jmlr jmlr2008 jmlr2008-83 knowledge-graph by maker-knowledge-mining

83 jmlr-2008-Robust Submodular Observation Selection

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Author: Andreas Krause, H. Brendan McMahan, Carlos Guestrin, Anupam Gupta

Abstract: In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations which are robust against a number of possible objective functions. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efﬁcient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NP-complete problems admit efﬁcient algorithms. We show how our algorithm can be extended to handle complex cost functions (incorporating non-unit observation cost or communication and path costs). We also show how the algorithm can be used to near-optimally trade off expected-case (e.g., the Mean Square Prediction Error in Gaussian Process regression) and worst-case (e.g., maximum predictive variance) performance. We show that many important machine learning problems ﬁt our robust submodular observation selection formalism, and provide extensive empirical evaluation on several real-world problems. For Gaussian Process regression, our algorithm compares favorably with state-of-the-art heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDP-based algorithms. c 2008 Andreas Krause, H. Brendan McMahan, Carlos Guestrin and Anupam Gupta. K RAUSE , M C M AHAN , G UESTRIN AND G UPTA Keywords: observation selection, experimental design, active learning, submodular functions, Gaussi

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. [sent-17, score-0.617]

2 We show that many important machine learning problems ﬁt our robust submodular observation selection formalism, and provide extensive empirical evaluation on several real-world problems. [sent-26, score-0.638]

3 K RAUSE , M C M AHAN , G UESTRIN AND G UPTA Keywords: observation selection, experimental design, active learning, submodular functions, Gaussian processes 1. [sent-31, score-0.538]

4 In sensor placement for contamination detection in water distribution networks (cf. [sent-39, score-0.534]

5 While NP-hard, the problem of selecting an optimal set of k observations maximizing a single submodular objective can be approximately solved using a simple greedy forward-selection algorithm, which is guaranteed to perform near-optimally (Nemhauser et al. [sent-46, score-0.719]

6 In this paper, we address the fundamental problem of nonmyopically selecting observations which are robust against such an adversarially chosen submodular objective function. [sent-49, score-0.66]

7 In particular: • We present S ATURATE, an efﬁcient algorithm for the robust submodular observation selection problem. [sent-50, score-0.638]

8 • We extensively evaluate our algorithm on several real-world tasks, including minimizing the maximum posterior variance in Gaussian Process regression, ﬁnding experiment designs which are robust with respect to parameter uncertainty, and sensor placement for outbreak detection. [sent-54, score-0.652]

9 In Section 2, we formulate the robust submodular observation selection problem, and in Section 3, we analyze its hardness. [sent-56, score-0.638]

10 In Section 6, we discuss how many important machine learning problems are instances of our robust submodular observation selection formalism. [sent-58, score-0.638]

11 1), and then introduce the robust submodular observation selection (RSOS) problem (Section 2. [sent-68, score-0.638]

12 (1978) prove a convenient characterization of submodular functions: F is submodular if and only if for all A ⊆ B ⊆ V and s ∈ V \ B it holds that F(A ∪{s})−F(A ) ≥ F(B ∪ {s}) − F(B ). [sent-86, score-1.022]

13 (1978) states that for submodular functions, the greedy algorithm achieves a constant factor approximation: 2764 ROBUST S UBMODULAR O BSERVATION S ELECTION Theorem 2 (Nemhauser et al. [sent-97, score-0.638]

14 1978) In the case of any normalized, monotonic submodular function F, the set AG obtained by the greedy algorithm achieves at least a constant fraction (1 − 1/e) of the objective value obtained by the optimal solution, that is, F(AG ) ≥ (1 − 1/e) max F(A ). [sent-98, score-0.731]

15 Consider a case with two submodular functions, F1 and F2 , where the set of observations is V = {s1 , s2 ,t1 ,t2 }. [sent-130, score-0.511]

16 As trivial lower / and upper bounds for the RSOS problem we can initially set cmin = 0 ≤ mini Fi (0), and cmax = mini Fi (V ), due to monotonicity of the Fi . [sent-179, score-0.386]

17 The key insight is that these truncated functions Fi,c remain monotonic and submodular (Fujito, 2000). [sent-184, score-0.579]

18 Since monotonic submodular functions are closed under convex combinations, F c is also submodular and monotonic. [sent-187, score-1.09]

19 (6) Problems of the form minA |A | such that F(A ) = F(V ), where F is a monotonic submodular function, are called submodular covering problems. [sent-192, score-1.09]

20 Since F c satisﬁes these requirements, the M IN C OVERc Problem (6) is an instance of such a submodular covering problem. [sent-193, score-0.535]

