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

International audienceIn this communication, the problem of blind source separation in chemical analysis and more precisely in the fluo-rescence spectroscopy framework is addressed. Classically multi-linear Canonical Polyadic (CP or Candecomp/Parafac) decompositions algorithms are used to perform that task. Yet, as the constituent vectors of the loading matrices should be nonnegative since they stand for nonnegative quantities (spectra and concentrations), we focus on NonNegative CP decomposition algorithms (NNCP). In the unconstrained case, two types of trilinear (or triadic) decomposition model have been studied. Here, our aim is to investigate different strategies concerning the choice of models and optimization schemes in the case of a nonnegativity constraint. Computer simulations are performed on synthetic data to illustrate the robustness of the proposed approaches versus overfactoring problems but also the critical importance of the use of regularization terms

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

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

Overview of Constrained PARAFAC Models

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models

Multimodal Data Fusion: An Overview of Methods, Challenges and Prospects

International audienceIn various disciplines, information about the same phenomenon can be acquired from different types of detectors, at different conditions, in multiple experiments or subjects, among others. We use the term "modality" for each such acquisition framework. Due to the rich characteristics of natural phenomena, it is rare that a single modality provides complete knowledge of the phenomenon of interest. The increasing availability of several modalities reporting on the same system introduces new degrees of freedom, which raise questions beyond those related to exploiting each modality separately. As we argue, many of these questions, or "challenges" , are common to multiple domains. This paper deals with two key questions: "why we need data fusion" and "how we perform it". The first question is motivated by numerous examples in science and technology, followed by a mathematical framework that showcases some of the benefits that data fusion provides. In order to address the second question, "diversity" is introduced as a key concept, and a number of data-driven solutions based on matrix and tensor decompositions are discussed, emphasizing how they account for diversity across the datasets. The aim of this paper is to provide the reader, regardless of his or her community of origin, with a taste of the vastness of the field, the prospects and opportunities that it holds