nips nips2009 nips2009-41 knowledge-graph by maker-knowledge-mining

41 nips-2009-Bayesian Source Localization with the Multivariate Laplace Prior

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Author: Marcel V. Gerven, Botond Cseke, Robert Oostenveld, Tom Heskes

Abstract: We introduce a novel multivariate Laplace (MVL) distribution as a sparsity promoting prior for Bayesian source localization that allows the speciﬁcation of constraints between and within sources. We represent the MVL distribution as a scale mixture that induces a coupling between source variances instead of their means. Approximation of the posterior marginals using expectation propagation is shown to be very efﬁcient due to properties of the scale mixture representation. The computational bottleneck amounts to computing the diagonal elements of a sparse matrix inverse. Our approach is illustrated using a mismatch negativity paradigm for which MEG data and a structural MRI have been acquired. We show that spatial coupling leads to sources which are active over larger cortical areas as compared with an uncoupled prior. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We represent the MVL distribution as a scale mixture that induces a coupling between source variances instead of their means. [sent-2, score-0.646]

2 Approximation of the posterior marginals using expectation propagation is shown to be very efﬁcient due to properties of the scale mixture representation. [sent-3, score-0.287]

3 We show that spatial coupling leads to sources which are active over larger cortical areas as compared with an uncoupled prior. [sent-6, score-0.54]

4 Let q, p, and t denote the number of sensors, sources and time points, respectively. [sent-8, score-0.223]

5 Sensor readings Y ∈ Rq×t and source currents S ∈ Rp×t are related by Y = XS + E (1) where X ∈ Rq×p is a lead ﬁeld matrix that represents how sources project onto the sensors and E ∈ Rq×t represents sensor noise. [sent-9, score-0.77]

6 Unfortunately, localizing distributed sources is an ill-posed inverse problem that only admits a unique solution when additional constraints are deﬁned. [sent-10, score-0.313]

7 In a Bayesian setting, these constraints take the form of a prior on the sources [3, 19]. [sent-11, score-0.306]

8 Popular choices of prior source amplitude distributions are Gaussian or Laplace priors, whose MAP estimates correspond to minimum norm and minimum current estimates, respectively [18]. [sent-12, score-0.442]

9 In contrast, minimum current estimates lead to focal source estimates that may be scattered too much throughout the brain volume [9]. [sent-14, score-0.521]

10 In this paper, we take the Laplace prior as our point of departure for Bayesian source localization (instead of using just the MAP estimate). [sent-15, score-0.428]

11 Here, in contrast, we assume a multivariate Laplace distribution over all sources, which allows sources to be coupled. [sent-17, score-0.357]

12 Since the posterior cannot be computed exactly, we formulate an efﬁcient expectation propagation 1 algorithm [12] which allows us to approximate the posterior of interest for very large models. [sent-20, score-0.21]

13 Efﬁciency arises from the block diagonal form of the approximate posterior covariance matrix due to properties of the scale mixture representation. [sent-21, score-0.301]

14 The computational bottleneck then reduces to computation of the diagonal elements of a sparse matrix inverse, which can be solved through Cholesky decomposition of a sparse matrix and application of the Takahashi equation [17]. [sent-22, score-0.232]

15 2 Bayesian source localization In a Bayesian setting, the goal of source localization is to estimate the posterior p(S | Y, X, Σ, Θ) ∝ p(Y | S, X, Σ)p(S | Θ) (2) where the likelihood term p(Y | S) = t N (yt | Xst , Σ) factorizes over time and Σ represents sensor noise. [sent-25, score-0.921]

16 1 The source localization problem can be formulated as a (Bayesian) linear regression problem where the source currents s play the role of the regression coefﬁcients and rows of the lead ﬁeld matrix X can be interpreted as covariates. [sent-31, score-0.84]

17 In the following, we deﬁne a multivariate Laplace distribution, represented in terms of a scale mixture, as a convenient prior that incorporates both spatio-temporal and sparsity constraints. [sent-32, score-0.276]

18 L (s | λ) = (4) Eltoft et al [5] deﬁned the multivariate Laplace distribution as a scale mixture of a multivariate √ Gaussian given by zΣ1/2 s where s is a standard normal multivariate Gaussian, Σ is a positive deﬁnite matrix, and z is drawn from a univariate exponential distribution. [sent-36, score-0.604]

19 In contrast, we use an alternative formulation of the multivariate Laplace distribution that couples the variances of the sources rather than the source currents themselves. [sent-38, score-0.809]

