DATE: Wednesday 11 September 2024 at 11:00

SPEAKER: Pierre-Antoine Absil (UCLouvain)

TITLE: Feasible and Infeasible Optimization on Manifolds

ABSTRACT: This talks gives an introduction to the area of optimization on manifolds - also termed Riemannian optimization - and its applications in engineering and the sciences. Such applications arise when the optimization problem can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold", in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in applications include the rotation group SO(3) (e.g., for the generation of rigid body motions from sample points), the set of fixed-rank matrices (appearing for example in low-rank models for recommender systems), the set of 3x3 symmetric positive-definite matrices (e.g., for the interpolation and denoising of diffusion tensors in brain imaging), and the shape manifold (involved notably in morphing tasks).

In the recent years, the practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. In this talk, we give an overview of geometric optimization algorithms and their applications, with an emphasis on recently developed infeasible optimization methods, in the sense that the iterates do not belong to the manifold but converge to it.

**************************************************************************************************

DATE: October

SPEAKER: TBA

TITLE: TBA

ABSTRACT: TBA

**************************************************************************************************

DATE: Somewhere between Tuesday 12 Novembre 2024 and Thursday 14 November 2024

SPEAKER: Claire Brécheteau (Nantes Université)

TITLE: TBA

ABSTRACT: TBA

**************************************************************************************************

DATE: December

SPEAKER: TBA

TITLE: TBA

ABSTRACT: TBA

**************************************************************************************************

DATE: Thursday 23 January 2025

SPEAKER: Eddie Aamari (École Normale Supérieure de Paris)

TITLE: A theory of stratification learning: clustering-by-dimensionality with reconstruction

ABSTRACT: Given i.i.d. random variables X_1, ..., X_n in R^D drawn from a stratified mixture \cup_k M_k of immersed C2-manifolds of different dimensions d_k with k at most K, we study the minimax estimation of the family M_k and the associated unsupervised clustering problem. We provide a constructive algorithm allowing to estimate each mixture component M_k at its optimal dimension-specific rate (log n /n)^{2/d_k} adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers which also identifies the number of layers K, the dimensions d_k, assign each point X_i to a layer accurately, and estimate tangent spaces optimally. The results hold regardless of any reach assumption on the M_k's nor on intersection configurations M_k \cap M_{k'}. They open the way to a broad clustering framework, where each mixture component (or stratum) M_k models a cluster, emanating from a specific nonlinear correlation phenomenon leaving only d_k local degrees of freedom.