I am very happy to invite you to my PhD defense, on November 28th at 2:00 PM (CET) at Centre Inria d'Université Côte d'Azur (Euler Violet room).

Zoom link (ID: 858 4773 4398 / Passcode: 779231)

A flag is a sequence of nested linear subspaces of increasing dimension. It can also be defined as a sequence of mutually-orthogonal subspaces, whose dimensions form the type. The set of flags of the same type forms a smooth, compact and connected Riemannian manifold. As abstract as these flag manifolds may seem, this thesis aims to demonstrate that they are truly important in statistics.

The eigenspaces of a real symmetric matrix form a flag, whose type corresponds to the multiplicities of the eigenvalues. Consequently, flag manifolds should naturally be involved in certain essential statistical methods such as principal component analysis, which is based precisely on the spectral decomposition of the empirical covariance matrix. However, their use in statistics remains very limited, in favor of simpler spaces such as Stiefel and Grassmann manifolds, to which the principal components and subspaces used in dimension reduction belong, respectively.

A first fundamental contribution of this thesis is the discovery of a new type of parsimony in covariance matrices. Our study of flag manifolds reveals that the number of covariance parameters decreases quadratically with the multiplicities of the eigenvalues. By virtue of the principle of parsimony, we show that empirical eigenvalues whose relative distance is below a certain threshold should be equalized. This result has an important impact in statistics: it implies a transition from principal component analysis to principal subspace analysis, with clear gains in interpretability.

Several extensions of our principal subspace analysis are proposed. In particular, we reformulate the choice of flag type as an optimization problem on the space of covariance matrices, stratified by the multiplicities of the eigenvalues. A lasso-like relaxation on the eigenvalues drastically improves the speed of model selection. Other methodologies—such as hierarchical clustering of eigenvalues and Bayesian approximation of marginal likelihood—are also explored.

In order to improve expressiveness, we extend our principal subspace analysis to mixture models. Since learning parameters via a classic expectation–maximization algorithm makes model selection difficult, we propose a variant that automatically estimates and groups eigenvalues. We obtain theoretical guarantees on the monotonicity of the objective function during iterations, which makes our approach promising for learning parsimonious mixture models.

Finally, we show that certain dimensionality reduction methods suffer from a problem: the representations they produce at different dimensions are not nested. Extending our methodology via a simple and generic principle—involving optimization on flag manifolds—allows us to naturally obtain consistent representations.

Keywords: Covariance matrices, Flag manifolds, Parsimony, Principal component analysis, Riemannian geometry, Statistics.

President

Florence Forbes, Directrice de Recherche, Centre Inria de l'Université Grenoble Alpes

Reviewers

Pierre-Antoine Absil, Professor, UCLouvain

Ian Dryden, Professor, University of Nottingham

Examiners

Frédéric Pascal, Professor, CentraleSupélec

Armin Schwartzman, Professor, UC San Diego

Supervisor

Xavier Pennec, Directeur de recherche, Centre Inria d’Université Côte d’Azur

Guests

Charles Bouveyron, Professor, Centre Inria d'Université Côte d'Azur

Pierre-Alexandre Mattei, Chargé de Recherche, Centre Inria d'Université Côte d'Azur

 🚩Start at 2:00 PM (CET) (I advise arriving 15 minutes in advance, so that you can register at the reception, which will guide you to the room)

📈 Presentation (45 minutes)

❓ Jury questions (~1 hour)

⚖️ Jury deliberation (~30 minutes)

🙏 Jury decision (hopefully positive!) + acknowledgments (the best part, please be there)

🍾 Thesis pot in Inria (~5:00 PM)

🪩 Party in Antibes (~8:00 PM, please DM me if you want to attend)