On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal

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

International audienceIn 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

Exploring the feasibility of tensor decomposition for analysis of fNIRS signals: a comparative study with grand averaging method

The analysis of functional near-infrared spectroscopy (fNIRS) signals has not kept pace with the increased use of fNIRS in the behavioral and brain sciences. The popular grand averaging method collapses the oxygenated hemoglobin data within a predefined time of interest window and across multiple channels within a region of interest, potentially leading to a loss of important temporal and spatial information. On the other hand, the tensor decomposition method can reveal patterns in the data without making prior assumptions of the hemodynamic response and without losing temporal and spatial information. The aim of the current study was to examine whether the tensor decomposition method could identify significant effects and novel patterns compared to the commonly used grand averaging method for fNIRS signal analysis. We used two infant fNIRS datasets and applied tensor decomposition (i.e., canonical polyadic and Tucker decompositions) to analyze the significant differences in the hemodynamic response patterns across conditions. The codes are publicly available on GitHub. Bayesian analyses were performed to understand interaction effects. The results from the tensor decomposition method replicated the findings from the grand averaging method and uncovered additional patterns not detected by the grand averaging method. Our findings demonstrate that tensor decomposition is a feasible alternative method for analyzing fNIRS signals, offering a more comprehensive understanding of the data and its underlying patterns