jmlr jmlr2008 jmlr2008-60 knowledge-graph by maker-knowledge-mining

60 jmlr-2008-Minimal Nonlinear Distortion Principle for Nonlinear Independent Component Analysis

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Author: Kun Zhang, Laiwan Chan

Abstract: It is well known that solutions to the nonlinear independent component analysis (ICA) problem are highly non-unique. In this paper we propose the “minimal nonlinear distortion” (MND) principle for tackling the ill-posedness of nonlinear ICA problems. MND prefers the nonlinear ICA solution with the estimated mixing procedure as close as possible to linear, among all possible solutions. It also helps to avoid local optima in the solutions. To achieve MND, we exploit a regularization term to minimize the mean square error between the nonlinear mixing mapping and the best-ﬁtting linear one. The effect of MND on the inherent trivial and non-trivial indeterminacies in nonlinear ICA solutions is investigated. Moreover, we show that local MND is closely related to the smoothness regularizer penalizing large curvature, which provides another useful regularization condition for nonlinear ICA. Experiments on synthetic data show the usefulness of the MND principle for separating various nonlinear mixtures. Finally, as an application, we use nonlinear ICA with MND to separate daily returns of a set of stocks in Hong Kong, and the linear causal relations among them are successfully discovered. The resulting causal relations give some interesting insights into the stock market. Such a result can not be achieved by linear ICA. Simulation studies also verify that when doing causality discovery, sometimes one should not ignore the nonlinear distortion in the data generation procedure, even if it is weak. Keywords: nonlinear ICA, regularization, minimal nonlinear distortion, mean square error, best linear reconstruction

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 HK Department of Computer Science and Engineering The Chinese University of Hongkong Hong Kong Editor: Aapo Hyv¨ rinen a Abstract It is well known that solutions to the nonlinear independent component analysis (ICA) problem are highly non-unique. [sent-7, score-0.414]

2 In this paper we propose the “minimal nonlinear distortion” (MND) principle for tackling the ill-posedness of nonlinear ICA problems. [sent-8, score-0.717]

3 MND prefers the nonlinear ICA solution with the estimated mixing procedure as close as possible to linear, among all possible solutions. [sent-9, score-0.484]

4 To achieve MND, we exploit a regularization term to minimize the mean square error between the nonlinear mixing mapping and the best-ﬁtting linear one. [sent-11, score-0.563]

5 The effect of MND on the inherent trivial and non-trivial indeterminacies in nonlinear ICA solutions is investigated. [sent-12, score-0.561]

6 Moreover, we show that local MND is closely related to the smoothness regularizer penalizing large curvature, which provides another useful regularization condition for nonlinear ICA. [sent-13, score-0.475]

7 Experiments on synthetic data show the usefulness of the MND principle for separating various nonlinear mixtures. [sent-14, score-0.376]

8 Finally, as an application, we use nonlinear ICA with MND to separate daily returns of a set of stocks in Hong Kong, and the linear causal relations among them are successfully discovered. [sent-15, score-0.437]

9 Simulation studies also verify that when doing causality discovery, sometimes one should not ignore the nonlinear distortion in the data generation procedure, even if it is weak. [sent-18, score-0.494]

10 Keywords: nonlinear ICA, regularization, minimal nonlinear distortion, mean square error, best linear reconstruction 1. [sent-19, score-0.703]

11 However, nonlinear ICA does not necessarily lead to nonlinear BSS. [sent-24, score-0.682]

12 In Hyv¨ rinen and Paa junen (1999), it was shown that solutions to nonlinear ICA always exist and that they are highly non-unique. [sent-25, score-0.396]

13 Actually, one can easily construct a nonlinear transformation of some non-Gaussian independent variables to produce independent outputs. [sent-26, score-0.39]

14 Taleb and Jutten (1999) also gave an example in which nonlinear mixtures of independent variables are still independent. [sent-38, score-0.487]

15 One can see that nonlinear BSS is impossible without additional prior knowledge on the mixing model, since the independence assumption is not strong enough in the general nonlinear mixing case (Jutten and Taleb, 2000; Taleb, 2002). [sent-39, score-0.968]

16 If we constrain the nonlinear mixing mapping to have some speciﬁc forms, the indeterminacies in the results of nonlinear ICA can be reduced dramatically, and as a consequence, nonlinear ICA may lead to nonlinear BSS. [sent-40, score-1.705]

17 , 1973) to nonlinear functions, a particular class of nonlinear mixing mappings, which satisfy an addition theorem in the sense of the theory of functional equations, were considered in Eriksson and Koivunen (2002). [sent-45, score-0.825]

18 In particular, the post-nonlinear (PNL) mixing model (Taleb and Jutten, 1999), which assumes that the mixing mapping is a linear transformation followed by a component-wise nonlinear one, has drawn much attention. [sent-46, score-0.722]

19 In practice, the exact form of the nonlinear mixing procedure is probably unknown. [sent-47, score-0.484]

20 Consequently, in order to model arbitrary nonlinear mappings, one may need to resort to a ﬂexible nonlinear function approximator, such as the multi-layer perceptron (MLP) or the radial basis function (RBF) network, to represent the nonlinear separation system. [sent-48, score-1.088]

21 Moreover, the smoothness constraint,1 which is implicitly provided by MLP’s with small initial weights and with a relatively small number of hidden units, was believed to be a suitable regularization condition to achieve nonlinear BSS. [sent-50, score-0.459]

22 Variational Bayesian nonlinear ICA (Lappalainen and Honkela, 2000; Valpola, 2000) uses the MLP to model the nonlinear mixing transformation. [sent-56, score-0.825]

23 If we can have some additional knowledge about the nonlinear mixing transformation and incorporate it efﬁciently, the results of nonlinear ICA will be much more meaningful and reliable. [sent-58, score-0.874]

24 Hence, provided that the nonlinear ICA outputs are mutually independent, we would prefer the solution with the estimated data generation procedure of minimal nonlinear distortion (MND). [sent-60, score-0.862]

25 2456 M INIMAL N ONLINEAR D ISTORTION FOR N ONLINEAR ICA information can help to reduce the indeterminacies in nonlinear ICA greatly, and moreover, to avoid local optima in the solutions to nonlinear ICA. [sent-78, score-0.855]

