Lieu : IHP, Amphi Hermite
14.00 : Eddie Aamari (CNRS et Université Paris Diderot)
Titre : Approximation and Geometry of the Reach
Résumé : Various problems within computational geometry and manifold learning encode geometric regularity through the so-called reach, a generalized convexity parameter. The reach $\tau_M$ of a submanifold $M \subset \R^D$ is the maximal offset radius on which the projection onto $M$ is well defined. The quantity $\tau_M$ renders a certain minimal scale of $M$, giving bounds on both maximum curvature and possible bottleneck structures. In this talk, we will study the geometry of the reach through an approximation theory perspective. We present new geometric results on the reach of submanifolds without boundary. An estimator $\hat{\tau}_n$ of $\tau_M$ will be described, in an idealized i.i.d. framework where tangent spaces are known. The estimator $\hat{\tau}_n$ is showed to achieve uniform expected loss bounds over a $\mathcal{C}^3$-like model. Minimax upper and lower bounds are derived. We will conclude with an extension to a model in which tangent spaces are unknown.
15.00 : Jisu Kim (Carnegie Mellon University)
Titre : Persistent homology of kenel density estimator filtration on Rips complex
Résumé : When we observe a point cloud in the Euclidean space, the persistent homology of the upper level sets filtration of the density is one of the most important tools to understand topological features of the data generating distribution. The persistent homology of KDEs (kernel density estimators) for the density function is a natural way to estimate the target quantity. In practice, however, calculating the persistent homology of KDEs on d-dimensional Euclidean spaces requires to approximate the ambient space to a grid, which could be computationally inefficient when the dimension of the ambient space is high or topological features are in different scales. In this talk, I consider the persistent homologies of KDE filtrations on Rips complexes as alternative estimators. This paper shows the consistency result for the persistent homology of the upper level sets filtration of the density. I also describe a novel methodology to construct an asymptotic confidence set based on the bootstrap procedure. Unlike existing procedures, this method does not heavily rely on grid-approximations, scales to higher dimensions, and is adaptive to heterogeneous topological features.
16.00 : Marco Cuturi (CREST)
Titre : Regularization for Optimal Transport and Dynamic Time Warping Distances
Résumé : Machine learning deals with mathematical objects that have structure. Two common structures arising in applications are point clouds / histograms, as well as time series. Early progress in optimization (linear and dynamic programming) have provided powerful families of distances between these structures, namely Wasserstein distances and dynamic time warping scores. Because they rely both on the minimization of a linear functional over a (discrete) space of alignments and a continuous set of couplings respectively, both result in non-differentiable quantities. We show how two distinct smoothing strategies result in quantities that are better behaved and more suitable for machine learning applications, with applications to the computation of Fréchet means.