seminars

We study the Gaussian statistical models whose log-likelihood function has a unique complex critical point, i.e., has maximum likelihood degree one. We exploit the connection developed by Améndola et. al. between the models having maximum likelihood degree one and homaloidal polynomials. We study the spanning tree generating function of a graph and show this polynomial is homaloidal when the graph is chordal. When the graph is a cycle on n vertices, n≥4, we prove the polynomial is not homaloidal, and show that the maximum likelihood degree of the resulting model is the nth Eulerian number. These results support our conjecture that the spanning tree generating function is a homaloidal polynomial if and only if the graph is chordal. We also provide an algebraic formulation for the defining equations of these models. Using existing results, we provide a computational study on constructing new families of homaloidal polynomials. In the end, we analyze the symmetric determinantal representation of such polynomials and provide an upper bound on the size of the matrices involved.

In this talk, we will explore two methodologies designed to enhance the performance of Global Navigation Satellite Systems (GNSS): collaborative positioning techniques to improve positioning accuracy and robust signal processing to enhance resilience against jamming attacks. First, we will introduce the Massive User-Centric Single Difference (MUCSD) algorithm, which enhances GNSS accuracy through user collaboration. MUCSD leverages a network of receivers exchanging observables and noisy estimates of position and clock bias. Implemented as an iterative weighted least squares (WLS) estimator, MUCSD achieves a performance comparable to Differential GNSS (DGNSS) without the need for costly reference stations. Simulation results demonstrate that MUCSD outperforms DGNSS as the number of collaborative receivers increases, showcasing its scalability. Next, we will discuss the Robust Interference Mitigation (RIM) framework for snapshot architectures, addressing interferences such as continuous wave and chirp jamming signals. While studies on RIM typically assume the number of quantization bits allows for full signal representation, our study examines the impact of low quantization bits on baseline snapshot receiver performance in the presence of interference. Additionally, we will analyze the effect of quantization on the median absolute deviation (MAD) robust measure of statistical dispersion. By combining collaborative positioning techniques and robust interference mitigation, this talk will showcase advancements that improve GNSS precision and resilience.

This presentation will explore the current state of using local diffusion models in video and image generation, the interfaces and workflows commonly utilized in this area. In addition to highlighting the current capabilities of these models, we will also discuss their limitations and potential opportunities for future research and multidisciplinary collaboration.

ParaTuck-2 decomposition (PT2D) of 3-rd order tensors is a 2-level extension of the well-known CP (canonical polyadic) decomposition (CPD). It is relevant in several applications, such as chemometrics, telecommunications, and machine learning. As shown in (Harshman, Lundy, 1996), the PT2D enjoys strong uniqueness properties (up to scaling/permutation ambiguities, similarly to the CPD). However, there are very few results on theory and algorithms for the PT2D. In particular, common strategies, such as the alternating least squares, suffer from convergence and initialization issues. We propose an algebraic algorithm for the PT2D decomposition in the case when the ParaTuck-2 ranks are smaller than the frontal dimensions of the tensor. Our approach relies only on linear algebra operations and is based on finding the kernel of a structured matrix constructed from the tensor. It refines the previously known identifiability conditions. Yet another algorithm is proposed for the symmetric case, which appears in the implicit approach to the PARAFAC-2 model.