26 The minimal nonlinear distortion of the mixing system is achieved by the technique of regularization. [sent-79, score-0.58]

27 The objective function of nonlinear ICA with MND is the mutual information between outputs penalized by some terms measuring the level of “closeness to linear” of the mixing system. [sent-80, score-0.538]

28 The mean square error (MSE) between the nonlinear mixing system and its best-ﬁtting linear one provides such a regularization term. [sent-81, score-0.517]

29 To ensure that nonlinear ICA results in nonlinear BSS, one may also need to enforce the local MND of the nonlinear mapping averaged at every point, which turns out to be the smoothness regularizer exploiting second-order partial derivatives. [sent-82, score-1.17]

30 MND, as well as the smoothness regularizer, can be incorporated in various nonlinear ICA methods to improve the results. [sent-83, score-0.398]

31 The second one is nonlinear ICA based on kernels (Zhang and Chan, 2007a), in which the nonlinear separation system is modeled using some kernel methods. [sent-87, score-0.764]

32 We then explain why MND helps to alleviate the ill-posedness in nonlinear ICA solutions, by investigating the effect of MND on trivial and non-trivial indeterminacies in nonlinear ICA solutions systematically. [sent-88, score-0.902]

33 Finally, nonlinear ICA with MND is used to discover linear causal relations in the Hong Kong stock market and give encouraging results. [sent-91, score-0.46]

34 We also give experimental results on synthetic data, which illustrate that when performing ICA-based causality discovery on the data whose generation procedure involves nonlinear distortion, one should take into account the nonlinear effect in the ICA separation system, even if it is mild. [sent-92, score-0.804]

35 Nonlinear ICA with Minimal Nonlinear Distortion In this section we ﬁrst brieﬂy review the general nonlinear ICA problem, and then propose the minimal nonlinear distortion (MND) principle for regularization of nonlinear ICA. [sent-94, score-1.187]

36 , sn )T by a nonlinear transformation: x = F (s), (1) where F is an unknown real-valued n-component mixing function. [sent-102, score-0.484]

37 The general nonlinear ICA problem is to ﬁnd a mapping G : R n → Rn such that y = G (x) has statistically independent components. [sent-104, score-0.387]

38 In order to achieve nonlinear BSS, which aims at recovering the original sources si , we should resort to additional prior information or suitable regularization constraints. [sent-106, score-0.489]

39 Under the condition that the separation outputs yi are mutually independent, this principle prefers the solution with the estimated ˆ mixing transformation F as close as possible to linear. [sent-112, score-0.386]

40 Given a nonlinear mapping F , ∗ , which ﬁts F best among all afﬁne mappings A, is an ˆ its deviation from the afﬁne mapping A indicator of its “closeness to linear”, or the level of its nonlinear distortion. [sent-114, score-0.774]

41 We denote this measure by R and A MSE (θ), where θ denotes the set of unknown parameters in the ∗ = (x∗ , · · · , x∗ )T be the output of the afﬁne transformation from y by nonlinear ICA system. [sent-118, score-0.41]

42 i=1 i With RMSE measuring the level of nonlinear distortion, nonlinear ICA with MND can be formulated as the following constrained optimization problem. [sent-138, score-0.682]

43 2458 M INIMAL N ONLINEAR D ISTORTION FOR N ONLINEAR ICA it is not easy to guarantee the invertibility of the mapping provided by a ﬂexible nonlinear function approximator, like the MLP. [sent-147, score-0.387]

44 Hence when nonlinearity in F is not too strong, MND helps to guarantee the invertibility of the nonlinear ICA separation system G . [sent-149, score-0.494]

45 2 j=1 i=1 E(yi ) = −∑∑ (6) RMSE depends only on the inputs and the outputs of the nonlinear ICA system G (θ). [sent-173, score-0.396]

46 The choice of λc depends on the level of nonlinear distortion in the mixing procedure. [sent-185, score-0.58]

47 If the nonlinear distortion is considerable, we should use a very small value for λc to give the G network enough ﬂexibility. [sent-186, score-0.437]

48 In our experiments, we found that the separation performance of nonlinear ICA with MND is robust to the value of λc in a certain range. [sent-187, score-0.406]

49 3 Relation to Previous Works The MISEP method has been reported to solve some nonlinear BSS problems successfully, including separating a real-life nonlinear image mixture (Almeida, 2005, 2003). [sent-191, score-0.714]

50 Later, we will investigate the effect of MND on nonlinear ICA solutions theoretically, and compare various related nonlinear ICA methods empirically. [sent-202, score-0.703]

51 It is also useful to ensure nonlinear ICA to result in nonlinear BSS (Almeida, 2003). [sent-217, score-0.682]

52 In contrast, one may enforce the local MND of the nonlinear mapping averaged at F every point. [sent-223, score-0.387]

53 Incorporation of MND in Different Nonlinear ICA Methods Now we should choose a model for the nonlinear ICA separation system G (θ) and give the learning rule for nonlinear ICA with MND as well as nonlinear ICA with the smoothness constraint for G . [sent-245, score-1.145]

54 1 MISEP FOR N ONLINEAR ICA MISEP adopts the MLP to model the separation function G in the nonlinear ICA problem. [sent-252, score-0.406]

55 First, the separation system is a nonlinear transformation, which is modeled by the MLP. [sent-255, score-0.406]

56 With the Infomax principle, parameters in G and ψi are learned by maximizing the joint entropy of the outputs of the structure in Figure 2, which can be written as H(u) = H(x) + E{log | det J|}, where J = ∂u is the Jacobian of the nonlinear transformation from x to u. [sent-269, score-0.445]

57 3 MISEP WITH S MOOTHNESS C ONSTRAINT ON G The mapping provided by a MLP may not be smooth enough to make nonlinear ICA result in nonlinear BSS. [sent-314, score-0.786]

58 5 We have applied the MND principle and the smoothness regularizer to nonlinear ICA based on kernels; for details, see Zhang and Chan (2007a). [sent-348, score-0.477]

59 So it is quite necessary to explicitly enforce the smoothness constraint for nonlinear ICA based on kernels. [sent-350, score-0.398]

60 Investigation of the Effect of MND In this section we intend to explain why the MND principle, including the smoothness regularization, helps to alleviate the ill-posedness of nonlinear ICA from a mathematical viewpoint. [sent-352, score-0.398]

