Smoothed Separable Nonnegative Matrix Factorization

Given a set of data points belonging to the convex hull of a set of vertices, a key problem in data analysis and machine learning is to estimate these vertices in the presence of noise. Many algorithms have been developed under the assumption that there is at least one nearby data point to each vertex; two of the most widely used ones are vertex component analysis (VCA) and the successive projection algorithm (SPA). This assumption is known as the pure-pixel assumption in blind hyperspectral unmixing, and as the separability assumption in nonnegative matrix factorization. More recently, Bhattacharyya and Kannan (ACM-SIAM Symposium on Discrete Algorithms, 2020) proposed an algorithm for learning a latent simplex (ALLS) that relies on the assumption that there is more than one nearby data point for each vertex. In that scenario, ALLS is probalistically more robust to noise than algorithms based on the separability assumption. In this paper, inspired by ALLS, we propose smoothed VCA (SVCA) and smoothed SPA (SSPA) that generalize VCA and SPA by assuming the presence of several nearby data points to each vertex. We illustrate the effectiveness of SVCA and SSPA over VCA, SPA and ALLS on synthetic data sets, and on the unmixing of hyperspectral images.Comment: 27 pages, 11 figure

De 20 000 à 18 000 BP en Quercy : apports de la séquence du Cuzoul de Vers à la compréhension de l'évolution des comportements socio-économiques entre Solutréen récent et Badegoulien

Essai de synthèse des travaux menés autour du gisement du Cuzoul de Vers (Lot)

Stratégies d'optimisation pour la séparation aveugle de sources parcimonieuses grande échelle

During the last decades, Blind Source Separation (BSS) has become a key analysis tool to study multi-valued data. The objective of this thesis is however to focus on large-scale settings, for which most classical algorithms fail. More specifically, it is subdivided into four sub-problems taking their roots around the large-scale sparse BSS issue: i) introduce a mathematically sound robust sparse BSS algorithm which does not require any relaunch (despite a difficult hyper-parameter choice); ii) introduce a method being able to maintain high quality separations even when a large-number of sources needs to be estimated; iii) make a classical sparse BSS algorithm scalable to large-scale datasets; and iv) an extension to the non-linear sparse BSS problem. The methods we propose are extensively tested on both simulated and realistic experiments to demonstrate their quality. In-depth interpretations of the results are proposed.Lors des dernières décennies, la Séparation Aveugle de Sources (BSS) est devenue un outil de premier plan pour le traitement de données multi-valuées. L’objectif de ce doctorat est cependant d’étudier les cas grande échelle, pour lesquels la plupart des algorithmes classiques obtiennent des performances dégradées. Ce document s’articule en quatre parties, traitant chacune un aspect du problème: i) l’introduction d’algorithmes robustes de BSS parcimonieuse ne nécessitant qu’un seul lancement (malgré un choix d’hyper-paramètres délicat) et fortement étayés mathématiquement; ii) la proposition d’une méthode permettant de maintenir une haute qualité de séparation malgré un nombre de sources important: iii) la modification d’un algorithme classique de BSS parcimonieuse pour l’application sur des données de grandes tailles; et iv) une extension au problème de BSS parcimonieuse non-linéaire. Les méthodes proposées ont été amplement testées, tant sur données simulées que réalistes, pour démontrer leur qualité. Des interprétations détaillées des résultats sont proposées

Stacked Sparse Blind Source Separation for Non-Linear Mixtures

Linear Blind Source Separation (BSS) has known a tremendous success in fields ranging from biomedical imaging to astrophysics. In this work, we however propose to depart from the usual linear setting and tackle the case in which the sources are mixed by an unknown non-linear function. We propose a stacked sparse BSS method enabling a sequential decomposition of the data through a linear-by-part approximation. Beyond separating the sources, the introduced StackedAMCA can under discussed conditions further learn the inverse of the unknown non-linear mixing, enabling to reconstruct the sources despite a severely ill-posed problem. The quality of the method is demonstrated on two experiments , and a comparison is performed with state-of-the art non-linear BSS algorithms

Stacked Sparse Non-Linear Blind Source Separation

Linear Blind Source Separation (BSS) has known a tremendous success in fields ranging from biomedical imaging to astrophysics. In this work, we however depart from the usual linear setting and tackle the case in which the sources are mixed by an unknown non-linear function. We propose to use a sequential decomposition of the data enabling its approximation by a linear-by-part function. Beyond separating the sources, the introduced StackedAMCA can further empirically learn in some settings an approximation of the inverse of the unknown non-linear mixing, enabling to reconstruct the sources despite a severely ill-posed problem. The quality of the method is demonstrated experimentally, and a comparison is performed with state-of-the art non-linear BSS algorithms

