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101 nips-2005-Is Early Vision Optimized for Extracting Higher-order Dependencies?

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Author: Yan Karklin, Michael S. Lewicki

Abstract: Linear implementations of the efﬁcient coding hypothesis, such as independent component analysis (ICA) and sparse coding models, have provided functional explanations for properties of simple cells in V1 [1, 2]. These models, however, ignore the non-linear behavior of neurons and fail to match individual and population properties of neural receptive ﬁelds in subtle but important ways. Hierarchical models, including Gaussian Scale Mixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspects of neural processing such as normalization and context effects [6,7]. Previously, it had been assumed that the lower level representation is independent of the hierarchy, and had been ﬁxed when training these models. Here we examine the optimal lower-level representations derived in the context of a hierarchical model and ﬁnd that the resulting representations are strikingly different from those based on linear models. Unlike the the basis functions and ﬁlters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive ﬁelds and collectively span a broad range of spatial scales. Our work uniﬁes several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 These models, however, ignore the non-linear behavior of neurons and fail to match individual and population properties of neural receptive ﬁelds in subtle but important ways. [sent-8, score-0.318]

2 Hierarchical models, including Gaussian Scale Mixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspects of neural processing such as normalization and context effects [6,7]. [sent-9, score-0.452]

3 Previously, it had been assumed that the lower level representation is independent of the hierarchy, and had been ﬁxed when training these models. [sent-10, score-0.149]

4 Here we examine the optimal lower-level representations derived in the context of a hierarchical model and ﬁnd that the resulting representations are strikingly different from those based on linear models. [sent-11, score-0.563]

5 Unlike the the basis functions and ﬁlters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive ﬁelds and collectively span a broad range of spatial scales. [sent-12, score-0.99]

6 Our work uniﬁes several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy. [sent-13, score-0.735]

7 1 Introduction Efﬁcient coding hypothesis has been proposed as a guiding computational principle for the analysis of early visual system and motivates the search for good statistical models of natural images. [sent-14, score-0.511]

8 Early work revealed that image statistics are highly non-Gaussian [8, 9], and models such as independent component analysis (ICA) and sparse coding have been developed to capture these statistics to form efﬁcient representations of natural images. [sent-15, score-0.913]

9 It has been suggested that these models explain the basic computational goal of early visual cortex, as evidenced by the similarity between the learned parameters and the measured receptive ﬁelds of simple cells in V1. [sent-16, score-0.463]

10 ∗ To whom correspondence should be addressed In fact, it is not clear exactly how well these methods predict the shapes of neural receptive ﬁelds. [sent-17, score-0.318]

11 There has been no thorough characterization of ICA and sparse coding results for different datasets, pre-processing methods, and speciﬁc learning algorithms employed, although some of these factors clearly affect the resulting representation [10]. [sent-18, score-0.508]

12 When ICA or sparse coding is applied to natural images, the resulting basis functions resemble Gabor functions [1, 2] — 2D sine waves modulated by Gaussian envelopes — which also accurately model the shapes of simple cell receptive ﬁelds [11]. [sent-19, score-1.17]

13 Often, these results are visualized in a transformed space, by taking the logarithm of the pixel intensities, sphering (whitening) the image space, or ﬁltering the images to ﬂatten their spectrum. [sent-20, score-0.192]

14 When analyzed in the original image space, the learned ﬁlters (the models’ analogues of neural receptive ﬁelds) do not exhibit the multi-scale properties of the visual system, as they tend to cluster at high spatial frequencies [10, 12]. [sent-21, score-0.778]

15 Neural receptive ﬁelds, on the other hand, span a broad range of spatial scales, and exhibit distributions of spatial phase and other parameters unmatched by ICA and SC results [13,14]. [sent-22, score-0.625]

16 Therefore, as models of early visual processing, these models fail to predict accurately either the individual or the population properties of cortical visual neurons. [sent-23, score-0.331]

17 Linear efﬁcient coding methods are also limited in the type of statistical structure they can capture. [sent-24, score-0.313]

18 Applied to natural images, their coefﬁcients contain signiﬁcant residual dependencies that cannot be accounted for by the linear form of the models. [sent-25, score-0.184]

19 Several solutions have been proposed, including multiplicative Gaussian Scale Mixtures [4] and generative hierarchical models [5, 6]. [sent-26, score-0.31]

20 These models capture some of the observed dependencies; but their analysis so far has been focused on the higher-order structure learned by the model. [sent-27, score-0.26]