61 There are two types of indeterminacies in solutions to nonlinear ICA, namely trivial indeterminacies and non-trivial indeterminacies. [sent-353, score-0.713]

62 Trivial indeterminacies mean that the estimate of s j produced by nonlinear ICA may be any nonlinear function of s j ; non-trivial indeterminacies mean that the outputs of nonlinear ICA, although mutually independent, are still a mixing of the original sources. [sent-354, score-1.533]

63 1 For Trivial Indeterminacies Let us assume in this section that, in the solutions of nonlinear ICA, each component depends only on one of the sources. [sent-357, score-0.38]

64 Now let us consider a particular kind of nonlinear mixtures, in which each observed nonlinear mixture xi is assumed to be generated by xi = fi1 (s1 ) + fi2 (s2 ) + · · · + fin (sn ), (15) where fi j are invertible functions. [sent-363, score-0.714]

65 We call such nonlinear mixtures distorted source (DS) mixtures, since each observation is a linear mixture of nonlinearly distorted sources. [sent-364, score-0.579]

66 For this nonlinear mixing model, we have the following theorem on the effect of MND on trivial indeterminacies in its nonlinear ICA solutions. [sent-365, score-1.024]

67 Here the following assumptions are made: A1: In the output of nonlinear ICA, each component depends only on one of the sources and is zero-mean. [sent-366, score-0.46]

68 The difference between nonlinear ICA based on kernels discussed here and the kernel-based nonlinear BSS method by Harmeling et al. [sent-370, score-0.699]

69 Under assumptions A1 & A2, the estimate of s j produced by nonlinear ICA with MND is the ﬁrst principal component of f∗ j (s j ) = [ f1 j (s j ), · · · , fn j (s j )]T , multiplied by a constant. [sent-376, score-0.379]

70 The following theorem discusses the effect of MND on trivial indeterminacies of nonlinear ICA solutions in this case. [sent-382, score-0.561]

71 In particular, it states that by incorporating MND into the nonlinear ICA system, trivial indeterminacies in nonlinear ICA solutions are overcome; it shows how the outputs of the nonlinear ICA system, as the estimate of the sources, are related to the original sources s i and the mixing system F . [sent-383, score-1.498]

72 Under assumptions A1 & A2, the estimate of s j produced by nonlinear ICA with MND is the ﬁrst principal ˜ ˜ ˜ component of D∗ j (s j ) = [D1 j (s j ), · · · , Dn j (s j ), ]T , multiplied by a constant. [sent-388, score-0.379]

73 Under the condition that nonlinear distortion in the mixing mapping ˜ ˜ F is not strong, Di j (s j ) would not be far from linear. [sent-390, score-0.626]

74 To summarize, Theorems 1 and 2 show that trivial the ﬁrst principal component (PC) of D indeterminacies in nonlinear ICA solutions can be overcome by the MND principle; and when the mixing mapping is not strong, the nonlinear distortion in the nonlinear ICA outputs w. [sent-392, score-1.597]

75 1 R EMARK In the proof of Theorems 1 and 2, we have made use of the fact that mutual information is invariant to any component-wise strictly monotonic nonlinear transformation of the variables. [sent-398, score-0.413]

76 To model the trivial transformations, we apply a separate nonlinear function approximator (such as a MLP) to each output of nonlinear ICA to generate the ﬁnal nonlinear ICA result. [sent-405, score-1.09]

77 This provides a way to tackle the trivial indeterminacies; after performing nonlinear ICA with any nonlinear ICA method, if we know that there only exist trivial indeterminacies, we can adopt the above technique to determine the trivial transformations. [sent-408, score-0.823]

78 2 For Non-Trivial Indeterminacies Now let us investigate the effect of MND on non-trivial indeterminacies in nonlinear ICA solutions. [sent-410, score-0.493]

79 That is, y is always a solution to nonlinear ICA of the nonlinear mixture x = F (s). [sent-422, score-0.714]

80 Consider the mixing mapping x = F (s) which can be decomposed as a non-trivial transformation of s shown in Figure 3 (denote by z its output), followed by a nonlinear transformation x = F L (z) which is close enough to linear. [sent-453, score-0.628]

81 In this situation, the output of nonlinear ICA with MND would be an estimate of z, and if no additional knowledge of the mixing mapping is given, it is impossible to recover the original sources s i . [sent-454, score-0.631]

82 x-mark points show linear mixtures of the sources which ﬁt the nonlinear mixtures best. [sent-478, score-0.714]

83 The experiments in Zhang and Chan (2007a) have empirically shown that both MND and the smoothness constraint are useful to ensure nonlinear ICA based on kernels to result in nonlinear BSS, when nonlinear distortion in the mixing procedure is not very strong. [sent-482, score-1.336]

84 After obtaining nonlinear factor analysis solutions using the package, we applied linear ICA (FastICA by Hyv¨ rinen 1999 was used) to achieve a nonlinear BSS. [sent-509, score-0.737]

85 In addition, in order to show the necessity of nonlinear ICA methods for separating nonlinear mixtures, linear ICA (FastICA was adopted) was also used to separate the nonlinear mixtures. [sent-510, score-1.023]

86 Besides, we apply a ﬂexible nonlinear transformation h to y i to minimize the MSE between h(yi ) and si , and use the SNR of h(yi ) relative to si as another performance measure. [sent-516, score-0.458]

87 e, ’approach’, ’symm’, ’g’, ’tanh’); 2468 M INIMAL N ONLINEAR D ISTORTION FOR N ONLINEAR ICA Three kinds of nonlinear mixtures were investigated. [sent-532, score-0.487]

88 They are distorted source (DS) mixtures, post-nonlinear (PNL) mixtures, and generic nonlinear (GN) mixtures which are generated by a MLP. [sent-533, score-0.528]

89 To see the level of nonlinear distortion in the mixing transformation, we also give the scatter plot of the afﬁne transformation of si which ﬁts xi the best. [sent-543, score-0.735]

90 9 In this case, the proposed nonlinear ICA with MND (labelled by MND) also gives almost the best results; especially for the MLP without direct connections, the result of nonlinear ICA with MND is clearly the best. [sent-578, score-0.702]

