Colloquium and Seminars

Self-interacting Markov chains arise in a range of models and applications. For example, they can be used to approximate the quasi-stationary distributions of irreducible Markov chains and to model random walks with edge or vertex reinforcement. The term self-interacting Markov chain is something of a misnomer, as such processes interact with their full path history at each time instant, and therefore are non-Markovian. Under conditions on the self-interaction mechanism, we establish a large deviation principle for the empirical measure of self-interacting chains on finite spaces. In this setting, the rate function takes a strikingly different form than the classical Donsker-Varadhan rate function associated with the empirical measure of a Markov chain; the rate function for self-interacting chains is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function.

We start the talk by presenting general results of strong density of sub-algebras of bounded Lipschitz functions in metric Sobolev spaces. We apply such results to show the density of smooth cylinder functions in Sobolev spaces of functions on the Wasserstein space $\mathcal P_2$ endowed with a finite positive Borel measure. As a byproduct, we obtain the infinitesimal Hilbertianity of Wassertein Sobolev spaces. By taking advantage of these results, we further address the challenging problem of the numerical approximation of Wassertein Sobolev functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces introduced above and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude.

We will discuss our work on the convergence of iterative hard threshold algorithms for sparse signal recovery problems. For classification problems with nonseparable data this algorithm can be thought of minimizing the so-called ReLU loss. It seems to be very effective (statistically optimal, simple iterative method) for a large class of models of nonseparable data - sparse generalized linear models. It is also robust to adversarial perturbation. Based on joint work with Namiko Matsumoto.

In this talk, we study the emergence of spatially localised coherent structures induced by a compact region of spatial heterogeneity that is motivated by numerical studies into the formation of tornados. While one-dimensional localised patterns induced by spatial heterogeneities have been well studied, proving the existence of fully localised patterns in higher dimensions remains an open problem in pattern formation. We present a general approach to prove the existence of fully localised two-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. This includes patterns with radial and dihedral symmetries, but also extends to patterns beyond these standard rotational symmetry groups. In order to demonstrate the approach, we consider the planar Swift--Hohenberg equation whose linear bifurcation parameter is modified with a radially-symmetric step function. In this case the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. The introduction of a spatial heterogeneity results in an infinite family of bifurcation points with finite dimensional kernels, allowing one to prove local and global bifurcation theorems. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue to large amplitude via analytic global bifurcation theory. Notably, the primary (possibly stable) bifurcating branch in the Swift--Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity. We also discuss how one can use geometric singular perturbation theory to prove the persistence of the patterns to smooth spatial heterogeneities. This work is in collaboration with Daniel Hill and Matthew Turner.