21 Using Wolsey’s result and the observation that α can be chosen independently of the truncation threshold c, we ﬁnd: Lemma 4 Given integral valued1 monotonic submodular functions F1 , . [sent-196, score-0.606]

22 GPC (F c , c) / A ← 0; while F c (A ) < c do foreach s ∈ V \ A do δs ← F c (A ∪ {s}) − F c (A ); A ← A ∪ {argmaxs δs }; end Algorithm 1: The greedy submodular partial cover (GPC) algorithm. [sent-218, score-0.638]

23 We call Algorithm 2, which formalizes this procedure, the submodular saturation algorithm (S ATURATE), as the algorithm considers the truncated objectives Fi,c , and chooses sets which saturate all these objectives. [sent-219, score-0.723]

24 The total number of submodular function evaluations is O |V |2 m log m min Fi (V ) i . [sent-223, score-0.511]

25 1, if we had an exact algorithm for submodular coverage, then we would set α = 1, and S ATURATE would return the optimal solution to the RSOS problem. [sent-227, score-0.511]

26 Since, in our experience, the greedy algorithm for optimizing submodular functions works very effectively (cf. [sent-228, score-0.706]

27 , Fm , k, α) / cmin ← 0; cmax ← mini Fi (V ); Abest ← 0; 1 while (cmax − cmin ) ≥ m do c ← (cmin + cmax )/2; 1 Deﬁne F c (A ) ← m ∑i min{Fi (A ), c}; A ← GPC(F c , c); if |A | > αk then cmax ← c; else cmin ← c; Abest = A end end Algorithm 2: The Submodular Saturation algorithm. [sent-235, score-0.673]

28 In addition, for many submodular functions Fi , such as the variance reduction, it is often cheaper to compute F c (XA ∪s ) − F c (XA ) instead of F c (XA ∪s ). [sent-242, score-0.591]

29 Taken together, Theorem 3 and Theorem 6, provide strong theoretical evidence that S ATURATE achieves best possible theoretical guarantees for the problem of maximizing the minimum over a set of submodular functions. [sent-253, score-0.571]

30 For a particular location s, Das and Kempe (2008) show that the variance reduction Fs (often) is a monotonic submodular function. [sent-269, score-0.639]

31 In these cases, we can associate, with each parameter setting θ, a different submodular objective function Fθ , for example, Fθ (A ) = I(XA ; XU | θ), and we might want to select a set A which simultaneously performs well for all possible parameter values. [sent-290, score-0.56]

32 where Σ(θ0 ) is the joint covariance of observations and parameters after linearization around θ0 ; thus, Fθ0 is the sum of marginal parameter variance reductions, which are (often) individually monotonic and submodular (Das and Kempe, 2008), and so Fθ0 is monotonic and submodular as well. [sent-308, score-1.199]

33 Our goal in sensor placement is then to select a set of sensors A such that the minimum penalty 2774 ROBUST S UBMODULAR O BSERVATION S ELECTION reduction is as large as possible, that is, we want to select A ∗ = argmax min Fi (A ). [sent-331, score-0.529]

34 |A |≤k i∈I In other words, an adversary observes our sensor placement A , and then decides on an intrusion i for which our utility Fi (A ) is as small as possible. [sent-332, score-0.393]

35 , the utility for placing a set of sensors), and the set B ⊆ V of failing sensors, we can deﬁne a new function FB (A ) = F(A \ B ), corresponding to the (reduced) utility of placement A after the sensor failures. [sent-340, score-0.388]

36 Hence, the problem of optimizing sensor placements which are robust to sensor failures results in a problem of simultaneously maximizing a collection of submodular functions, for example, for the worst-case failure of k < k sensors we solve A ∗ = argmax min FB (A ). [sent-342, score-1.518]

37 Krause and Guestrin (2005) show, that in a large class of graphical models, the information gain F(A ) is in fact a submodular function. [sent-348, score-0.511]

38 1 I MPROVED G UARANTEES FOR S ENSOR FAILURES As discussed above, in principle, we could ﬁnd a placement robust to single sensor failures by using S ATURATE to (approximately) solve A ∗ = argmax min Fs (A ). [sent-360, score-0.523]

39 , we place far fewer sensors than we have possible sensor locations). [sent-364, score-0.381]

40 This modiﬁed objective is still monotonic and submodular ˆ ˆ when restricted to sets of size at most k. [sent-368, score-0.604]

41 This more general result holds, since the analysis of the greedy algorithm for submodular covering of Wolsey (1982), which we used to prove Lemma 4, applies to the more general setting of non-uniform cost functions. [sent-405, score-0.687]