91 But in this paper we assume that the form of the mixing procedure is unknown, and treat it as a general nonlinear ICA problem. [sent-582, score-0.484]

92 3 For Generic Nonlinear Mixtures We used a 2-2-2 MLP to generate nonlinear mixtures from sources. [sent-598, score-0.487]

93 They are not large such that the mixing mapping is invertible and the nonlinear distortion produced by the MLP would not be very strong. [sent-601, score-0.626]

94 Apparently nonlinear ICA with MND gives the best separation results in this case. [sent-605, score-0.406]

95 Summed over all the three cases discussed above, we can see that MISEP with MND produces promising results for the general nonlinear ICA problem, provided that nonlinearity in the mixing mapping is not very strong. [sent-606, score-0.618]

96 1 we have discussed the effect of the MND principle on trivial indeterminacies of nonlinear ICA solutions. [sent-611, score-0.575]

97 In particular, Theorem 1 states that for DS mixtures, if there are only trivial indeterminacies, each output of nonlinear ICA with MND is the PC of the contributions of the corresponding source to all mixtures. [sent-612, score-0.43]

98 2472 M INIMAL N ONLINEAR D ISTORTION FOR N ONLINEAR ICA (a) (b) 6 8 Best−fitting linear mixture Generic nonlinear mixture 6 4 4 2 x2 s2 2 0 0 −2 −2 −4 −4 −6 −8 −2 −1 0 s1 1 −6 −5 2 0 x 5 1 Figure 9: (a) Scatter plot of the sources si . [sent-637, score-0.538]

99 To tackle possible mild nonlinearity in the data generation procedure, we use nonlinear ICA with MND, instead of linear ICA, to separate the observed data. [sent-676, score-0.45]

100 As the nonlinear distortion is mild, it can be neglected and consequently, linear causal relations among the observed data can be discovered. [sent-677, score-0.505]