21 Meanwhile, the lower-level representation is either chosen a priori [4] or adapted separately, in the absence of the hierarchy [6] or with a ﬁxed hierarchical structure speciﬁed in advance [5]. [sent-28, score-0.485]

22 Here we examine whether the optimal lower-level representation of natural images is different when trained in the context of such non-linear hierarchical models. [sent-29, score-0.574]

23 We also illustrate how the model not only describes sparse marginal densities and magnitude dependencies, but captures a variety of joint density functions that are consistent with previous observations and theoretical conjectures. [sent-30, score-0.496]

24 We show that learned lower-level representations are strikingly different from those learned by the linear models: they are more multi-scale, spanning a wide range of spatial scales and phases of the Gabor sinusoid relative to the Gaussian envelope. [sent-31, score-0.574]

25 2 Fully adaptable scale mixture model A simple and scalable model for natural image patches is a linear factor model, in which the data x are assumed to be generated as a linear combination of basis functions with additive noise x = Au + . [sent-33, score-0.648]

26 (2) The coefﬁcients u are assumed to be mutually independent, and often modeled with sparse distributions (e. [sent-35, score-0.281]

27 Laplacian) that reﬂect the non-Gaussian statistics of natural scenes [8,9], P (ui ) ∝ exp(− P (u) = i |ui |) . [sent-37, score-0.175]

28 i (3) We can then adapt the basis functions A to maximize the expected log-likelihood of the data L = log P (x|A) over the data ensemble, thereby learning a compact, efﬁcient representation of structure in natural images. [sent-38, score-0.409]

29 This is the model underlying the sparse coding algorithm [2] and closely related to independent component analysis (ICA) [1]. [sent-39, score-0.528]

30 An alternative to ﬁxed sparse priors for u (3) is to use a Gaussian Scale Mixture (GSM) model [3]. [sent-40, score-0.259]

31 This type of model can also account for the observed dependencies among coefﬁcients u, for example, by expressing them as pair-wise dependencies among the multiplier variables λ [4, 15]. [sent-42, score-0.29]

32 A more general model, proposed in [6, 16], employs a hierarchical prior for P (u) with adapted parameters tuned to the global patterns in higher-order dependencies. [sent-43, score-0.327]

33 In fact, if the conditional density P (u|v) is Gaussian, this Hi2 erarchical Scale Mixture (HSM) is equivalent to a GSM model, with λ = σu and P (u|λ) = P (u|v) = N (0, exp(Bv)), with the added advantage of a more ﬂexible representation of higher-order statistical regularities in B. [sent-46, score-0.218]

34 Whereas previous GSM models of natural images focused on modeling local relationships between coefﬁcients of ﬁxed linear transforms, this general hierarchical formulation is fully adaptable, allowing us to recover the optimal lower-level representation A, as well as the higher-order components B. [sent-47, score-0.584]

35 The optimal lower-level basis A is computed similarly to the sparse coding algorithm — the goal is to minimize reconstruction ˆ ˆ error of the inferred MAP estimate u. [sent-50, score-0.573]

36 However, u is estimated not with a ﬁxed sparsifying prior, but with a concurrently adapted hierarchical prior. [sent-51, score-0.327]

37 (9) 2 Marginalizing over the latent higher-order variables in the hierarchical models leads to sparse distributions similar to the Laplacian and other density functions assumed in ICA. [sent-53, score-0.705]

38 Gaussian, qu = 2 Laplacian, qu = 1 Gen Gauss, qu = 0. [sent-54, score-0.243]

39 7 HSM, B = [0;0] HSM, B = [1;1] HSM, B = [2;2] HSM, B = [1;−1] HSM, B = [2;−2] HSM, B = [1;−2] Figure 1: This model can describe a variety of joint density functions for coefﬁcients u. [sent-55, score-0.206]

40 For illustration, in the hierarchical models the dimensionality of v is 1, and the matrix B is simply a column vector. [sent-59, score-0.31]

41 Even with this simple hierarchy, the model can generate sparse star-shaped (bottom row) or radially symmetric (middle row) densities, as well as more complex non-symmetric densities (bottom right). [sent-61, score-0.353]

42 Bi-variate joint distributions of GSM coefﬁcients can capture non-linear dependencies in wavelet coefﬁcients [4]. [sent-68, score-0.326]

43 In the fully adaptable HSM, however, the joint density can take a variety of shapes that depend on the learned parameters B (ﬁgure 1). [sent-69, score-0.384]