42 The key insight of the S ATURATE algorithm is that the nonsubmodular robust optimization problem can be approximately solved by solving a submodular covering problem. [sent-418, score-0.635]

43 Let us summarize our analysis as follows: Proposition 7 Assume we have an algorithm which, given a monotonic submodular function F and a cost function g, returns a solution A such that F(A ) = F(V ) and g(A ) ≤ αF min A :F(A )=F(V ) g(A ), where αF depends on the function F. [sent-425, score-0.58]

44 In these problems, instead of a general submodular objective function, the special case of a modular (additive) function F is optimized: A ∗ = argmax F(A ). [sent-433, score-0.587]

45 Since modular functions are a special case of submodular functions, such problems can be approximately solved using Proposition 7. [sent-439, score-0.535]

46 Note that F cwc ,cac is a submodular function, and hence (11) is a submodular covering problem, which can be approximately solved using the greedy algorithm. [sent-461, score-1.234]

47 The total number of submodular function evaluations is O |V |2 m log(∑i Fi (V )) . [sent-465, score-0.511]

48 We consider three data sets: [T] temperature data from a deployment of 52 sensors at Intel Research Berkeley, [P] Precipitation data from the Paciﬁc Northwest of the United States (Widmann and Bretherton, 1999) and [L] temperature data from the NIMS sensor node (Harmon et al. [sent-472, score-0.381]

49 Note, that, whereas the maximum variance reduction is non-submodular, the average variance reduction is (often) submodular (Das and Kempe, 2008), and hence the greedy algorithm can 5. [sent-570, score-0.806]

50 While the classical E-optimality is not submodular (Krause et al. [sent-605, score-0.511]

51 , 2007b), Bayesian A-optimality is (usually) submodular (Das and Kempe, 2008; Krause et al. [sent-606, score-0.511]

52 , 2006) was organized as an international challenge to ﬁnd the best sensor placements for a real (but anonymized) metropolitan water distribution network, consisting of 12,527 nodes. [sent-664, score-0.445]

53 The objective functions Z1 and Z2 are in fact submodular for a ﬁxed intrusion scenario (Leskovec et al. [sent-670, score-0.614]

54 Z2 Figure 10: (a,b) compare S ATURATE, greedy and SA in the water network setting, when optimizing worst-case detection time (Z1 , (a)) and affected population (Z2 , (b)). [sent-711, score-0.4]

55 We also use the greedy algorithm to optimize sensor placements, ignoring possible sensor failures. [sent-745, score-0.617]

56 sensor failure, and in the case of a single, worst-case sensor failure. [sent-747, score-0.49]

57 However, as the number of sensors increases, the placement scores optimized using S ATURATE for sensor failures quickly approach those of Greedy in the case of no sensor failures. [sent-750, score-0.777]

58 Hence, even if only a small number of sensors are placed, S ATURATE can quickly exploit redundancy and ﬁnd sensor placements, which perform well both with and without sensor failures. [sent-751, score-0.626]

59 On the other hand, when not taking sensor failures into account, such failures can drastically diminish the utility of a placed set of sensors. [sent-752, score-0.386]

60 We then use the greedy algorithm to optimize sensor placement of increasing sizes for the single model Σ, optimizing the average variance reduction objective function. [sent-768, score-0.642]

61 While the average RMS error is roughly equal for both placements, the maximum RMS error is larger for the greedy sensor placement, as compared to the robust placement of S ATURATE, especially for small numbers of sensors (six and less sensors). [sent-792, score-0.701]

62 Related Work In this section, we review related work in submodular function optimization, robust discrete optimization, robust methods in statistics, sensor placement, game theory and machine learning. [sent-827, score-0.956]

63 (1978) and Wolsey (1982) analyze the greedy algorithm for optimizing monotonic submodular functions. [sent-830, score-0.726]

64 Lov´ sz (1983) discusses the relationship between a submodular functions and convexity. [sent-831, score-0.535]

65 He also shows that under certain conditions, the minimum of two submodular functions remains submodular (and hence can be efﬁciently optimized using the greedy algorithm). [sent-832, score-1.173]

66 Fujito (2000) uses submodularity of truncated functions to ﬁnd sets with partial submodular coverage; however, they do not consider the case of multiple objectives, which we address in this paper. [sent-834, score-0.573]

67 (2001) consider covering problems for a generalization of submodular functions; they use a similar binary search technique combined with multiple applications of the greedy algorithm. [sent-836, score-0.662]

68 Their approach does not apply to maximizing the minimum over a set of submodular functions. [sent-837, score-0.543]

69 Golovin and Streeter (2008) present an algorithm for online maximization of a single submodular set function. [sent-838, score-0.511]

