Algorithms for nonnegative matrix factorization with the beta-divergence

This paper describes algorithms for nonnegative matrix factorization (NMF) with the beta-divergence (beta-NMF). The beta-divergence is a family of cost functions parametrized by a single shape parameter beta that takes the Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito divergence as special cases (beta = 2,1,0, respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization (MM) algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a beta-dependent power exponent. The monotonicity of the heuristic algorithm can however be proven for beta in (0,1) using the proposed auxiliary function. Then we introduce the concept of majorization-equalization (ME) algorithm which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The paper also describes how the proposed algorithms can be adapted to two common variants of NMF : penalized NMF (i.e., when a penalty function of the factors is added to the criterion function) and convex-NMF (when the dictionary is assumed to belong to a known subspace).Comment: \`a para\^itre dans Neural Computatio

Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods

The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage. The main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self-tuning adaptive scheme allowing to achieve the minimal computation cost. The connection between this algorithm and some existing strategies is discussed and its efficiency is illustrated on a linear inverse problem of image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio

Efficient binary reconstruction for non destructive evaluation using gammagraphy

International audienceThe localization and the sizing of 3D flaws within a homogeneous metallic media is a major task for non destructive evaluation (NDE). This paper adresses the problem of the reconstruction of such flaws using an efficient binary algorithm. Basically, the method rests on the fact that a simple binary constraint suffices for an accurate and robust reconstructions in the context of NDE. A heuristic minimization, computationally attractive, is designed in order to provide fast reconstructions. The proposed algorithm is compared with standard binary (the iterated conditional mode algorithm) and non binary (penalized approach with convex potentials Gibbs random fields) reconstruction techniques

On Algorithms Based on Joint Estimation of Currents and Contrast in Microwave Tomography

This paper deals with improvements to the contrast source inversion method which is widely used in microwave tomography. First, the method is reviewed and weaknesses of both the criterion form and the optimization strategy are underlined. Then, two new algorithms are proposed. Both of them are based on the same criterion, similar but more robust than the one used in contrast source inversion. The first technique keeps the main characteristics of the contrast source inversion optimization scheme but is based on a better exploitation of the conjugate gradient algorithm. The second technique is based on a preconditioned conjugate gradient algorithm and performs simultaneous updates of sets of unknowns that are normally processed sequentially. Both techniques are shown to be more efficient than original contrast source inversion.Comment: 12 pages, 12 figures, 5 table

Application and validation of spatial mixture modelling for the joint detection-estimation of brain activity in fMRI.

International audienceWithin-subject analysis in event-related functional Magnetic Resonance Imaging (fMRI) first relies on (i) a detection step to localize which parts of the brain are activated by a given stimulus type, and second on (ii) an estimation step to recover the temporal dynamics of the brain response. Recently, a Bayesian detection-estimation approach that jointly addresses (i)-(ii) has been proposed in [1]. This work is based on an independent mixture model (IMM) and provides both a spatial activity map and an estimate of brain dynamics. In [2], we accounted for spatial correlation using a spatial mixture model (SMM) based on a binary Markov random field. Here, we assess the SMM robustness and flexibility on simulations which diverge from the priors and the generative BOLD model and further extend comparison between SMM and IMM on real fMRI data, focusing on a region of interest in the auditory cortex

Automatic Road Crack Detection by Selection of Minimal Paths

National audienceAbstract – Automatic detection of road cracks from pavement images has become an important challenge in many countries. Among the different methods proposed in the literature, this paper proposes to combine shortest-paths previously estimated in the image. The proposed method takes account of both photometric and geometric characteristics of cracks simultaneously and requires a few informations a priori. It has been tested on image data sets collected by a dynamic imaging system

Primal-dual interior point optimization for a regularized reconstruction of NMR relaxation time distributions

International audienceThis paper deals with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) spectroscopy. This large scale and ill-posed inverse problem is solved by the iterative minimization of a regularized objective function allowing to encode some prior assumptions on the sought distribution. The numerical optimization of the criterion is performed using a primal-dual interior point algorithm allowing to handle the non-negativity constraint. The performances of the proposed approach are illustrated through the processing of real data from a two-dimensional NMR experiment

Reconstruction d'un spectre RMN 2D par maximum d'entropie

International audienceLa résonance magnétique nucléaire (RMN) est une méthode moderne de spectroscopie utilisée pour l'analyse de la composition de produits biologiques. Nous nous intéressons dans cet article à l'estimation d'un spectre corrélation T1-T2 à partir de mesures RMN. Les difficultés de l'estimation sont liées au caractère mal-posé de ce problème inverse et à la taille importante des données à traiter. La méthode d'estimation est fondée sur le Maximum d'Entropie et nous proposons deux algorithmes de reconstruction itérative ; le premier est fondé sur l'algorithme de Bryan et Skilling et le second utilise le gradient conjugué non-linéaire. Par ailleurs, la structure du modèle d'observation est avantageusement exploitée pour alléger le coût de calcul sans employer les approximations proposées récemment par Vankataramanan et al. De plus, nous proposons le rajout d'une étape de recherche de pas adaptée à la fonction entropique afin d'assurer une décroissance du critère. Les algorithmes sont évalués sur un exemple synthétique et leur applicabilité est illustrée sur des données réelle