Considered as the last great frontier of cardiac electrophysiology, atrial fibrillation (AF) is the most common sustained arrhythmia encountered in clinical practice, responsible for high hospitalization rates and a significant proportion of brain strokes in the Western world. Analyzing AF electrophysiological complexity noninvasively requires the extraction of the atrial activity (AA) signal from the electrocardiogram (ECG). To perform this task, most approaches including classical average beat subtraction need sufficiently long ECG records, thus limiting real-time analysis. Matrix factorizations can also be used for AA signal estimation by exploiting the spatial diversity of the multi-lead ECG, but require some constraints to guarantee uniqueness that may lack physiological grounds and hinder results interpretation.
This talk will review recent results obtained at the I3S Laboratory, UMR 7271, Université Côte d'Azur, CNRS, on tensor decompositions for noninvasive AA signal extraction in AF ECGs, which guarantee uniqueness under milder constraints on their factors. Specifically, the block term decomposition (BTD) has been shown to be particularly suitable to address this biomedical problem, as atrial and ventricular cardiac activity sources can be modeled by matrices with special structure. The structure of these matrices ensures model uniqueness while their rank is linked to signal complexity. In this framework, we have put forward the Hankel and Löwner BTD as AA extraction tools in AF ECG episodes, with validation in a population of persistent AF patients and several challenging types of ECG segments, including short beat-to-beat intervals and low-amplitude fibrillatory waves. Accurate AA extraction can be achieved from ECG segments as short as a single heartbeat. We have also developed a robust computational algorithm - the so-called alternating group lasso BTD (BTD-AGL) - to simultaneously recover the model structure (number of block terms and multilinear rank of each term) and the model factors. In addition, tensor modeling allows us to derive a novel index to quantify AF complexity nonivasively, useful to characterize stepwise catheter ablation, a first-line therapeutic option for the treatment of persistent forms of the arrhythmia. The index correlates with the expected decrease in AF complexity over ablation steps and is predictive of AF recurrence, which presents clear clinical interest.

Many problems in imaging and data science require to reconstruct, from partial observations, highly concentrated signals, such as pointwise sources or contour lines. This work introduces a novel algorithm for recovering measures supported on such structured domains, given a finite number of their moments. Our approach is based on the traditional singular value decomposition methodology of subspace methods, but lifts their restriction to the framework of Dirac masses, and is able to recover geometrically faithful discrete approximations of measures with density. The crucial step consists in the approximate joint diagonalization of a few non-commuting matrices, which we perform using a quasi-Newton algorithm. Experiments show that our method performs well, not only in the setting of well separated Dirac masses, as predicted by the standard theory of the truncated moment problem, but also in the case of continuous measures, which is not covered by theoretical guarantees and where usual methods empirically fail. We illustrate its applicability in optimal transport problems, where the coupling measure is often localized on the graph of some function.

La chimie analytique sur des artefacts archéologiques est une partie essentielle des recherches en archéologie modernes et, d'année en année, l'amélioration des instruments a permis de générer des données à une fréquence spatiale et temporelle élevée. En particulier, l'imagerie spectrale Raman peut être appliquée avec succès à la recherche en archéologie en raison de sa simplicité de mise en œuvre, afin d'étudier les sociétés humaines du passé par l'analyse de leurs vestiges provenant de fouilles. Cette technique spectrale permet d'obtenir simultanément des informations spatiales et spectrales en préservant l'intégrité de l'échantillon. Cependant, en raison de la complexité inhérente des échantillons en archéologie (ancienneté, fragilité, manque ou absence totale d'informations sur leur composition), l'interprétation chimique peut s'avérer complexe. Des problèmes spécifiques de sélectivité spectrale liés à des composés chimiques inattendus, peuvent apparaître en raison de leur condition de conservation. En outre, la détection des composés mineurs devient difficile car les composés majeurs imposent leurs contributions dans les spectres acquis. Il est donc important de développer de nouvelles approches chimiométriques afin de surmonter ces inconvénients et de découvrir ainsi toutes les informations chimiques réelles contenues dans l'ensemble des données spectrales acquises. Dans le cadre des progrès constants dans le développement d'outils mathématiques et statistiques performants, une approche chimiométrique pertinente a été introduite dans ce contexte. Cette approche vise à extraire des sources spectrales distinctes d’un jeu de données en imagerie Raman sur un échantillon archéologique : un fragment de mosaïque. L'objectif est d'extraire des informations spectrales sélectives par l'analyse du regroupement de pixels afin d'améliorer l'étape d'optimisation initiale au sein de l'algorithme MCR-ALS (Multivariate Curve Resolution and Alternating Least-Squares), une technique bien connue de démélange des signaux. Le principe sous-jacent de l'algorithme MCR-ALS est que les spectres acquis sont considérés comme des combinaisons linéaires de spectres « purs » de tous les composés chimiques individuels présents dans le système étudié. Il est parfois difficile d'obtenir les résultats souhaités par le biais de l'algorithme, en particulier si les estimations initiales des profils spectraux ou de concentrations sont inexactes en raison de signaux complexes, de bruit dans les données ou encore d'un manque de sélectivité spectrale, ce qui entraîne une déficience de rang (c'est-à-dire une mauvaise estimation du nombre total de signaux « purs »). C'est pourquoi une approche basée sur des regroupements de pixels, combiné à de multiples approches de projection orthogonale (OPA) sur les spectres, a été mise au point pour améliorer l'estimation du rang de la matrice de départ, et donc, l'étape d'initialisation de l'approche MCR-ALS avant l'optimisation.