70 An interesting question for future work would be to investigate whether our approach for maximizing the minimum over a collection of submodular functions can be generalized to an online setting as well. [sent-839, score-0.567]

71 A large part of the theory of optimizing submodular functions is concerned with minimizing instead of maximizing a single submodular function. [sent-840, score-1.145]

72 Queyranne (1995) present the ﬁrst algorithm for minimizing symmetric submodular functions; Iwata et al. [sent-841, score-0.534]

73 (2001) and Schrijver (2000) present combinatorial algorithms for minimizing arbitrary (not necessarily symmetric) submodular functions. [sent-842, score-0.534]

74 2 Robust Discrete Optimization Robust optimization of submodular functions is an instance of a robust discrete optimization problem. [sent-844, score-0.635]

75 However, their results do not transfer to our setting of robust submodular optimization, since in this case, even though non-robust solutions are (1 − 1/e) approximable, the non-robust formulation does not admit any approximation guarantees (cf. [sent-853, score-0.639]

76 (2006) consider the problem of robust sensor placements in water distribution networks. [sent-882, score-0.545]

77 They formulate Mixed Integer Programs for selecting sensor placements robust against uncertainty in adversarial strategies and in water demands. [sent-883, score-0.611]

78 (2006) consider different notions of robustness in the context of water distribution networks, intended to remove some of the pessimistic assumptions of purely robust sensor placements. [sent-887, score-0.45]

79 However, they do not discuss any implications regarding approximation guarantees, and do not consider the case of arbitrary submodular functions. [sent-895, score-0.511]

80 ,Am ) i Feige (2006) develop a randomized 2-approximation for subadditive and 1 − 1/e approximation for submodular valuation functions. [sent-931, score-0.511]

81 However, their approach does not apply to our robust submodular observation selection setting. [sent-939, score-0.638]

82 The algorithm of Queyranne (1995) for minimizing symmetric submodular functions has been used for learning graphical models by Narasimhan and Bilmes (2004) and for clustering by Narasimhan et al. [sent-944, score-0.558]

83 2793 K RAUSE , M C M AHAN , G UESTRIN AND G UPTA We are not aware of any work on optimizing the minimum over a collection of submodular functions. [sent-946, score-0.555]

84 A note by Seeger (2004) proves that the greedy algorithm in the IVM optimizes a submodular function. [sent-956, score-0.638]

85 In previous work, the authors demonstrated that several important observation selection objectives are submodular (Krause et al. [sent-973, score-0.583]

86 (2007) consider the problem of planning informative paths for multiple robots, where the informativeness is modeled using a submodular objective function, and a constraint on path lengths connecting the locations is speciﬁed. [sent-981, score-0.642]

87 4, both approaches can be made robust with respect to a worst-case submodular function. [sent-984, score-0.611]

88 Conclusions In this paper, we considered the RSOS problem of robustly selecting observations which are informative with respect to a worst-case submodular objective function. [sent-986, score-0.583]

89 In each of these settings, the individual objectives are submodular and can be approximated well using, for example, the greedy algorithm; the robust objective, however, is not submodular. [sent-988, score-0.783]

90 We apply Wolsey’s result to the monotonic submodular function F c . [sent-1018, score-0.555]

91 hold that mini Fi (Abest ) ≥ mini Fi (A For the running time, since at the ﬁrst iteration, cmax − cmin ≤ mini Fi (V ), and cmax − cmin is 1 halved during each iteration, it follows that after 1 + log2 m mini Fi (V ) iterations, cmax − cmin < m , at which point the algorithm terminates. [sent-1024, score-0.964]

92 i The guarantee α is obtained from the analysis of the greedy submodular coverage algorithm of Wolsey (1982), similar to Lemma 4. [sent-1030, score-0.675]

93 Robust optimization of contaminant sensor placement for community water systems. [sent-1129, score-0.465]

94 A combinatorial strongly polynomial algorithm for minimizing submodular functions. [sent-1264, score-0.534]

95 An analysis of the approximations for maximizing submodular set functions. [sent-1387, score-0.543]

96 Battle of water sensor networks: A design challenge for engineers and algorithms. [sent-1394, score-0.41]

97 The battle of the water sensor networks (BWSN): A design challenge for engineers and algorithms. [sent-1442, score-0.41]

98 A combinatorial algorithm minimizing submodular functions in strongly polynomial time. [sent-1507, score-0.558]

99 Formulation and optimization of robust sensor placement problems for contaminant warning systems. [sent-1564, score-0.46]