44 Note that this model can produce sparse, star-shaped distributions as in the linear models, or radially symmetric distributions that cannot be described by the linear models. [sent-70, score-0.193]

45 Such joint density proﬁles have been observed empirically in the responses of phase-offset wavelet coefﬁcients to natural images and have inspired polar transformation and quadrature pair models [17] (as well as connections to phaseinvariant neural responses). [sent-71, score-0.494]

46 The model described here can capture these joint densities and others, but rather than assume this structure a priori, it learns it automatically from the data. [sent-72, score-0.247]

47 3 Methods To examine how the lower-level representation is affected by the hierarchical model structure, we compared A learned by the sparse coding algorithm [2] and the HSM described above. [sent-73, score-0.908]

48 The models were trained on 20 × 20 image patches sampled from 40 images of out- door scenes in the Kyoto dataset [12]. [sent-74, score-0.283]

49 We applied a low-pass radially symmetric ﬁlter to the full images to eliminate high corner frequencies (artifacts of the square sampling lattice), and removed the DC component from each image patch, but did no further pre-processing. [sent-75, score-0.384]

50 1, and the basis functions were initialized to small random values and adapted on stochastically sampled batches of 300 patches. [sent-78, score-0.262]

51 The parameters of the hierarchical model were estimated in a similar fashion. [sent-81, score-0.302]

52 01, because emergence of the variance patterns requires some stabilization in the basis functions in A. [sent-85, score-0.235]

53 Because encoding in the sparse coding and in the hierarchical model is a non-linear process, it is not possible to compare the inverse of A to physiological data. [sent-86, score-0.766]

54 4 Results The shapes of basis functions and ﬁlters obtained with sparse coding have been previously analyzed and compared to neural receptive ﬁelds [10, 14]. [sent-89, score-1.02]

55 In the original space, sparse coding basis functions have very particular shapes: except for a few large, low frequency functions, all are localized, odd-symmetric, and span only a single period of the sinusoid (ﬁgure 2, top left). [sent-91, score-0.785]

56 The estimated ﬁlters are similar but smaller (ﬁgure 2, bottom left), with peak spatial frequencies clustered at higher frequencies (ﬁgure 3). [sent-92, score-0.35]

57 In the hierarchical model, the learned representation is strikingly different (ﬁgure 2, right panels). [sent-93, score-0.509]

58 Both the basis and the ﬁlters span a wider range of spatial scales, a result previously unobserved for models trained on non-preprocessed images, and one that is more consistent with physiological data [13, 14]. [sent-94, score-0.399]

59 Also, the shapes of the basis functions are different — they more closely resemble Gabor functions, although they tend to be less smooth than the sparse coding basis functions. [sent-95, score-0.925]

60 We also compared the distributions of spatial phases for ﬁlters obtained with sparse coding and the hierarchical model (ﬁgure 4). [sent-97, score-0.942]

61 While sparse coding ﬁlters exhibit a strong tendency for odd-symmetric phase proﬁles, the hierarchical model results in a much more uniform distribution of spatial phases. [sent-98, score-0.968]

62 Although some phase asymmetry has been observed in simple cell receptive ﬁelds, their phase properties tend to be much more uniform than sparse coding ﬁlters [14]. [sent-99, score-0.854]

63 In the hierarchical model, the higher-order representation B is also adapted to the statistical structure of natural images. [sent-100, score-0.564]

64 sparse or Gaussian) can determine the type of structure captured in B, we discovered that it does not affect the nature of the lower-level representation. [sent-103, score-0.225]

65 Shown are subsets of the learned basis functions and the estimates for the ﬁlters obtained with reverse correlation. [sent-108, score-0.332]

66 These functions are displayed in the original image space. [sent-109, score-0.151]

67 25 0° 180° 0 0° Figure 3: Scatter plots of peak frequencies and orientations of the Gabor functions ﬁtted to the estimated ﬁlters. [sent-114, score-0.203]

68 Although both SC and HSM ﬁlters exhibit predominantly high spatial frequencies, the hierarchical model yields a representation that tiles the spatial frequency space much more evenly. [sent-116, score-0.673]

69 SC phase HSM phase SC freq HSM freq 40 40 50 50 20 20 25 25 0 0 π/4 π/2 0 0 π/4 π/2 0 0. [sent-117, score-0.272]

70 50 Figure 4: The distributions of phases and frequencies for Gabor functions ﬁtted to sparse coding (SC) and hierarchical scale model (HSM) ﬁlters. [sent-125, score-1.056]

