By Petr Tichavský, Massoud Babaie-Zadeh, Olivier J.J. Michel, Nadège Thirion-Moreau
This ebook constitutes the lawsuits of the thirteenth overseas convention on Latent Variable research and sign Separation, LVA/ICA 2017, held in Grenoble, France, in Feburary 2017. The fifty three papers awarded during this quantity have been rigorously reviewed and chosen from 60 submissions. They have been equipped in topical sections named: tensor ways; from resource positions to room houses: studying equipment for audio scene geometry estimation; tensors and audio; audio sign processing; theoretical advancements; physics and bio sign processing; latent variable research in commentary sciences; ICA idea and purposes; and sparsity-aware sign processing.
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Extra info for Latent Variable Analysis and Signal Separation: 13th International Conference, LVA/ICA 2017, Grenoble, France, February 21-23, 2017, Proceedings
Example text
H. Phan et al. 2 Toeplitzation Definition 1 (Three-way Toeplitzation). Tensorization of the signal x(t) of length L to an order-3 tensor X of size I × J × K, where I + J + K = L + 2, X(i, j, k) = x(I + J + k − i − j) . (10) For this tensorization, each horizontal slice X(i, :, :) is a Toeplitz matrix composed of vectors [x(I + 1 − i), . . , x(I + J − i)] and [x(I + J − i), . . , y(I + J + K − 1 − i)]. Lemma 3. Order-3 Toeplitz tensors tensorized from a sinusoid signal is of multilinear rank-(2, 2, 2), and can be represented as X= 1 G ×1 U1 ×2 U2 ×3 U3 sin (ω) (11) 3 where G is a tensor of size 2 × 2 × 2 G(:, :, 1) = sin(ω(I1 + 2) + 2φ) − sin(ω(I1 + 1) + 2φ) , sin(ωI1 + 2φ) − sin(ω(I1 + 1) + 2φ) G(:, :, 2) = sin(ωI1 + 2φ) − sin(ω(I1 + 1) + 2φ) , − sin(ω(I1 − 1) + 2φ) sin(ωI1 + 2φ) and the three factor matrices are given by ⎡ ⎢ U1 = ⎢ ⎣ ⎤ ⎡ ⎤ ⎡ ⎤ −x(I − i + 3) x(I − i + 2) x(I − i + 2) −x(I − i + 1) (12) x(1) x(2) x(I) x(I + 1) x(I + J − 1) x(I + J − 2) ⎥ ⎢ ⎥ ⎢ ⎥ .
U2 = ⎣ ⎦ , U3 = ⎣ ⎦ . . . x(I) x(I + 1) x(I + J − 1) x(I + J) x(L) x(L − 1) Proof. The proof can be seen from the fact that G ×1 x(i) x(i + 1) = sin(ω) and [x(I + j − 1), x(I + j)] −x(I − i + 3) x(I − i + 2) x(I − i + 2) x(I − i + 1) x(I + J + k − 2) x(I + J + k − 3) = sin2 (ω) x(I + J + k − i − j). In Appendix, we show that the three-way folding and Toeplitzation of x(t) = t also yield multilinear rank-(2, 2, 2) tensors or tensors of rank-3. A more general result is that the tensors of x(t) = tn have multilinear rank of (n + 1, n + 1, n + 1).
1 Volterra Kernels, Tensors and Tensor Decomposition The Volterra Model for Nonlinear Systems We consider single-input-single-output systems that map an input u at time instance k onto an output y at time instance k. The Volterra series expansion 18 P. Dreesen et al. generalizes the well-known convolution operator for linear dynamical systems to the nonlinear case. Essentially the Volterra model of a nonlinear system expresses the output y(k) as a polynomial function of time-shifted input variables u(k), u(k − 1), .