We consider the task of learning individual specific intensities of survival processes from static and longitudinal data. Modeling the intensities as solutions to non-parametric unknown differential equations allows us to provide a precise bias-variance decomposition of a signature-based estimator. This estimator yields excellent performance on a large array of both simulated and real datasets from finance, predictive maintenance and churn prediction.

Part 1: Deep Nonnegative Matrix Factorization with β-Divergences
Our first topic revolves around the Deep Nonnegative Matrix Factorization (deep NMF), a novel and promising facet of unsupervised learning. Deep NMF has emerged as a potent technique for extracting multi-layered features spanning various scales. However, conventional deep NMF models have primarily relied on the least squares error as their evaluation metric, which may not be the most suitable gauge for assessing the quality of approximations across diverse datasets. For data types such as audio signals and documents, β-divergences have gained recognition as a more fitting alternative. In this seminar, we present new models and algorithms that harness β-divergences to enhance deep NMF, with an emphasis on the notion of identifiability.

Part 2: Heavy-Tailed Stochastic Optimization for Deep Neural Networks
Our second topic concerns stochastic optimization, with a particular focus on recent discoveries concerning the nature of stochastic gradient noise in deep neural network training. Contrary to the conventional assumption of Gaussian noise, empirical evidences show that gradient noise often exhibits heavy-tailed characteristics. We introduce an efficient mechanism for optimizers to handle this noise behavior. Additionally, we showcase an extension of our recently introduced stochastic optimizer, referred to as NAG-GS, specifically tailored for the training of Vision Transformers.

Matrix and tensor decomposition plays a crucial role in addressing various real-world problems related to topics such as statistical inference, data acquisition, and data restoration. In this talk, we start with low-rank matrix/tensor models for the data completion problem [1,2]. We tackle this problem in the framework of low-rank matrix/tensor decomposition with a least-squares model. These rank-constrained problems are known not only for their low computational complexity but also the capability of extracting the most important information in the data. We discuss a type of Riemannian gradient-based algorithms that exploit the structure of these rank-constrained models. Secondly, we present a novel application of low-rank matrix methods in the context of causal structure learning. We will show how low-rank matrix decomposition, in combination with a sparse mask operator, can be used to efficiently find directed acyclic graphs (DAGs) proximal to a given graph (with cycles). Furthermore, for learning causal DAGs from observational data, we present a sparse matrix decomposition method [4] and discuss its efficiency through experiments on synthetic and real-world data. [1] S. Dong, P.-A. Absil, and K. A. Gallivan, Riemannian gradient descent methods for graph-regularized matrix completion. Linear Algebra and its Applications 623 (2021), 193-235 [2] S. Dong, B. Gao, Y. Guan, and F. Glineur, New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition, SIAM Journal on Matrix Analysis and Applications 43 (2) (2022), 840-866 [3] S. Dong and M. Sebag, From graphs to DAGs: a low-complexity model and a scalable algorithm, European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML-PKDD), 2022 [4] S. Dong, K. Uemura, A. Fujii, S. Chang, Y. Koyanagi, K. Maruhashi, and M. Sebag, Learning large causal structures from inverse covariance matrix via matrix decomposition, arXiv preprint arXiv:2211.14221, 2023