71 The phase units specify the phase of the sinusoid in relation to the peak of the Gaussian envelope of the Gabor function; 0 is even-symmetric, π/2 is odd-symmetric. [sent-126, score-0.276]

72 group co-localized lower-level basis functions and separately represent spatial contrast and oriented image structure. [sent-128, score-0.402]

73 As reported previously [6,16], with a sparse prior on v, the model learns higher-order components that individually capture complex spatial, orientation, and scale regularities in image data. [sent-129, score-0.553]

74 5 Discussion We have demonstrated that adapting a general hierarchical model yields lower-level representations that are signiﬁcantly different than those obtained using ﬁxed priors and linear generative models. [sent-130, score-0.463]

75 The resulting basis functions and ﬁlters are multi-scale and more consistent with several observed characteristics of neural receptive ﬁelds. [sent-131, score-0.453]

76 It is interesting that the learned representations are similar to the results obtained when ICA or sparse coding is applied to whitened images (i. [sent-132, score-0.755]

77 The hierarchical model is performing a similar scaling operation through the inference of higher-order variables v that scale the priors on basis function coefﬁcients u. [sent-136, score-0.542]

78 Thus the model can rely on a generic “white” lower level representation, while employing an adaptive mechanism for normalizing the space, which accounts for non-stationary statistics on an image-by-image basis [6]. [sent-137, score-0.205]

79 The ﬂexibility of the hierarchical model allows us to learn a lower-level representation that is optimal in the context of the hierarchy. [sent-139, score-0.409]

80 Thus, we expect the learned parameters to deﬁne a better statistical model for natural images than other approaches in which the lower-level representation or the higher-order dependencies are ﬁxed in advance. [sent-140, score-0.525]

81 For example, the ﬂexible marginal distributions, illustrated in ﬁgure 1, should be able to capture a wider range of statistical structure in natural images. [sent-141, score-0.251]

82 One way to quantify the beneﬁt of an adapted lowerlevel representation is to apply the model to problems like image de-noising and ﬁlling-in missing pixels. [sent-142, score-0.257]

83 Finally, although the results presented here are more consistent with the observed properties of neural receptive ﬁelds, several discrepancies remain. [sent-144, score-0.287]

84 For example, our results, as well as those of other statistical models, fail to account for the prevalence of low spatial frequency receptive ﬁelds observed in V1. [sent-145, score-0.428]

85 This could be a result of the speciﬁc choice of the distribution assumed by the model, although the described hierarchical framework makes few assumptions about the joint distribution of basis function coefﬁcients. [sent-146, score-0.509]

86 More likely, the non-stationary statistics of the natural scenes play a role in determining the properties of the learned representation. [sent-147, score-0.309]

87 This provides a strong motivation for training models with an “over-complete” basis, in which the number of basis functions is greater than the dimensionality of the input data [19]. [sent-149, score-0.245]

88 In this case, different subsets of the basis functions can adapt to optimally represent different image contexts, and the population properties of such over-complete representations could be signiﬁcantly different. [sent-150, score-0.435]

89 It would be particularly interesting to investigate representations learned in these models in the context of a hierarchical model. [sent-151, score-0.52]

90 Emergence of simple-cell receptive-ﬁeld properties by learning a sparse code for natural images. [sent-164, score-0.32]

91 Random cascades on wavelet trees and their use in analyzing and modeling natural images. [sent-180, score-0.178]

92 A hierarchical bayesian model for learning non-linear statistical regularities in non-stationary natural signals. [sent-193, score-0.491]

93 Independent component ﬁlters of natural images compared with simple cells in primary visual cortex. [sent-217, score-0.318]

94 An evaluation of the two-dimensional gabor ﬁlter model of simple receptive ﬁelds in cat striate cortex. [sent-224, score-0.394]

95 Sparse coding of natural images using an overcomplete set of limited capacity units. [sent-230, score-0.483]

96 Spatial frequency selectivity of cells in macaque visual cortex. [sent-239, score-0.16]

97 Spatial structure and symmetry of simple-cell receptive ﬁelds in macaque primary visual cortex. [sent-244, score-0.316]

98 Image denoising using Gaussian scale mixtures in the wavelet domain. [sent-253, score-0.199]

99 Nonlinear neurons and highorder statistics: New approaches to human vision and electronic image processing. [sent-264, score-0.167]

100 Sparse coding with an overcomplete basis set: A strategy employed by V1? [sent-282, score-0.422]

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