{"id":6337,"date":"2023-09-26T10:27:34","date_gmt":"2023-09-26T08:27:34","guid":{"rendered":"https:\/\/samovar.telecom-sudparis.eu\/?p=6337"},"modified":"2023-09-26T11:48:56","modified_gmt":"2023-09-26T09:48:56","slug":"seminaire-sop-09-novembre-2023-a-13h30-salle-f201-evry","status":"publish","type":"post","link":"https:\/\/samovar.telecom-sudparis.eu\/index.php\/2023\/09\/26\/seminaire-sop-09-novembre-2023-a-13h30-salle-f201-evry\/","title":{"rendered":"S\u00e9minaire SOP jeudi 9 novembre 2023 \u00e0 13h30 salle F201, Evry"},"content":{"rendered":"\n
\n

Lieu :<\/strong> Salle F201, 9 rue Charles Fourier Evry 91011<\/p>\n\n\n\n

Date :<\/strong> Jeudi 9 novembre 2023<\/p>\n\n\n\n

Programme<\/strong><\/p>\n<\/div><\/div>\n\n\n\n

13h30-14h30 : Arthur Marmin, Ma\u00eetre de Conf\u00e9rences \u00e0 Aix-Marseille Universit\u00e9<\/p>\n\n\n\n

14h30-15h30 : Sholom Schechtman, Maitre de conf\u00e9rences \u00e0 T\u00e9l\u00e9com SudParis<\/p>\n\n\n\n

1er expos\u00e9 d’Arthur Marmin<\/strong>, Ma\u00eetre de Conf\u00e9rences \u00e0 Aix-Marseille Universit\u00e9<\/strong><\/p>\n\n\n\n

Titre :<\/strong> Majorization-minimization for Sparse Nonnegative Matrix Factorization with the \u03b2-divergence<\/p>\n\n\n\n

R\u00e9sum\u00e9 :<\/strong> Nonnegative matrix factorization (NMF) consists in decomposing a data matrix with nonnegative entries into the product of two nonnegative matrices. NMF has found many applications such as feature extraction in image processing and text mining, audio source separation, blind unmixing in hyperspectral imaging, and user recommendation. In this context, the two factor matrices represent a dictionary of basis vectors and an activation matrix respectively. In this talk, I will present new multiplicative updates for NMF with the \u03b2-divergence and sparse regularization of one of the two factors. It is well known that the norm of the other factor then needs to be controlled in order to avoid an ill-posed formulation. Standard practice consists in constraining the columns to have unit norm, which leads to a nontrivial optimization problem. The approach I will present leverages a reparametrization of the original problem into the optimization of an equivalent scale-invariant objective function. From there, I will derive block-descent majorization-minimization algorithms that result in simple multiplicative updates for either l1-regularization or the more \u201caggressive\u201d log-regularization. In contrast with other state-of-the-art methods, the obtained algorithms are universal in the sense that they can be applied to any \u03b2-divergence (i.e., any value of \u03b2) and that they come with convergence guarantees for both the objective function and the sequence of generated iterates. I will also present two state-of-the-art methods (an heuristic method and a Lagrangian one) and I will report numerical comparisons using various datasets: face images, an audio spectrogram, hyperspectral data, and song play counts.<\/p>\n\n\n\n

2\u00e8me expos\u00e9 de Sholom Schechtman<\/strong>, Maitre de conf\u00e9rences \u00e0 T\u00e9l\u00e9com SudParis<\/strong><\/p>\n\n\n\n

Titre :<\/strong> L’\u00e9vitement des points-selles actifs d’une fonction faiblement convexe par la descente de sous-gradient stochastique<\/p>\n\n\n\n

R\u00e9sum\u00e9<\/strong>: Il est connu que m\u00eame sur une fonction non-convexe et non-diff\u00e9rentiable, sous des hypoth\u00e8ses faibles, l’algorithme du sous-gradient stochastique (SGD) converge vers les points critique de la fonction objectif. Cependant, cet ensemble des points critiques est g\u00e9n\u00e9ralement plus grand que l’ensemble des minima locaux et continent des \u201cpoints selles\u201d – des points envers lesquels la convergence est ind\u00e9sirable. L’objectif de cet expos\u00e9 est de montrer que si la fonction objectif est faiblement convexe alors le SGD \u00e9vite un certain type de point – les points-selles actifs.<\/p>\n\n\n\n

En premi\u00e8re partie, nous commencerons par une introduction aux fonctions semi-alg\u00e9briques, ou plus g\u00e9n\u00e9ralement d\u00e9finissables dans une structure o-minimale, une classe qui contient la grande majorit\u00e9 de fonctions utilis\u00e9es en optimisation. Nous montrerons que ces fonctions, bien que non-lisses, contiennent une structure diff\u00e9rentiable sous-jacente. Finalement, nous pr\u00e9senterons un r\u00e9sultat nouveau – la formule de projection renforc\u00e9e (am\u00e9lioration de la formule de projection de Bolte, Daniilidis, Lewis et Shiota), qui donne une description simple des sous-gradients d’une telle fonction.<\/p>\n\n\n\n

En deuxi\u00e8me partie, nous \u00e9tudierons la forme typique des points critiques de telles fonctions et montrerons que si la fonction est d\u00e9finissable et faiblement convexe alors ses points critiques g\u00e9n\u00e9riques peuvent \u00eatre seulement des minima locaux ou des points-selles actifs.<\/p>\n\n\n\n

Enfin, en derni\u00e8re partie nous montrerons que, sous des hypoth\u00e8ses faibles, le SGD sur une fonction faiblement convexe et d\u00e9finissable \u00e9vite (ne converge pas) vers un point-selle actif.<\/p>\n","protected":false},"excerpt":{"rendered":"

Lieu : Salle F201, 9 rue Charles Fourier Evry 91011 Date : Jeudi 9 novembre 2023 Programme 13h30-14h30 : Arthur Marmin, Ma\u00eetre de Conf\u00e9rences \u00e0 Aix-Marseille Universit\u00e9 14h30-15h30 : Sholom Schechtman, Maitre de conf\u00e9rences \u00e0 T\u00e9l\u00e9com SudParis 1er expos\u00e9 d’Arthur Marmin, Ma\u00eetre de Conf\u00e9rences \u00e0 Aix-Marseille Universit\u00e9 Titre : Majorization-minimization for Sparse Nonnegative Matrix Factorization with the \u03b2-divergence 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