Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification

Abstract

14 pagesInternational audienceTwo main drawbacks can be stated in the alternating least square (ALS) algorithm used to fit the canonical decomposition (CAND) of multi-way arrays. First its slow convergence caused by the presence of collinearity between factors in the multi-way array it decomposes. Second its blindness to Hermitian symmetries of the considered arrays. Enhanced line search (ELS) scheme was found to be a good way to cope with the slow convergence of the ALS algorithm together with a partial use of the Hermitian symmetry. However, to our knowledge, required equations to perform the latter scheme are only given in the case of third and fifth order arrays. Therefore, our first contribution consists in generalizing the ELS procedure to the case of complex arrays of any order greater than three. Our second contribution is another improvement of the ALS scheme, able to profit from Hermitianity and positive semi-definiteness of the considered arrays. It consists in resorting to the CAND first of a third order array having one unitary loading matrix and second of several rank-1 arrays. An iterative algorithm is then proposed alternating between Procrustes problem solving and the computation of rank-one matrix approximations in order to achieve the CAND of the third order array

ICA-Based EEG denoising: a comparative analysis of fifteen methods

“…Second, it is strictly convex with respect to the linear combination of the independent sources. Then, SIM-BEC solves the ICA problem by looking for the maxima of that cumulant index-based objective function [25 [32] methods also perform ICA using cumulants of the data [18]. SOBI, SOBI rob and TFBSS use SO cumulants, CoM 2 and JADE use both the SO and FO cumulants, and FOBIUM JAD , ICAR 3 , FOOBI 1 and 4-CANDHAP c only use the FO cumulants of the data.…”

“…Next, representative methods of two classes, including the most used ICA techniques in signal processing, are briefly described and studied in terms of performance and numerical complexity: techniques based on the Differential Entropy (DE) such as (extended) InfoMax [16,17], PICA [19] and two different implementations of FastICA [18, ch.6] versus cumulant-based methods. Among cumulantbased techniques, representative algorithms of three subfamilies are studied: i) the techniques using only SO statistics of the data such as SOBI [14,15], SOBI rob [20], TFBSS [21], ii) the algorithms based on SO and FO statistics such as JADE [22], CoM 2 [23], and iii) the methods requiring only HO statistics such as ERICA [24], SIMBEC [25], FOBIUM JAD [26,27], ICAR 3 [28,29], FOOBI 1 [30], 4-CANDHAP c [31,32]. Quantitative results are obtained on simulated epileptic data generated with a physiologically-plausible model [33][34][35].…”

Fast Alternating LS Algorithms for High Order CANDECOMP/PARAFAC Tensor Factorizations

“…CP algorithms available in the literature compute all for , either sequentially (in alternating algorithms [1], [2], [5], [19], [42], [45], [46]) or simultaneously (as in all-atonce algorithms [21], [24], [26]- [29], line-search [20], [21]). This section will present a fast method to compute the gradients recursively for .…”

“…This confirms that FastALS not only has a lower computational cost, but also requires less memory than CP_ALS. Other alternating algorithms for CPD with/without additional constrains such as nonnegativity, orthogonality [38], [45], [46], [48], [49] can be accelerated in a similar way.…”

Fast and unique Tucker decompositions via multiway blind source separation

“…On the other hand, in the (2D) matrix case, very efficient low-rank constrained matrix factorization techniques (also called penalized matrix factorizations) have been developed, such as principal component analysis (PCA), ICA [25][26][27][28], sparse component analysis (SCA) [29,30], smooth component analysis (SmoCA) [5] and nonnegative matrix factorization (NMF) [5,31], just to name a few. These matrix factorization techniques have their own bias, advantages, and are widely applied to blind source separation (BSS), dimensionality reduction, data compression and feature extraction, by exploiting various assumptions and a priori knowledge.…”