Mathematics in Independent Component Analysis

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38 Chapter 1. Statistical machine learning of biomedical data IEEE SIGNAL PROCESSING LETTERS, VOL. X, NO. XX, XXXX 2 IEEE SIGNAL PROCESSING LETTERS, VOL. X, NO. XX, XXXX 2 x2 x2 α α x1 x1 α α F (α) F (α) 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 ∆(α) mean ∆(α) mean ∆(α) median ∆(α) median 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 α 0.6 0.8 α (a) circle histogram for α = 0.4 (b) comparison of mean and median Fig. 1. Mean- versus median-based clustering. We consider the mixture x of two independent gamma-distributed sources (γ = 0.5, 10 5 (a) circle histogram for α = 0.4 (a) circle histogram for α = 0.4 (b) comparison of mean and median Fig. 1. Mean- versus median-based clustering. We consider the mixture x of two independent gamma-distributed sources samples) using a mixing matrix A with columns inclined by α and (π/2 − α) respectively. (a) shows the mixture (γ = 0.5, 10 density for α = 0.4 after projection onto the circle. For α ∈ [0, π/4), (b) compares the error when estimating A by the mean and the median of the projected density in the receptive field F (α) = (−π/4, π/4) of the known column a1 of A. The former is the k-means convergence criterion. 5 (b) comparison of mean and median Figure 1.18: Mean- versus median-based clustering. We consider the mixture x(t) of two in- samples) using a mixing matrix A with columns inclined by α and (π/2 − α) respectively. (a) shows the mixture dependent gamma-distributed sources (γ = 0.5, 10 density for α = 0.4 after projection onto the circle. For α ∈ [0, π/4), (b) compares the error when estimating A by the mean and the median of the projected density in the receptive field F (α) = (−π/4, π/4) of the known column a1 of A. The former is the k-means convergence criterion. 5 samples) using a mixing matrix A with columns inclined by α and (π/2 − α) respectively. (a) shows the mixture density for α = 0.4 after projection onto the circle. For α ∈ [0, π/4), (b) compares the error when estimating A by the mean and the median of the projected density in the receptive field F (α) = (−π/4, π/4) of the known column a1 of A. The former is the k-means convergence criterion. one oneGaussian Gaussiancomponent, component, it itisis well-known well-knownthat that A Ais isdetermined determineduniquely uniquelyby by x [3]. [3]. However, However, how how to to do do i(t) ∈ [1 : n] such that w this algorithmically is far from i(t)(t) or w this algorithmically is far fromobvious, obvious, and andalthough althoughquite quite a afew fewalgorithms algorithmshave havebeen beenproposed proposedrecently recently [4]–[6], [4]–[6], performance performanceisis yet yetlimited. limited. The The most most commonly commonly used used overcomplete algorithms algorithms rely rely on on sparse sparse ′ i(t) (t) is the neuron closest to x(t). Then set wi(t)(t + 1) := π(wi(t)(t) + η(t)π(σy(t) − wi(t)(t))) and w ′ i(t) (t + 1) := −w i(t)(t + 1), where σ := 1 if w i(t)(t) is closest to y(t), σ := −1 otherwise. sources (after possible sparsification by preprocessing), which can be identified by clustering, usually by k-means or some extension [5], [6]. But apart from the fact that theoretical justifications have not been found, mean-based clustering only identifies the correct A if the data density approaches a delta distribution. In figure 1, we illustrate the deficiency of mean-based clustering; we get an error of up to 5◦ sources (after possible sparsification by preprocessing), which can be identified by clustering, usually by k-means or some extension [5], [6]. But apart from the fact that theoretical justifications have not been found, mean-based clustering only identifies the correct A if the data density approaches a delta distribution. In figure 1, we illustrate the deficiency of mean-based clustering; we get an error of up to 5 per mixing angle, which is rather substantial considering the sparse density and the simple, complete case of m = n = 2. Moreover the figure indicates that median-based clustering performs much better. Indeed, mean-based clustering does not possess any equivariance property (performance independent of A). In the following we propose a novel median-based clustering method and prove its equivariance (lemma 1.2) and convergence. For brevity, the proofs are given for the case of arbitrary n, but m = 2, although they