Study of different strategies for the Canonical Polyadic decomposition of nonnegative third order tensors with application to the separation of spectra in 3D fluorescence spectroscopy

Vu, Xuan; Chaux, Caroline; Maire, Sylvain; Thirion-Moreau, Nadège

doi:10.1109/mlsp.2014.6958896

Cited by 4 publications

(5 citation statements)

References 22 publications

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“…Indeed, from lemma 1, we can easily conclude that α 2,1 is greater than A * . If we replace α and A by D T F and DD T F, respectively, then we obtain equation (3).…”

Section: A Problem Formulationmentioning

confidence: 99%

Low rank canonical polyadic decomposition of tensors based on group sparsity

Han¹,

Albera²,

Kachenoura³

et al. 2017

2017 25th European Signal Processing Conference (EUSIPCO)

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Abstract-A new and robust method for low rank Canonical Polyadic (CP) decomposition of tensors is introduced in this paper. The proposed method imposes the Group Sparsity of the coefficients of each Loading (GSL) matrix under orthonormal subspace. By this way, the low rank CP decomposition problem is solved without any knowledge of the true rank and without using any nuclear norm regularization term, which generally leads to computationally prohibitive iterative optimization for largescale data. Our GSL-CP technique can be then implemented using only an upper bound of the rank. It is compared in terms of performance with classical methods, which require to know exactly the rank of the tensor. Numerical simulated experiments with noisy tensors and results on fluorescence data show the advantages of the proposed GSL-CP method in comparison with classical algorithms.

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“…Indeed, from lemma 1, we can easily conclude that α 2,1 is greater than A * . If we replace α and A by D T F and DD T F, respectively, then we obtain equation (3).…”

Section: A Problem Formulationmentioning

confidence: 99%

Low rank canonical polyadic decomposition of tensors based on group sparsity

Han¹,

Albera²,

Kachenoura³

et al. 2017

2017 25th European Signal Processing Conference (EUSIPCO)

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“…When simulated data are used, the true tensor rank

\overset{R}{true}

is known, while the tensor rank that will be used for the model and thus for the decomposition is denoted by

\overset{R}{true}

. In the case of simulated data, we can consider the 2 error indices that have already been used by Vu et al instead of the reconstruction error

∥ scriptT - \overset{T}{true} ∥_{F}^{2}

, which is classically used with real experimental data. The first error index, denoted by E 1 , measures the error of estimation but discarding the overfactoring part.…”

Section: Numerical Simulationsmentioning

confidence: 99%

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

Chaux

Thirion-Moreau

et al. 2017

Journal of Chemometrics

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In this article, we address the problem of tensor factorization subject to certain constraints. We focus on the canonical polyadic decomposition, also known as parallel factor analysis. The interest of this multilinear 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 nonconvex and sometimes even nondifferentiable, making their optimization more difficult and 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 approaches to derive a new penalized nonnegative third‐order canonical polyadic decomposition algorithm. Its interest, efficiency, robustness, and flexibility are illustrated thanks to computer simulations performed on both simulated and real experimental 3D fluorescence spectroscopy data.

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“…We choose the same α A = α B = α C =10 −3 and l 1 ‐norm regularization terms for each factor during the first half of the iterations, then in the second half, they are discarded to avoid a bound in the performances. We choose l 1 ‐norm regularization terms because it was shown in that they lead to the best results. The same random initializations are used for all the algorithms.…”

Section: Nonnegative Cp Decomposition Of Three‐way Arraysmentioning

confidence: 99%

“…Moreover, regularization terms (that might be different from one loading matrix to another one) are also added in order to reinforce certain properties (sparseness or smoothness of the solution). As shown in [43], they improve the robustness of algorithms versus model errors such as a possible overestimation of the tensor rank (which corresponds to the number of fluorescent compounds effectively present in the studied samples, yet, this rank is unknown and it can only be estimated). Without these regularization terms, spurious compounds might appear.…”

Section: Introductionmentioning

confidence: 99%

A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

Royer

Thirion-Moreau

Comon

et al. 2015

Journal of Chemometrics

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International audienceWe consider blind source separation in chemical analysis focussing on the 3D fluorescence spectroscopy framework. We present an alternative method to process the Fluorescence Excitation-Emission Matrices (FEEM): first, a preprocessing is applied to eliminate the Raman and Rayleigh scattering peaks that clutter the FEEM. To improve its robustness versus possible improper settings, we suggest to associate the classical Zepp's method with a morphological image filtering technique. Then, in a second stage, the Canonical Polyadic (CP or Cande-comp/Parafac) decomposition of a nonnegative 3-way array has to be computed. In the fluorescence spectroscopy context, the constituent vectors of the loading matrices should be nonnegative (since standing for spectra and concentrations). Thus, we suggest a new NonNegative third order CP decomposition algorithm (NNCP) based on a non linear conjugate gradient optimisation algorithm with regularization terms and periodic restarts. Computer simulations performed on real experimental data are provided to enlighten the effectiveness and robustness of the whole processing chain and to validate the approach

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Study of different strategies for the Canonical Polyadic decomposition of nonnegative third order tensors with application to the separation of spectra in 3D fluorescence spectroscopy

Cited by 4 publications

References 22 publications

Low rank canonical polyadic decomposition of tensors based on group sparsity

Low rank canonical polyadic decomposition of tensors based on group sparsity

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

A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

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