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Therefore, if transforms that achieve independence have non-diagonal Jacobian, scalar-wise restrictions in the original (coupled coefﬁcients) domain y are not allowed. 50 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING y2 1 −0,5 −0,5 0,4 R= r2 y1 r1 R−1 = 0,83 −0,83 0,67 −1,67 Figure 1: Insensitivity regions in different representation domains, y (left) and r (right), related by a non-diagonal transform ∇R and its inverse ∇R−1 . Figure 1 illustrates this situation. The shaded region in the right plot (r domain) represents the n-dimensional box determined by the ε f insensitivities in each dimension ( f =1,2), in which a scalar-wise approach is appropriate due to independence among signal coefﬁcients. Given that the particular ∇R transform is not diagonal, the corresponding shaded region in the left plot (the original y domain) is not aligned along the axes of the representation. This has negative implications: note that for the highlighted points, smaller distortions in both dimensions in the y domain (as implied by SVM with tighter but scalar ε f insensitivities) do not necessarily imply lying inside the insensitivity region in the ﬁnal truly independent (and meaningful) r domain. Therefore, the original y domain is not suitable for the direct application of conventional SVM, and consequently, non-trivial coupled insensitivity regions are required. Summarizing, in the image coding context, the condition for an image representation y to be strictly suitable for conventional SVM learning is that the transform that maps the original representation y to an independent coefﬁcient representation r must be locally diagonal. As will be reviewed below, independence among coefﬁcients (and the transforms to obtain them) may be deﬁned in both statistical and perceptual terms (Hyvarinen et al., 2001; Malo et al., 2001; Epifanio et al., 2003; Malo et al., 2006). On the one hand, a locally diagonal relation to a statistically independent representation is desirable because independently induced distortions (as the conventional SVM approach does) will preserve the statistics of the distorted signal, that is, it will not introduce artiﬁcial-looking artifacts. On the other hand, a locally diagonal relation to a perceptually 51 ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO independent representation is desirable because independently induced distortions do not give rise to increased subjective distortions due to non-trivial masking or facilitation interactions between the distortions in each dimension (Watson and Solomon, 1997). In this work, we show that conventional linear domains do not fulﬁll the diagonal Jacobian condition in either the statistical case or in the perceptual case. This theoretical result is experimentally conﬁrmed by comparing SVM learning in previously reported linear domains (Robinson and Kecman, 2003; G´ mez-P´ rez et al., 2005) and in a recently proposed non-linear perceptual domain o e that simultaneously reduces the statistical and the perceptual relations (Malo et al., 2006), thus, this non-linear perceptual domain is closer to fulﬁlling the proposed condition. The rest of the paper is structured as follows. Section 2 reviews the fact that linear coefﬁcients of the image representations commonly used for SVM training are neither statistically independent nor perceptually independent. Section 3 shows that transforms for obtaining statistical and/or perceptual independence from linear domains have non-diagonal Jacobian. This suggests that there is room to improve the performance of conventional SVM learning reported in linear domains. In Section 4, we propose the use of a perceptual representation for SVM training because it strictly fulﬁlls the diagonal Jacobian condition in the perceptual sense and increases the statistical independence among coefﬁcients, bringing it closer to fulﬁlling the condition in the statistical sense. The experimental image coding results conﬁrm the superiority of this domain for SVM training in Section 5. Section 6 presents the conclusions and ﬁnal remarks. 2. Statistical and Perceptual Relations Among Image Coefﬁcients Statistical independence among the coefﬁcients of a signal representation refers to the fact that the joint PDF of the class of signals to be considered can be expressed as a product of the marginal PDFs in each dimension (Hyvarinen et al., 2001). Simple (second-order) descriptions of statistical dependence use the non-diagonal nature of the covariance matrix (Clarke, 1985; Gersho and Gray, 1992). More recent and accurate descriptions use higher-order moments, mutual information, or the non-Gaussian nature (sparsity) of marginal PDFs (Hyvarinen et al., 2001; Simoncelli, 1997). Perceptual independence refers to the fact that the visibility of errors in coefﬁcients of an image may depend on the energy of neighboring coefﬁcients, a phenomenon known in the perceptual literature as masking or facilitation (Watson and Solomon, 1997). Perceptual dependence has been formalized just up to second order, and this may be described by the non-Euclidean nature of the perceptual metric matrix (Malo et al., 2001; Epifanio et al., 2003; Malo et al., 2006). 2.1 Statistical Relations In recent years, a variety of approaches, known collectively as “independent component analysis” (ICA), have been developed to exploit higher-order statistics for the purpose of achieving a unique linear solution for coefﬁcient independence (Hyvarinen et al., 2001). The basis functions obtained when these methods are applied to images are spatially localized and selective for both orientation and spatial frequency (Olshausen and Field, 1996; Bell and Sejnowski, 1997). Thus, they are similar to basis functions of multi-scale wavelet representations. Despite its name, linear ICA does not actually produce statistically independent coefﬁcients when applied to photographic images. Intuitively, independence would seem unlikely, since images are not formed from linear superpositions of independent patterns: the typical combination rule for the elements of an image is occlusion. Empirically, the coefﬁcients of natural image decom52 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING | f | = 10.8 cpd | f | = 24.4 cpd −20 −10 −10 fy (cpd) −30 −20 fy (cpd) −30 0 0 10 10 20 20 30 30 −30 −20 −10 0 f (cpd) 10 20 30 −30 x −20 −10 0 f (cpd) 10 20 30 x Figure 2: Statistical interaction of two particular coefﬁcients of the local Fourier Transform with their neighbors in a natural image database. The absolute value of the frequency of these coefﬁcients is | f | = 10.8 and | f | = 24.4 cycles/degree (cpd). positions in spatially localized oscillating basis functions are found to be fairly well decorrelated (i.e., their covariance is almost zero). However, the amplitudes of coefﬁcients at nearby spatial positions, orientations, and scales are highly correlated (even with orthonormal transforms) (Simoncelli, ´ 1997; Buccigrossi and Simoncelli, 1999; Wainwright et al., 2001; Hyvarinen et al., 2003; Guti errez et al., 2006; Malo et al., 2006; Malo and Guti´ rrez, 2006). This suggests that achieving statistical e independence requires the introduction of non-linearities beyond linear ICA transforms. Figure 2 reproduces one of many results that highlight the presence of statistical relations of natural image coefﬁcients in block PCA or linear ICA-like domains: the energy of spatially localized oscillating ﬁlters is correlated with the energy of neighboring ﬁlters in scale and orientation (see Guti´ rrez et al., 2006). A remarkable feature is that the interaction width increases with frequency, e as has been reported in other domains, for example, wavelets (Buccigrossi and Simoncelli, 1999; Wainwright et al., 2001; Hyvarinen et al., 2003), and block-DCT (Malo et al., 2006). In order to remove the remaining statistical relations in the linear domains y, non-linear ICA methods are necessary (Hyvarinen et al., 2001; Lin, 1999; Karhunen et al., 2000; Jutten and Karhunen, 2003). Without lack of generality, non-linear ICA transforms can be schematically understood as a two-stage process (Malo and Guti´ rrez, 2006): e T R (( y hh x hh (( r, (1) R−1 T−1 where x is the image representation in the spatial domain, and T is a global unitary linear transform that removes second-order and eventually higher-order relations among coefﬁcients in the spatial domain. Particular examples of T include block PCA, linear ICAs, DCT or wavelets. In the ICA literature notation, T is the separating matrix and T−1 is the mixing matrix. The second transform 53 ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO R is an additional non-linearity that is introduced in order to remove the statistical relations that still remain in the y domain. 