can be readily extended to higher sensor signal dimensions. Corresponding algorithms are proposed and experimentally validated. ◦ We showed that instead of finding the mean in the receptive field (as in k-means), the algorithm searches for the corresponding median. For this we had to study the end points of geometric matrix-recovery, so we assumed that the algorithm had already converged. The idea then was to formulate a condition which the end points have to satisfy and to show that the solutions are among them. per Themixing anglesangle, γ1, . . which . , γn is ∈ rather [0, π) substantial are said to considering satisfy the overcomplete sparse density and geometric the simple, convergence complete condition (GCC) if they are the medians of y restricted to their receptive fields i.e. if γi is case of m = n = 2. Moreover the figure indicates that median-based clustering performs much better. the median of py|F (γi). Moreover, a constant random vector ˆω ∈ R Indeed, mean-based clustering does not possess any equivariance property (performance independent of A). In the following we propose a novel median-based clustering method and prove its equivariance (lemma 1.2) and convergence. For brevity, the proofs are given for the case of arbitrary n, but m = 2, although they can be readily extended to higher sensor signal dimensions. Corresponding algorithms are proposed and experimentally validated. n is called fixed point if E(ζ(ye − ˆω)) = 0. We showed that the median-condition is equivariant meaning that it does no depend on A. Hence any algorithm based on such a condition is equivariant, as confirmed by figure 1.18. If we set ξ(ω) := (cos ω, sin ω) ⊤ , then θ(ξ(ω)) = ω and the following holds: Theorem 1.5.2 (Convergence of overcomplete median-based clustering). The set Φ of fixed points of geometric matrix-recovery contains an element (ˆω1, . . . , ˆωn) such that the resulting matrix (ξ(ˆω1) . . . ξ(ˆωn)) solves the overcomplete BSS problem. The stable fixed points in the above set Φ can be found by the geometric matrix-recovery algorithm.
1.5. Machine learning for data preprocessing 39 samples (a) setup centroids Grassmann clustering (b) division division update (c) update (d) after one iteration Figure 1.19: Illustration of the batch k-means algorithm. One of the most commonly used partitional clustering techniques is the k-means algorithm, which in its batch form partitions the data set into k disjoint clusters by simply iterating between cluster assignments and cluster updates (Bishop, 1995). In general, its goal can be described as follows: Given a set A of points in some metric space (M, d), find a partition of A into disjoint nonempty subsets Bi, � i Bi = A, together with centroids ci ∈ M so as to minimize the sum of the squares of the distances of each point of A to the centroid ci of the cluster Bi containing it. A common approach to minimizing such energy functions is partial optimization with respect to the division matrix and the centroids. The batch k-means algorithm employs precisely this strategy, see figure 1.19. After an initial, random choice of centroids c1, . . . , ck, it iterates between the following two steps until convergence measured by a suitable stopping criterion: • cluster assignment: for each sample x(t) determine an index i(t) = argmin i d(x(t), ci) • cluster update: within each cluster Bi := {a(t)|i(t) = i} determine the centroid ci by ci := argmin c � d(a, c) 2 a∈Bi (1.17) Solving (1.17) is straight-forward in the Euclidean case, however nontrivial in other metric spaces. In Gruber and Theis (2006), see chapter 18, we generalized the concept of k-means by applying it not to the standard Euclidean space but to the manifold of subvectorspaces of R n of a fixed dimension p, also known as the Grassmann manifold Gn,p. Important examples include projective space i.e. the manifold of lines and the space of all hyperplanes. We represented an element of Gn,p by p orthonormal vectors (v1, . . . , vp). Concatenating these into an (n × p)-matrix V, this matrix is unique except for right multiplication by an orthogonal matrix. We therefore wrote [V] ∈ Gn,p for the subspace. This allowed us to define a distance d([V], [W]) := 2 −1/2 �VV ⊤ − WW ⊤ �F on the Grassmannian known as the projection F-norm.
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Mathematics in Independent Component Analysis

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