Heuristics for Efficient Sparse Blind Source Separation

International audience—Sparse Blind Source Separation (sparse BSS) is a key method to analyze multichannel data in fields ranging from medical imaging to astrophysics. However, since it relies on seeking the solution of a non-convex penalized matrix factorization problem, its performances largely depend on the optimization strategy. In this context, Proximal Alternating Linearized Minimization (PALM) has become a standard algorithm which, despite its theoretical grounding, generally provides poor practical separation results. In this work, we first investigate the origins of these limitations, which are shown to take their roots in the sensitivity to both the initialization and the regularization parameter choice. As an alternative, we propose a novel strategy that combines a heuristic approach with PALM. We show its relevance on realistic astrophysical data

Distributed sparse BSS for large-scale datasets

Blind Source Separation (BSS) [1] is widely used to analyze multichannel data stemming from origins as wide as astrophysics to medicine. However, existent methods do not efficiently handle very large datasets. In this work, we propose a new method coined DGMCA (Distributed Generalized Morphological Component Analysis) in which the original BSS problem is decomposed into subproblems that can be tackled in parallel, alleviating the large-scale issue. We propose to use the RCM (Riemannian Center of Mass-[6][7]) to aggregate during the iterative process the estimations yielded by the different subproblems. The approach is made robust both by a clever choice of the weights of the RCM and the adaptation of the heuristic parameter choice proposed in [4] to the parallel framework. The results obtained show that the proposed approach is able to handle large-scale problems with a linear acceleration performing at the same level as GMCA and maintaining an automatic choice of parameters. I. LARGE-SCALE BLIND SOURCE SEPARATION Given m row observations of size t stacked in a matrix Y assumed to follow a linear model Y = AS + N, the objective of BSS [1] is to estimate the matrices A (size m × n) and S (size n × t) up to a mere permutation and scaling indeterminacy. In this model, A mixes the n row sources in S, the observations being entached by some unknwown noise N (size m × t). We will assume that n ≤ m. While ill-posed, this problem can be regularized assuming the sparsity of S [2]. The estimation will then turn into the minization of: ˆ A, ˆ S = arg min A,S 1 2 Y − AS 2 F +Λ S 1 +i X:X k 2 =1, ∀k (A) , (1) with ·· F the Frobenius norm, Λ the regularization parameters and iC(·) the indicator function of the set C. The first term is a data fidelity one, the second enforces the sparsity and the last avoids degenerated solutions with A 2 F → 0 by enforcing unit columns. To tackle Eq. (1), the GMCA [4] algorithm has known a tremendous success due to an automatic decreasing parameter strategy making it robust. However, in this work we will assume that the data Y are large-scale in the sense that t can have huge values (e.g. up to 10 9 samples), which make the treatement of Y as a whole intractable. In this context, using GMCA or most other algorithms is prohibitive. II. PROPOSED METHOD This difficulty motivates the construction of J subproblems (j) of the type Yj = ASj + Nj where j denotes a subset of tj columns of the corresponding matrices. We use disjoints sets with j |tj| = t

Provably robust blind source separation of linear-quadratic near-separable mixtures

International audienceIn this work, we consider the problem of blind source separation (BSS) by departing from the usual linear model and focusing on the linear-quadratic (LQ) model. We propose two provably robust and computationally tractable algorithms to tackle this problem under separability assumptions which require the sources to appear as samples in the data set. The first algorithm generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS, and is referred to as SNPALQ. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions of the mixing are mitigated, thus improving the separation quality. SNPALQ is shown to be able to recover the ground truth factors that generated the data, even in the presence of noise. The second algorithm is a brute-force (BF) algorithm, which is used as a post-processing step for SNPALQ. It enables to discard the spurious (mixed) samples extracted by SNPALQ, thus broadening its applicability. The BF is in turn shown to be robust to noise under easier-to-check and milder conditions than SNPALQ. We show that SNPALQ with and without the BF postprocessing is relevant in realistic numerical experiments

Successive Nonnegative Projection Algorithm for Linear Quadratic Mixtures (iTWIST 2020)

International audienceIn this work, we tackle the problem of hyperspectral (HS) unmixing by departing from the usual linear model and focusing on a Linear-Quadratic (LQ) one. The proposed algorithm, referred to as Successive Nonnegative Projection Algorithm for Linear Quadratic mixtures (SNPALQ), extends the Successive Nonnegative Projection Algorithm (SNPA), designed to address the unmixing problem under a linear model. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions in the mixing are mitigated, thus improving the separation quality. The approach is shown to be relevant in a realistic numerical experiment

Three-dimensional simulation of a fire in a simplified gallery of the Chauvet-Pont d’Arc cave

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