNs based on Topographic ICA, and show that learning complex invariant features allows us to achieve highly competitive results for both the NORB and CIFAR-10 datasets. 1

6 0.5068664 103 nips-2010-Generating more realistic images using gated MRF's

7 0.49237868 224 nips-2010-Regularized estimation of image statistics by Score Matching

8 0.48047942 111 nips-2010-Hallucinations in Charles Bonnet Syndrome Induced by Homeostasis: a Deep Boltzmann Machine Model

9 0.4801563 17 nips-2010-A biologically plausible network for the computation of orientation dominance

10 0.45638493 140 nips-2010-Layer-wise analysis of deep networks with Gaussian kernels

11 0.42329806 82 nips-2010-Evaluation of Rarity of Fingerprints in Forensics

12 0.42121816 245 nips-2010-Space-Variant Single-Image Blind Deconvolution for Removing Camera Shake

13 0.41461661 31 nips-2010-An analysis on negative curvature induced by singularity in multi-layer neural-network learning

14 0.41440696 28 nips-2010-An Alternative to Low-level-Sychrony-Based Methods for Speech Detection

15 0.4031297 188 nips-2010-On Herding and the Perceptron Cycling Theorem

16 0.37558845 207 nips-2010-Phoneme Recognition with Large Hierarchical Reservoirs

17 0.37014696 3 nips-2010-A Bayesian Framework for Figure-Ground Interpretation

18 0.36587721 101 nips-2010-Gaussian sampling by local perturbations

19 0.36345732 57 nips-2010-Decoding Ipsilateral Finger Movements from ECoG Signals in Humans

20 0.36284319 143 nips-2010-Learning Convolutional Feature Hierarchies for Visual Recognition

similar papers computed by lda model

lda for this paper:

topicId topicWeight

[(12, 0.011), (13, 0.039), (17, 0.018), (24, 0.041), (27, 0.069), (30, 0.074), (35, 0.035), (45, 0.179), (50, 0.042), (52, 0.034), (60, 0.018), (61, 0.236), (70, 0.023), (77, 0.032), (90, 0.04)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.7965486 156 nips-2010-Learning to combine foveal glimpses with a third-order Boltzmann machine

Author: Hugo Larochelle, Geoffrey E. Hinton

2 0.78055716 188 nips-2010-On Herding and the Perceptron Cycling Theorem

Author: Andrew Gelfand, Yutian Chen, Laurens Maaten, Max Welling

Abstract: The paper develops a connection between traditional perceptron algorithms and recently introduced herding algorithms. It is shown that both algorithms can be viewed as an application of the perceptron cycling theorem. This connection strengthens some herding results and suggests new (supervised) herding algorithms that, like CRFs or discriminative RBMs, make predictions by conditioning on the input attributes. We develop and investigate variants of conditional herding, and show that conditional herding leads to practical algorithms that perform better than or on par with related classiﬁers such as the voted perceptron and the discriminative RBM. 1

3 0.66769618 155 nips-2010-Learning the context of a category

Author: Dan Navarro

Abstract: This paper outlines a hierarchical Bayesian model for human category learning that learns both the organization of objects into categories, and the context in which this knowledge should be applied. The model is ﬁt to multiple data sets, and provides a parsimonious method for describing how humans learn context speciﬁc conceptual representations.

4 0.6666472 260 nips-2010-Sufficient Conditions for Generating Group Level Sparsity in a Robust Minimax Framework

Author: Hongbo Zhou, Qiang Cheng

Abstract: Regularization technique has become a principled tool for statistics and machine learning research and practice. However, in most situations, these regularization terms are not well interpreted, especially on how they are related to the loss function and data. In this paper, we propose a robust minimax framework to interpret the relationship between data and regularization terms for a large class of loss functions. We show that various regularization terms are essentially corresponding to different distortions to the original data matrix. This minimax framework includes ridge regression, lasso, elastic net, fused lasso, group lasso, local coordinate coding, multiple kernel learning, etc., as special cases. Within this minimax framework, we further give mathematically exact deﬁnition for a novel representation called sparse grouping representation (SGR), and prove a set of sufﬁcient conditions for generating such group level sparsity. Under these sufﬁcient conditions, a large set of consistent regularization terms can be designed. This SGR is essentially different from group lasso in the way of using class or group information, and it outperforms group lasso when there appears group label noise. We also provide some generalization bounds in a classiﬁcation setting. 1

5 0.66650581 51 nips-2010-Construction of Dependent Dirichlet Processes based on Poisson Processes

Author: Dahua Lin, Eric Grimson, John W. Fisher

Abstract: We present a novel method for constructing dependent Dirichlet processes. The approach exploits the intrinsic relationship between Dirichlet and Poisson processes in order to create a Markov chain of Dirichlet processes suitable for use as a prior over evolving mixture models. The method allows for the creation, removal, and location variation of component models over time while maintaining the property that the random measures are marginally DP distributed. Additionally, we derive a Gibbs sampling algorithm for model inference and test it on both synthetic and real data. Empirical results demonstrate that the approach is effective in estimating dynamically varying mixture models. 1

6 0.6650843 109 nips-2010-Group Sparse Coding with a Laplacian Scale Mixture Prior

7 0.66485208 117 nips-2010-Identifying graph-structured activation patterns in networks

8 0.66330409 277 nips-2010-Two-Layer Generalization Analysis for Ranking Using Rademacher Average

9 0.66174489 201 nips-2010-PAC-Bayesian Model Selection for Reinforcement Learning

10 0.66060251 183 nips-2010-Non-Stochastic Bandit Slate Problems

11 0.6602087 238 nips-2010-Short-term memory in neuronal networks through dynamical compressed sensing

12 0.6598596 268 nips-2010-The Neural Costs of Optimal Control

13 0.65960908 248 nips-2010-Sparse Inverse Covariance Selection via Alternating Linearization Methods

14 0.65913498 7 nips-2010-A Family of Penalty Functions for Structured Sparsity

15 0.65873617 148 nips-2010-Learning Networks of Stochastic Differential Equations

16 0.65837777 280 nips-2010-Unsupervised Kernel Dimension Reduction

17 0.65638816 63 nips-2010-Distributed Dual Averaging In Networks

18 0.65631497 85 nips-2010-Exact learning curves for Gaussian process regression on large random graphs

19 0.65577275 88 nips-2010-Extensions of Generalized Binary Search to Group Identification and Exponential Costs

20 0.65530562 186 nips-2010-Object Bank: A High-Level Image Representation for Scene Classification & Semantic Feature Sparsification