A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

In this talk, we discuss a method for manifold learning that relies on a symmetric version of the optimal transport problem with a quadratic regularisation. We show that the solution of such a problem yields a sparse and adaptive affinity matrix that can be interpreted as a generalisation of the bistochastic kernel normalisation. We prove that the resulting kernel is consistent with a Laplace-type operator in the continuous limit, discuss geometric interpretations and establish robustness to heteroskedastic noise. The performance on certain simulated and real data examples will be shown. Some open questions will be raised across the talk.

Regression methods provide a simple mechanism for evaluating empirical functions over scattered data points, allowing to forecast future values. In particular, kernel-based regressions, such as the Nadaraya-Watson estimator are suitable for cases in which the relationship between the data points and the function is not linear. Successive applications of the Nadarya-Watson estimator, also known as Laplacian Pyramids model or iterative bias reduction, yield a multiscale model that generates a smoothed version of a function by using Gaussian kernels to smooth the residuals. 

In this talk, I will review some recent extensions of this regression for spatio-temporal forecasting and bio-medical applications.

To any group G we  associate the space Ch(G) of all characters on G. After defining this space and discussing its interesting properties, I’ll turn to discuss dynamics on such spaces. Our main result is that the action of any​ arithmetic group on the character space of its amenable/solvable radical is stiff, i.e., any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader, Boutonnet, Houdayer, and Peterson, this stiffness result is used to deduce dichotomy statements (and ‘charmenability’) for (not necessarily semi-simple) higher rank arithmetic groups pertaining to their normal subgroups, dynamical systems, representation theory and more. The talk is based on a joint work with Uri Bader. 

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

Data in high dimensional Euclidean space may often be described with fewer parameters than the ambient space, perhaps lying on or near a submanifold of lower intrinsic dimension. In part one of this talk, we will discuss a new method for estimating this intrinsic dimension from the data: a version of local PCA which is calibrated on quadratic embeddings which may better approximate a manifold at larger scales. The second part of the talk will focus on a technique of random embeddings to reduce ambient dimension. The topics of this talk are joint work with Anna Gilbert.

This is the third lecture in the seminar.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!