Le Thi Khanh Hien’s research while affiliated with University of Mons and other places

Deep nonnegative matrix factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales. However, all existing deep NMF models and algorithms have primarily centered their evaluation on the least squares error, which may not be the most appropriate metric for assessing the quality of approximations on diverse data sets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that ß-divergences offer a more suitable alternative. In this article, we develop new models and algorithms for deep NMF using some ß-divergences, with a focus on the Kullback-Leibler divergence. Subsequently, we apply these techniques to the extraction of facial features, the identification of topics within document collections, and the identification of materials within hyperspectral images.

In this paper, we propose an inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with nonlinear coupling constraints. Distinctive features of our proposed method, when compared with other alternating direction methods of multipliers for solving non-convex problems with nonlinear coupling constraints, include: (i) we apply the inertial technique to the update of primal variables and (ii) we apply a non-standard update rule for the multiplier by scaling the multiplier by a factor before moving along the ascent direction where a relaxation parameter is allowed. Subsequential convergence and global convergence are presented for the proposed algorithm.

We develop an inexact primal-dual first-order smoothing framework to solve a class of non-bilinear saddle point problems with primal strong convexity. Compared with existing methods, our framework yields a significant improvement over the primal oracle complexity, while it has competitive dual oracle complexity. In addition, we consider the situation where the primal-dual coupling term has a large number of component functions. To efficiently handle this situation, we develop a randomized version of our smoothing framework, which allows the primal and dual sub-problems in each iteration to be solved by randomized algorithms inexactly in expectation. The convergence of this framework is analyzed both in expectation and with high probability. In terms of the primal and dual oracle complexities, this framework significantly improves over its deterministic counterpart. As an important application, we adapt both frameworks for solving convex optimization problems with many functional constraints. To obtain an $\varepsilon$-optimal and $\varepsilon$-feasible solution, both frameworks achieve the best-known oracle complexities (in terms of their dependence on $\varepsilon$).

A significant limitation of one-class classification anomaly detection methods is their reliance on the assumption that unlabeled training data only contains normal instances. To overcome this impractical assumption, we propose two novel classification-based anomaly detection methods. Firstly, we introduce a semi-supervised shallow anomaly detection method based on an unbiased risk estimator. Secondly, we present a semi-supervised deep anomaly detection method utilizing a nonnegative (biased) risk estimator. We establish estimation error bounds and excess risk bounds for both risk minimizers. Additionally, we propose techniques to select appropriate regularization parameters that ensure the nonnegativity of the empirical risk in the shallow model under specific loss functions. Our extensive experiments provide strong evidence of the effectiveness of the risk-based anomaly detection methods.

In this paper, we propose an inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with \emph{nonlinear coupling constraints}. Distinctive features of our proposed method, when compared with other alternating direction methods of multipliers for solving non-convex problems with nonlinear coupling constraints, include: (i) we apply the inertial technique to the update of primal variables and (ii) we apply a non-standard update rule for the multiplier by scaling the multiplier by a factor before moving along the ascent direction where a relaxation parameter is allowed. Subsequential convergence and global convergence are presented for the proposed algorithm.

In this paper, we propose an algorithmic framework, dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous ADMM that use specific surrogate functions in the MM step, but also lead to new efficient ADMM schemes. To the best of our knowledge, in the nonconvex nonsmooth setting, ADMM used in combination with the MM principle to update each block of variables, and ADMM combined with inertial terms for the primal variables have not been studied in the literature. Under standard assumptions, we prove the subsequential convergence and global convergence for the generated sequence of iterates. We illustrate the effectiveness of iADMM on a class of nonconvex low-rank representation problems.

This paper considers a multiblock nonsmooth nonconvex optimization problem with nonlinear coupling constraints. By developing the idea of using the information zone and adaptive regime proposed in [J. Bolte, S. Sabach and M. Teboulle, Nonconvex Lagrangian-based optimization: Monitoring schemes and global convergence, Mathematics of Operations Research, 43: 1210--1232, 2018], we propose a multiblock alternating direction method of multipliers for solving this problem. We specify the update of the primal variables by employing a majorization minimization procedure in each block update. An independent convergence analysis is conducted to prove subsequential as well as global convergence of the generated sequence to a critical point of the augmented Lagrangian. We also establish iteration complexity and provide preliminary numerical results for the proposed algorithm.