20 For an uncoupled multivariate Laplace distribution, this generalization reads L (s | λ) = 2 N si | 0, u2 + vi N vi | 0, 1/λ2 N ui | 0, 1/λ2 i dudv (5) i such that each source current si gets assigned scale variables ui and vi . [sent-41, score-3.115]

21 We can interpret the scale variables corresponding to source i as indicators of its relevance: the larger (the posterior estimate 2 of) u2 + vi , the more relevant the corresponding source. [sent-42, score-0.802]

22 2 f s1 s2 ··· sp g1 g2 ··· gp ··· up u1 v1 u2 v2 h1 vp h2 Figure 1: Factor graph representation of Bayesian source localization with a multivariate Laplace prior. [sent-44, score-0.508]

23 Factors gi correspond to the coupling between sources and scales. [sent-46, score-0.445]

24 Factors h1 and h2 represent the (identical) multivariate Gaussians on u and v with prior precision matrix J. [sent-47, score-0.288]

25 This deﬁnition yields a coupling in the magnitudes of the source currents through their variances. [sent-50, score-0.604]

26 Note that this approach is deﬁning the multivariate Laplace with the help of a multivariate exponential distribution [10]. [sent-52, score-0.292]

27 3 Approximate inference Our goal is to compute posterior marginals for sources si as well as scale variables ui and vi in order to determine source relevance. [sent-57, score-1.753]

28 Using EP, we will approximate p(z) with q (z) ∝ t0 (z) i ti (z), where ¯ the ti (z) are Gaussian functions as well. [sent-68, score-0.36]

29 Equation (6) introduces 2p auxiliary Gaussian variables (u, v) that are coupled to the si ’s by p non-Gaussian factors, thus, we have to approximate p terms. [sent-70, score-0.457]

30 The multivariate Laplace distribution deﬁned in [5] introduces one auxiliary variable and couples all the si sj terms to it, therefore, it would lead to p2 non-Gaussian terms to be approximated. [sent-71, score-0.547]

31 Moreover, as we will see below, the a priori independence of u and v and 3 the form of the terms ti (z) results in an approximation of the posterior with the same block-diagonal structure as that of t0 (z). [sent-72, score-0.225]

32 ¯ ¯i ¯ In each step, EP updates ti with t∗ by deﬁning q \i ∝ t0 (z) \i tj , minimizing KL ti q \i q ∗ with ¯i respect to q ∗ and setting t∗ ∝ q ∗ /q \i . [sent-73, score-0.355]

33 It can be shown that when ti depends only on a subset of ¯ variables zi (in our case on zi = (si , ui , vi )) then so does ti . [sent-74, score-1.289]

34 The minimization of the KL divergence then boils down to the minimization of KL ti (zi )q \i (zi ) q ∗ (zi ) with respect to q ∗ (zi ) ¯ ¯i and ti is updated to t∗ (zi ) ∝ q ∗ (zi )/q \i (zi ). [sent-75, score-0.33]

35 , q ∗ (si , ui , vi ) is a Gaussian with the same mean and covariance matrix as q i (zi ) ∝ ti (zi )q \i (zi ). [sent-78, score-0.982]

36 We will now work out the EP update for the i-th term approximation in more detail to show by ¯ induction that ti (si , ui , vi ) factorizes into independent terms for si , ui , and vi . [sent-84, score-2.08]

37 Since ui and vi play exactly the same role, it is also easy to see that the term approximation is always symmetric in ui and vi . [sent-85, score-1.565]

38 Let us suppose that q (si , ui , vi ) and consequently q \i (si , ui , vi ) factorizes into independent terms for si , ui , and vi , e. [sent-86, score-2.663]

39 , we can write 2 2 2 q \i (si , ui , vi ) = N (si | mi , σi )N (ui | 0, νi )N (vi | 0, νi ). [sent-88, score-0.797]

40 (9) ¯ By initializing ti (si , ui , vi ) = 1, we have q(z) ∝ t0 (z) and the factorization of q \i (si , ui , vi ) follows directly from the factorization of t0 (z) into independent terms for s, u, and v. [sent-89, score-1.707]

41 (10) (11) 2 Since q i (ui , vi ) only depends on u2 and vi and is thus invariant under sign changes of ui and vi , i we must have E [ui ] = E [vi ] = 0, as well as E [ui vi ] = 0. [sent-92, score-1.98]

42 Because of symmetry, we further have 2 2 2 E u2 = E vi = (E u2 + E vi )/2. [sent-93, score-0.806]

43 Since q i (ui , vi ) can be expressed as a function of u2 + vi i i i only, this variance can be computed from (11) using one-dimensional Gauss-Laguerre numerical quadrature [15]. [sent-94, score-0.893]