2.2 Perceptual Relations Perceptual dependence among coefﬁcients in different image representations can be understood by using the current model of V1 cortex. This model can also be summarized by the two-stage (linear and non-linear) process described in Equation (1). In this perceptual case, T is also a linear ﬁlter bank applied to the original input image in the spatial domain. This ﬁlter bank represents the linear behavior of V1 neurons whose receptive ﬁelds happen to be similar to wavelets or linear ICA basis functions (Olshausen and Field, 1996; Bell and Sejnowski, 1997). The second transform, R, is a non-linear function that accounts for the masking and facilitation phenomena that have been reported in the linear y domain (Foley, 1994; Watson and Solomon, 1997). Section 3.2 gives a parametric expression for the second non-linear stage, R: the divisive normalization model (Heeger, 1992; Foley, 1994; Watson and Solomon, 1997). This class of models is based on psychophysical experiments assuming that the last domain, r, is perceptually Euclidean (i.e., perfect perceptual independence). An additional conﬁrmation of this assumption is the success of (Euclidean) subjective image distortion measures deﬁned in that domain (Teo and Heeger, 1994). Straightforward application of Riemannian geometry to obtain the perceptual metric matrix in other domains shows that the coefﬁcients of linear domains x and y, or any other linear transform of them, are not perceptually independent (Epifanio et al., 2003). Figure 3 illustrates the presence of perceptual relations between coefﬁcients when using linear block frequency or wavelet-like domains, y: the cross-masking behavior. In this example, the visibility of the distortions added on top of the background image made of periodic patterns has to be assessed. This is a measure of the sensitivity of a particular perceptual mechanism to distortions in that dimension, ∆y f , when mechanisms tuned to other dimensions are simultaneously active, that is, y f = 0, with f = f . As can be observed, low frequency noise is more visible in high frequency backgrounds than in low frequency backgrounds (e.g., left image). Similarly, high frequency noise is more visible in low frequency backgrounds than in high frequency ones (e.g., right image). That is to say, a signal of a speciﬁc frequency strongly masks the corresponding frequency analyzer, but it induces a smaller sensitivity reduction in the analyzers that are tuned to different frequencies. In other words, the reduction in sensitivity of a speciﬁc analyzer gets larger as the distance between the background frequency and the frequency of the analyzer gets smaller. The response of each frequency analyzer not only depends on the energy of the signal for that frequency band, but also on the energy of the signal in other frequency bands (cross-masking). This implies that a different amount of noise in each frequency band may be acceptable depending on the energy of that frequency band and on the energy of neighboring bands. This is what we have called perceptual dependence among different coefﬁcients in the y domain. At this point, it is important to stress the similarity between the set of computations to obtain statistically decoupled image coefﬁcients and the known stages of biological vision. In fact, it has been hypothesized that biological visual systems have organized their sensors to exploit the particular statistics of the signals they have to process. See Barlow (2001), Simoncelli and Olshausen (2001), and Simoncelli (2003) for reviews on this hypothesis. In particular, both the linear and the non-linear stages of the cortical processing have been successfully derived using redundancy reduction arguments: nowadays, the same class of linear 54 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING 3 cpd 6 cpd 12 cpd 24 cpd Figure 3: Illustrative example of perceptual dependence (cross-masking phenomenon). Equal energy noise of different frequency content, 3 cycl/deg (cpd), 6 cpd, 12 cpd and 24 cpd, shown on top of a background image. Sampling frequency assumes that these images subtend an angle of 3 deg. stage T is used in transform coding algorithms and in vision models (Olshausen and Field, 1996; Bell and Sejnowski, 1997; Taubman and Marcellin, 2001), and new evidence supports the same idea for the second non-linear stage (Schwartz and Simoncelli, 2001; Malo and Guti´ rrez, 2006). e According to this, the statistical and perceptual transforms, R, that remove the above relations from the linear domains, y, would be very similar if not the same. 3. Statistical and Perceptual Independence Imply Non-diagonal Jacobian In this section, we show that both statistical redundancy reduction transforms (e.g., non-linear ICA) and perceptual independence transforms (e.g., divisive normalization), have non-diagonal Jacobian for any linear image representation, so they are not strictly suitable for conventional SVM training. 3.1 Non-diagonal Jacobian in Non-linear ICA Transforms One possible approach for dealing with global non-linear ICA is to act differentially by breaking the problem into local linear pieces that can then be integrated to obtain the global independent coefﬁcient domain (Malo and Guti´ rrez, 2006). Each differential sub-problem around a particular e point (image) can be locally solved using the standard linear ICA methods restricted to the neighbors of that point (Lin, 1999). Using the differential approach in the context of a two-stage process such as the one in Equation (1), it can be shown that (Malo and Guti´ rrez, 2006): e r = r0 + Z x x0 T (x ) dx = r0 + Z x x0 ∇R(Tx ) T dx , (2) where T (x ) is the local separating matrix for a neighborhood of the image x , and T is the global separating matrix for the whole PDF. Therefore, the Jacobian of the second non-linear stage is: ∇R(y) = ∇R(Tx) = T (x) T−1 . 55 (3) ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO As local linear independent features around a particular image, x, differ in general from global linear independent features, that is, T (x) = T, the above product is not the identity nor diagonal in general. 3.2 Non-diagonal Jacobian in Non-linear Perceptual Transforms The current response model for the cortical frequency analyzers is non-linear (Heeger, 1992; Watson and Solomon, 1997). The outputs of the ﬁlters of the ﬁrst linear stage, y, undergo a non-linear sigmoid transform in which the energy of each linear coefﬁcient is weighted by a linear Contrast Sensitivity Function (CSF) (Campbell and Robson, 1968; Malo et al., 1997) and is further normalized by a combination of the energies of neighbor coefﬁcients in frequency, r f = R(y) f = sgn(y f ) |α f y f |γ , β f + ∑n =1 h f f |α f y f |γ f (4) where α f (Figure 4[top left]) are CSF-like weights, β f (Figure 4[top right]) control the sharpness of the response saturation for each coefﬁcient, γ is the so called excitation exponent, and the matrix h f f determines the interaction neighborhood in the non-linear normalization of the energy. This interaction matrix models the cross-masking behavior (cf. Section 2.2). The interaction in this matrix is assumed to be Gaussian (Watson and Solomon, 1997), and its width increases with the frequency. Figure 4[bottom] shows two examples of this Gaussian interaction for two particular coefﬁcients in a local Fourier domain. Note that