100 An analysis of the greedy algorithm for the submodular set covering problem. [sent-1583, score-0.662]

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Abstract: Conventional SVM-based image coding methods are founded on independently restricting the distortion in every image coefﬁcient at some particular image representation. Geometrically, this implies allowing arbitrary signal distortions in an n-dimensional rectangle deﬁned by the ε-insensitivity zone in each dimension of the selected image representation domain. Unfortunately, not every image representation domain is well-suited for such a simple, scalar-wise, approach because statistical and/or perceptual interactions between the coefﬁcients may exist. These interactions imply that scalar approaches may induce distortions that do not follow the image statistics and/or are perceptually annoying. Taking into account these relations would imply using non-rectangular εinsensitivity regions (allowing coupled distortions in different coefﬁcients), which is beyond the conventional SVM formulation. In this paper, we report a condition on the suitable domain for developing efﬁcient SVM image coding schemes. We analytically demonstrate that no linear domain fulﬁlls this condition because of the statistical and perceptual inter-coefﬁcient relations that exist in these domains. This theoretical result is experimentally conﬁrmed by comparing SVM learning in previously reported linear domains and in a recently proposed non-linear perceptual domain that simultaneously reduces the statistical and perceptual relations (so it is closer to fulﬁlling the proposed condition). These results highlight the relevance of an appropriate choice of the image representation before SVM learning. Keywords: image coding, non-linear perception models, statistical independence, support vector machines, insensitivity zone c 2008 Gustavo Camps-Valls, Juan Guti´ rrez, Gabriel G´ mez-P´ rez and Jes´ s Malo. e o e u ´ ´ ´ C AMPS -VALLS , G UTI E RREZ , G OMEZ -P E REZ AND M ALO 1. Problem Statement: The Diagonal Jacobian Condition Image coding schemes based on support vector machines (SVM) have been successfully introduced in the literature. SVMs have been used in the spatial domain (Robinson and Kecman, 2000), in the block-DCT domain (Robinson and Kecman, 2003), and in the wavelet domain (Ahmed, 2005; Jiao et al., 2005). These coding methods take advantage of the ability of the support vector regression (SVR) algorithm for function approximation using a small number of parameters (signal samples, or ¨ support vectors) (Smola and Scholkopf, 2004). In all current SVM-based image coding techniques, a representation of the image is described by the entropy-coded weights associated to the support vectors necessary to approximate the signal with a given accuracy. Relaxing the accuracy bounds reduces the number of needed support vectors. In a given representation domain, reducing the number of support vectors increases the compression ratio at the expense of bigger distortion (lower image quality). By applying the standard SVR formulation, a certain amount of distortion in each sample of the image representation is allowed. In the original formulation, scalar restrictions on the errors are introduced using a constant ε-insensitivity value for every sample. ´ Recently, this procedure has been reﬁned by Gomez-P´ rez et al. (2005) using a proﬁle-dependent e SVR (Camps-Valls et al., 2001) that considers a different ε for each sample or frequency. This frequency-dependent insensitivity, ε f , accounts for the fact that, according to simple (linear) perception models, not every sample in linear frequency domains (such as DCT or wavelets) contributes to the perceived distortion in the same way. Despite different domains have been proposed for SVM training (spatial domain, block-DCT and wavelets) and different ε insensitivities per sample have been proposed, in conventional SVR formulation, the particular distortions introduced by regression in the different samples are not coupled. In all the reported SVM-based image coding schemes, the RBF kernel is used and the penalization parameter is ﬁxed to an arbitrarily large value. In this setting, considering n-sample signals as n-dimensional vectors, the SVR guarantees that the approximated vectors are conﬁned in n-dimensional rectangles around the original vectors. These rectangles are just n-dimensional cubes in the standard formulation or they have certain elongation if different ε f are considered in each axis, f . Therefore, in all the reported SVM-based coding methods, these rectangles are always oriented along the axes of the (linear) image representation. According to this, a common feature of these (scalar-wise) approaches is that they give rise to decoupled distortions in each dimension. P´ rez-Cruz et al. (2002) proposed a hyperspherical insensitivity zone to correct the penalization e factor in each dimension of multi-output regression problems, but again, restrictions to each sample were still uncoupled. This scalar-wise strategy is not the best option in domains where the different dimensions of the image representation are not independent. For instance, consider the situation where actually independent components, r f , are obtained from a given image representation, y, applying some eventually non-linear transform, R: R y −→ r. In this case, SVM regression with scalar-wise error restriction makes sense in the r domain. However, the original y domain will not be suitable for the standard SVM regression unless the matrix ∇R is diagonal (up to any permutation of the dimensions, that is, only one non-zero element per row). 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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