44 The ﬁrst and second moments of si conditioned upon ui and vi follow directly from (10). [sent-95, score-1.136]

45 Because both (10) and (11) are invariant under sign changes of ui and vi , we must have 2 E [si ui ] = E [si vi ] = 0. [sent-96, score-1.542]

46 Furthermore, since the conditional moments again depend only on u2 + vi , i 2 also E [si ] and E si can be computed with one-dimensional Gauss-Laguerre integration. [sent-97, score-0.768]

47 Summarizing, we have shown that if the old term approximations factorize into independent terms for si , ui , ¯i and vi , the new term approximation after an EP update, t∗ (si , ui , vi ) ∝ q ∗ (si , ui , vi )/q \i (si , ui , vi ), must do the same. [sent-98, score-3.427]

48 Furthermore, given the cavity distribution q \i (si , ui , vi ), all required moments can be computed using one-dimensional numerical integration. [sent-99, score-0.878]

49 The crucial observation here is that the terms ti (si , ui , vi ) introduce dependencies between si and (ui , vi ), as expressed in Eqs. [sent-100, score-1.636]

50 That is, also when the expectations are taken with respect to the exact p(s, u, v) we have E [ui ] = E [vi ] = E [ui vi ] = E [si ui ] = E [si vi ] = 0 and 2 2 E u2 = E vi . [sent-103, score-1.577]

51 The variance of the scales E u2 + vi determines the amount of regularization i i on the source parameter si such that large variance implies little regularization. [sent-104, score-1.079]

52 Calculating q \i (si , ui , vi ) 4 requires the computation of the marginal moments q(si ), q(ui ) and q(vi ). [sent-107, score-0.839]

53 This precision matrix has the block-diagonal form   XT X/σ 2 + Ks 0 0  K= (12) 0 λ2 J + Ku 0 2 0 0 λ J + Kv where J is a sparse precision matrix which determines the coupling, and Ks , Ku , and Kv = Ku are diagonal matrices that contain the contributions of the term approximations. [sent-109, score-0.297]

54 4 Experiments Returning to the source localization problem, we will show that the MVL prior can be used to induce constraints on the source estimates. [sent-113, score-0.74]

55 The lead ﬁeld matrix is deﬁned for the three x, y, and z orientations in each of the source locations and was normalized to correct for depth bias. [sent-124, score-0.447]

56 In the next section, we compare source estimates for the MMN difference wave that have been obtained when using either a decoupled or a coupled MVL prior. [sent-127, score-0.554]

57 For ease of exposition, we focus on a spatial prior induced by the coupling of neighboring sources. [sent-128, score-0.354]

58 Differences in the source estimates will therefore arise only from the form of the 11589 × 11589 sparse precision matrix J. [sent-130, score-0.46]

59 The ﬁrst estimate is obtained by assuming that there is no coupling between elements of the lead ﬁeld matrix, such that J = I. [sent-131, score-0.26]

60 The second estimate is obtained by assuming a coupling between neighboring sources i and j within the brain volume with ﬁxed strength c. [sent-133, score-0.577]

61 This coupling is speciﬁed through the unnormalized precision matrix ˆ ˆ ˆ ˆ ˆ ˆ J by assuming Jix ,jx = Jiy ,jy = Jiz ,jz = −c while diagonal elements Jii are set to 1 − j=i Jij . [sent-134, score-0.362]

62 2 This prior dictates that the magnitude of the variances of the source currents are coupled between sources. [sent-135, score-0.558]

63 For the coupling strength c, we use correlation as a guiding principle. [sent-136, score-0.271]

64 Speciﬁcally, correlation between sources si and sj is given by ˆ rij = J−1 ˆ / J−1 ij 1 2 ii ˆ J−1 1 2 . [sent-138, score-0.569]

65 Note that this also leads to more distant sources having non-zero correlations. [sent-141, score-0.248]

66 5 J L C Figure 2: Spatial coupling leads to the normalized precision matrix J with coupling of neighboring source orientations in the x, y, and z directions. [sent-143, score-0.954]

67 The correlation matrix C shows the correlations between the source orientations. [sent-145, score-0.411]

68 neighboring sources is motivated by the notion that we expect neighboring sources to be similarly though not equivalently involved for a given task. [sent-147, score-0.544]

69 Figure 2 demonstrates how a chosen coupling leads to a particular structure of J, where irregularities in J are caused by the structure of the imaged brain volume. [sent-149, score-0.304]