the width of the perceptual interaction neighborhood increases with the frequency in the same way as the width of the statistical interaction neighborhood shown in Figure 2. We used a value of γ = 2 in the experiments. Taking derivatives in the general divisive normalization model, Equation (4), we obtain ∇R(y) f f = sgn(y f )γ α f |α f y f |γ |α f y f |γ−1 α f |α f y f |γ−1 δf f − hf f β f + ∑n =1 h f f |α f y f |γ (β f + ∑n =1 h f f |α f y f |γ )2 f f , (5) which is not diagonal because of the interaction matrix, h, which describes the cross-masking between each frequency f and the remaining f = f . Note that the intrinsic non-linear nature of both the statistical and perceptual transforms, Equations (3) and (5), makes the above results true for any linear domain under consideration. Speciﬁcally, if any other possible linear domain for image representation is considered, y = T y, then the Jacobian of the corresponding independence transform, R , is ∇R (y ) = ∇R(y) T −1 , which, in general, will also be non-diagonal because of the non-diagonal and point-dependent nature of ∇R(y). To summarize, since no linear domain fulﬁlls the diagonal Jacobian condition in either statistical or perceptual terms, the negative situation illustrated in Figure 1 may occur when using SVM in these domains. Therefore, improved results could be obtained if SVM learning were applied after some transform achieving independent coefﬁcients, R. 4. SVM Learning in a Perceptually Independent Representation In order to conﬁrm the above theoretical results (i.e., the unsuitability of linear representation domains for SVM learning) and to assess the eventual gain that can be obtained from training SVR 56 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING 0.04 αf 0.015 βf 0.03 0.012 0.02 0.009 0.006 0.01 0.003 0 0 4 8 12 16 20 24 28 32 0 f (frequency cycles/degree) 4 8 | f | = 10.8 cpd 16 20 24 28 32 | f | = 24.4 cpd −30 −20 −20 −10 −10 f (cpd) −30 0 y 0 y f (cpd) 12 f (frequency cycles/degree) 10 10 20 20 30 −30 −20 −10 0 fx (cpd) 10 20 30 −30 30 −20 −10 0 fx (cpd) 10 20 30 Figure 4: Parameters of the perceptual model: α f (top left), β f (top right). Bottom ﬁgures represent perceptual interaction neighborhoods h f f of two particular coefﬁcients of the local Fourier domain. in a more appropriate domain, we should compare the performance of SVRs in previously reported linear domains (e.g., block-DCT or wavelets) and in one of the proposed non-linear domains (either the statistically independent domain or the perceptually independent domain). Exploration of the statistical independence transform may have academic interest but, in its present formulation, it is not practical for coding purposes: direct application of non-linear ICA as in Equation (2) is very time-consuming for high dimensional vectors since lots of local ICA computations are needed to transform each block, and a very large image database is needed for a robust and signiﬁcant computation of R. Besides, an equally expensive differential approach is also needed to compute the inverse R−1 for image decoding. In contrast, the perceptual non-linearity (and its inverse) are analytical. These analytical expressions are feasible for reasonable block sizes, and there are efﬁcient iterative methods that can be used for larger vectors (Malo et al., 2006). In this paper, we explore the use of a psychophysically-based divisive normalized domain: ﬁrst compute a block-DCT transform and then apply the divisive normalization model described above for each block. The results will be compared to the ﬁrst competitive SVM coding results (Robinson 57 ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO ´ and Kecman, 2003) and the posterior improvements reported by G omez-P´ rez et al. (2005), both e formulated in the linear block-DCT domain. As stated in Section 2, by construction, the proposed domain is perceptually Euclidean with perceptually independent components. The Euclidean nature of this domain has an additional beneﬁt: the ε-insensitivity design is very simple because a constant value is appropriate due to the constant perceptual relevance of all coefﬁcients. Thus, direct application of the standard SVR method is theoretically appropriate in this domain. Moreover, beyond its built-in perceptual beneﬁts, this psychophysically-based divisive normalization has attractive statistical properties: it strongly reduces the mutual information between the ﬁnal coefﬁcients r (Malo et al., 2006). This is not surprising according to the hypothesis that try to explain the early stages of biological vision systems using information theory arguments (Barlow, 1961; Simoncelli and Olshausen, 2001). Speciﬁcally, dividing the energy of each linear coefﬁcient by the energy of the neighbors, which are statistically related with it, cf. Figure 2, gives coefﬁcients with reduced statistical dependence. Moreover, as the empirical non-linearities of perception have been reproduced using non-linear ICA in Equation (2) (Malo and Guti´ rrez, 2006), the empirical die visive normalization can be seen as a convenient parametric way to obtain statistical independence. 5. Performance of SVM Learning in Different Domains In this section, we analyze the performance of SVM-based coding algorithms in linear and nonlinear domains through rate-distortion curves and explicit examples for visual comparison. In addition, we discuss how SVM selects support vectors in these domains to represent the image features. 5.1 Model Development and Experimental Setup In the (linear) block-DCT domain, y, we use the method introduced by Robinson and Kecman (2003) (RKi-1), in which the SVR is trained to learn a ﬁxed (low-pass) number of DCT coefﬁcients ´ (those with frequency bigger than 20 cycl/deg are discarded); and the method proposed by G omezP´ rez et al. (2005) (CSF-SVR), in which the relevance of all DCT coefﬁcients is weighted according e to the CSF criterion using an appropriately modulated ε f . In the non-linear domain, r, we use the SVR with constant insensitivity parameter ε (NL-SVR). In all cases, the block-size is 16×16, that is, y, r ∈ R256 . The behavior of JPEG standard is also included in the experiments for comparison purposes. As stated in Section 1, we used the RBF kernel and arbitrarily large penalization parameter in every SVR case. In all experiments, we trained the SVR models without the bias term, and modelled the absolute value of the DCT, y, or response coefﬁcients, r. All the remaining free parameters (ε-insensitivity and Gaussian width of the RBF kernel σ) were optimized for all the considered models and different compression ratios. In the NL-SVM case, the parameters of the divisive normalization used in the experiments are shown in Figure 4. After training, the signal is described by the uniformly quantized Lagrange multipliers of the support vectors needed to keep the regression error below the thresholds ε f . The last step is entropy coding of the quantized weights. The compression ratio is controlled by a factor applied to the thresholds, ε f . 