A new penalized nonnegative third order tensor decomposition using a block coordinate proximal gradient approach: application to 3D fluorescence spectroscopy

Abstract : In this article, we address the problem of tensor factorization subject to certain constraints. We focus on the Canonical Polyadic Decomposition (CPD) also known as Parafac. The interest of this multi-linear decomposition coupled with 3D fluorescence spectroscopy is now well established in the fields of environmental data analysis, biochemistry and chemistry. When real experimental data (possibly corrupted by noise) are processed, the actual rank of the " observed " tensor is generally unknown. Moreover, when the amount of data is very large, this inverse problem may become numerically ill-posed and consequently hard to solve. The use of proper constraints reflecting some a priori knowledge about the latent (or hidden) tracked variables and/or additional information through the addition of penalty functions can prove very helpful in estimating more relevant components rather than totally arbitrary ones. The counterpart is that the cost functions that have to be considered can be non convex and sometimes even non differentiable making their optimization more difficult, leading to a higher computing time and a slower convergence speed. Block alternating proximal approaches offer a rigorous and flexible framework to properly address that problem since they are applicable to a large class of cost functions while remaining quite easy to implement. Here, we suggest a new block coordinate variable metric forward-backward method which can be seen as a special case of Majorize-Minimize (MM) approaches to derive a new penalized nonnegative third order CPD algorithm. Its interest, efficiency, robustness and flexibility are illustrated thanks to computer simulations carried out on both simulated and real experimental 3D fluorescence spectroscopy data.
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Xuan Vu, Caroline Chaux, Nadège Thirion-Moreau, Sylvain Maire, Elfrida Carstea. A new penalized nonnegative third order tensor decomposition using a block coordinate proximal gradient approach: application to 3D fluorescence spectroscopy. Journal of Chemometrics, Wiley, 2017, 31 (4), pp.e2859. ⟨10.1002/cem.2859⟩. ⟨hal-01387439⟩

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