70 The correlation matrix C shows the correlations between the sources induced by the structure of J. [sent-153, score-0.351]

71 Zeros in the correlation matrix arise from the independence between source orientations x, y, and z. [sent-154, score-0.431]

72 5 Results Figure 3 depicts the difference wave that was obtained by subtracting the trial average for standard tones from the trial average for deviant tones. [sent-155, score-0.232]

73 6 Figure 4: Source estimates using a decoupled prior (top) or a coupled prior (bottom). [sent-165, score-0.323]

74 Figure 5: Relative variance using a decoupled prior (top) or a coupled prior (bottom). [sent-167, score-0.328]

75 We now proceed to localizing the sources of the activation induced by mismatch negativity. [sent-170, score-0.328]

76 Figure 4 depicts the localized sources when using either a decoupled MVL prior or a coupled MVL prior. [sent-171, score-0.481]

77 The coupled spatial prior leads to stronger source currents that are spread over a larger brain volume. [sent-172, score-0.632]

78 MVL source localization has correctly identiﬁed the source over left temporal cortex but does not capture the source over right temporal cortex that is also hypothesized to be present (cf. [sent-173, score-1.004]

79 Differences between the decoupled and the coupled prior become more salient when we look at the relative variance of the auxiliary variables as shown in Fig. [sent-178, score-0.319]

80 Relative variance is deﬁned here as posterior variance minus prior variance of the auxiliary variables, normalized to be between zero and one. [sent-180, score-0.303]

81 This measure indicates the change in magnitude of the variance of the auxiliary variables, and thus indirectly that of the sources via Eq. [sent-181, score-0.316]

82 Since only sources with non-zero contributions should have high variance, this measure can be used to indicate the relevance of a source. [sent-183, score-0.255]

83 Figure 5 7 shows that temporal sources in both left and right hemispheres are relevant. [sent-184, score-0.255]

84 The relevance of the temporal source in the right hemisphere becomes more pronounced when using the coupled prior. [sent-185, score-0.432]

85 6 Discussion In this paper, we introduced a multivariate Laplace prior as the basis for Bayesian source localization. [sent-186, score-0.471]

86 By formulating this prior as a scale mixture we were able to approximate posteriors of interest using expectation propagation in an efﬁcient manner. [sent-187, score-0.291]

87 Computation time is mainly inﬂuenced by the sparsity structure of the precision matrix J which is used to specify interactions between sources by coupling their variances. [sent-188, score-0.6]

88 It was shown that coupling of neighboring sources leads to source estimates that are somewhat more spatially smeared as compared with a decoupled prior. [sent-190, score-0.932]

89 However, posterior marginals can still be used to exclude irrelevant sources since these will typically have a mean activation close to zero with small variance. [sent-195, score-0.346]

90 Note that it is straightforward to impose other constraints since this only requires the speciﬁcation of suitable interactions between sources through J. [sent-199, score-0.275]

91 For instance, the spatial prior could be made more realistic by taking anatomical constraints into account or by the inclusion of coupling between sources over time. [sent-200, score-0.557]

92 Other constraints that can be implemented with our approach are the coupling of individual orientations within a source, or even the coupling of source estimates between different subjects. [sent-201, score-0.852]

93 Coupling of source orientations has been realized before in [9] through an 1 / 2 norm, although not using a fully Bayesian approach. [sent-202, score-0.336]

94 In future work, we aim to examine the effect of the proposed priors and optimize the regularization and coupling parameters via empirical Bayes [4]. [sent-203, score-0.248]

95 Other directions for further research are inclusion of the noise variance in the optimization procedure and dealing with the depth bias that often arises in distributed source models in a more principled way. [sent-204, score-0.379]

96 To obtain a generalization of the univariate Laplace distribution, we used a multivariate exponential distribution of the scales, to be compared with the multivariate log-normal distribution in [11]. [sent-207, score-0.339]

97 Since (the efﬁciency of) our method for approximate inference only depends on the sparsity of the multivariate scale distribution, and not on its precise form, it should be feasible to compute approximate marginals for the model presented in [11] as well. [sent-210, score-0.345]

98 Concluding, we believe the scale mixture representation of the multivariate Laplace distribution to be a promising approach to Bayesian distributed source localization. [sent-211, score-0.521]

99 Combining sparsity and rotau tional invariance in EEG/MEG source reconstruction. [sent-282, score-0.315]

100 Classes of multivariate exponential and multivariate geometric distributions derived from Markov processes. [sent-287, score-0.292]

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