58 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING 5.2 Model Comparison In order to assess the quality of the coded images, three different measures were used: the standard ´ (Euclidean) RMSE, the Maximum Perceptual Error (MPE) (Malo et al., 2000; G omez-P´ rez et al., e 2005; Malo et al., 2006) and the also perceptually meaningful Structural SIMilarity (SSIM) index (Wang et al., 2004). Eight standard 256×256 monochrome 8 bits/pix images were used in the experiments. Average rate-distortion curves are plotted in Figure 5 in the range [0.05, 0.6] bits/pix (bpp). According to these entropy-per-sample data, original ﬁle size was 64 KBytes in every case, while the compressed image sizes were in the range [0.4, 4.8] KBytes. This implies that the compression ratios were in the range [160:1, 13:1]. In general, a clear gain over standard JPEG is obtained by all SVM-based methods. According to the standard Euclidean MSE point of view, the performance of RKi-1 and CSF-SVR algorithms is basically the same (note the overlapped curves in Figure 5(a)). However, it is widely known that the MSE results are not useful to represent the subjective quality of images, as extensively reported elsewhere (Girod, 1993; Teo and Heeger, 1994; Watson and Malo, 2002). When using more appropriate (perceptually meaningful) quality measures (Figures 5(b)-(c)), the CSF-SVR obtains a certain advantage over the RKi-1 algorithm for all compression rates, which was already reported by G´ mez-P´ rez et al. (2005). In all measures, and for the whole considered entropy range, the o e proposed NL-SVR clearly outperforms all previously reported methods, obtaining a noticeable gain at medium-to-high compression ratios (between 0.1 bpp (80:1) and 0.3 bpp (27:1)). Taking into account that the recommended bit rate for JPEG is about 0.5 bpp, from Figure 5 we can also conclude that the proposed technique achieves the similar quality levels at a lower bit rate in the range [0.15, 0.3] bpp. Figure 6 shows representative visual results of the considered SVM strategies on standard images (Lena and Barbara) at the same bit rate (0.3 bpp, 27:1 compression ratio or 2.4 KBytes in 256×256 images). The visual inspection conﬁrms that the numerical gain in MPE and SSIM shown in Figure 5 is also perceptually signiﬁcant. Some conclusions can be extracted from this ﬁgure. ´ First, as previously reported by Gomez-P´ rez et al. (2005), RKi-1 leads to poorer (blocky) results e because of the crude approximation of the CSF (as an ideal low-pass ﬁlter) and the equal relevance applied to the low-frequency DCT-coefﬁcients. Second, despite the good performance yielded by the CSF-SVR approach to avoid blocking effects, it is worth noting that high frequency details are smoothed (e.g., see Barbara’s scarf). These effects are highly alleviated by introducing SVR in the non-linear domain. See, for instance, Lena’s eyes, her hat’s feathers or the better reproduction of the high frequency pattern in Barbara’s clothes. Figure 7 shows the results obtained by all considered methods at a very high compression ratio for the Barbara image (0.05 bpp, 160:1 compression ratio or 0.4 KBytes in 256×256 images). This experiment is just intended to show the limits of methods performance since it is out of the recommended rate ranges. Even though this scenario is unrealistic, differences among methods are still noticeable: the proposed NL-SVR method reduces the blocky effects (note for instance that the face is better reproduced). This is due to a better distribution of support vectors in the perceptually independent domain. 5.3 Support Vector Distribution The observed different perceptual image quality obtained with each approach is a direct consequence of support vector distribution in different domains. Figure 8 shows a representative example 59 ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO 20 JPEG RKi−1 CSF−SVR NL−SVR 18 RMSE 16 14 12 10 8 0.1 0.2 0.3 0.4 Entropy (bits/pix) 0.5 0.6 22 JPEG RKi−1 CSF−SVR NL−SVR 20 18 16 MPE 14 12 10 8 6 4 0.1 0.2 0.3 0.4 Entropy (bits/pix) 0.5 0.6 0.85 0.8 SSIM 0.75 0.7 0.65 JPEG 0.6 RKi−1 CSF−SVR 0.55 NL−SVR 0.5 0.1 0.2 0.3 0.4 Entropy (bits/pix) 0.5 0.6 Figure 5: Average rate distortion curves over eight standard images (Lena, Barbara, Boats, Einstein, Peppers, Mandrill, Goldhill, Camera man) using objective and subjective measures for the considered JPEG (dotted) and the SVM approaches (RKi-1 dash-dotted, CSF-SVR dashed and NL-SVR solid). RMSE distortion (top), Maximum Perceptual Error, MPE (middle) (Malo et al., 2000; G´ mez-P´ rez et al., 2005; Malo et al., 2006), and Structural o e SIMilarity index, SSIM (bottom) (Wang et al., 2004). 60 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING Rki Rki SVR+CSF SVR+CSF NL+SVR NL+SVR Figure 6: Examples of decoded Lena (left) and Barbara (right) images at 0.3 bits/pix. From top to bottom: JPEG, RKi-1, CSF-SVR, and NL-SVR. 61 ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO (a) (b) (c) (d) Figure 7: Examples of decoded Barbara images at a high compression ratio of 0.05 bits/pix (160:1) for (a) JPEG, (b) RKi-1, (c) CSF-SVR, and (d) NL-SVR. of the distribution of the selected support vectors by the RKi-1 and the CSF-SVR models working in the linear DCT domain, and the NL-SVM working in the perceptually independent non-linear domain r. Speciﬁcally, a block of Barbara’s scarf at different compression ratios is used for illustration purposes. The RKi-1 approach (Robinson and Kecman, 2003) uses a constant ε but, in order to consider the low subjective relevance of the high-frequency region, the corresponding coefﬁcients are neglected. As a result, this approach only allocates support vectors in the low/medium frequency regions. The CSF-SVR approach uses a variable ε according to the CSF and gives rise to a more natural concentration of support vectors in the low/medium frequency region, which captures medium to high frequency details at lower compression rates (0.5 bits/pix). Note that the number of support vectors is bigger than in the RKi-1 approach, but it selects some necessary high-frequency coefﬁcients to keep the error below the selected threshold. However, for bigger compression ratios (0.3 bits/pix), it misrepresents some high frequency, yet relevant, features (e.g., the peak from the stripes). The NL-SVM approach works in the non-linear transform domain, in which a more uniform coverage 62 O N THE S UITABLE D OMAIN FOR SVM T RAINING IN I MAGE C ODING (a) (b) (c) 0 200 10 100 0 200 5 5 15 0 0 20 30 25 20 10 20 30 fy (cpd) 10 0 fy (cpd) 30 10 fx (cpd) 0 200 5 10 100 0 200 5 10 100 15 25 10 30 20 30 f (cpd) y 10 0 30 10 0.5 20 30 25 20 5 1 0 0 20 30 0 1.5 15 15 0 fy (cpd) 30 0 fx (cpd) 20 25 20 0 fx (cpd) fx (cpd) 20 30 25 20 30 10 0.5 15 0 5 1 10 100 15 0 1.5 25 20 f (cpd) y 10 30 fy (cpd) 0 0 fx (cpd) fx (cpd) Figure 8: Signal in different domains and the selected support vectors by the SVM models in a block of the Barbara image at 0.3 bits/pix (top row) and 0.5 bits/pix (bottom row). Different domains are analyzed: (a) linear DCT using RKi-1, (b) linear DCT with CSF-SVM, and (c) non-linear perceptual domain with standard ε-SVM (NL-SVR). of the domain is done, accounting for richer (and perceptually independent) coefﬁcients to perform efﬁcient sparse signal reconstruction. It is important to remark that, for a given method (or domain), tightening ε f implies (1) considering more support vectors, and (2) an increase in entropy (top and bottom rows in Figure 8, 0.3 bpp to 0.5 bpp). However, note that the relevant measure is the entropy and not the number of support vectors: even though the number of selected support vectors in the r domain is higher, their variance is lower, thus giving rise to the same entropy after entropy coding. 6. Conclusions In this paper, we have reported a condition on the suitable domain for developing efﬁcient SVM image coding schemes. The so-called diagonal Jacobian condition states that SVM regression with scalar-wise error restriction in a particular domain makes sense only if the transform that maps this domain to an independent coefﬁcient representation is locally diagonal. We have demonstrated that, 63 ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO in general, linear domains do not fulﬁll this condition because non-trivial statistical and perceptual inter-coefﬁcient relations do exist in these domains. This theoretical ﬁnding has been experimentally conﬁrmed by observing that improved compression results are obtained when SVM is applied in a non-linear perceptual domain that starts from the same linear domain used by previously reported SVM-based image coding schemes. These results highlight the relevance of an appropriate image representation choice before SVM learning. Further work is tied to the use of SVM-based coding schemes in statistically, rather than perceptually, independent non-linear ICA domains. In order to do so, local PCA instead of local ICA may be used in the local-to-global differential approach (Malo and Guti´ rrez, 2006) to speed up the e non-linear computation. 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