Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of high-level assumptions allowing to describe a regime that seems to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.

Random walks on networks are widely used to model stochastic processes such as search strategies, transportation problems or disease propagation. A prominent example of such process is the dynamics of naive T cells within the lymph node while they are scanning for antigens. The observed T cells trajectories in small sub-volumes of the lymph node are well modeled as a random walk and they have been shown to follow the lymphatic conduit network as substrate for migration. One can then ask how does the connectivity patterns of the lymph node conduit network affect the T cells collective exploration behavior. In particular, does the network display properties that are uniform across the whole volume of the lymph node or can we distinguish some heterogeneities? We propose a workflow to accurately and efficiently define and compute these quantities on large networks, which enables us to characterize heterogeneities within a very large published dataset of Lymph Node Conduit Network. To establish the significance of our results, we compared the results obtained on the lymph node to null models of varying complexity. We identified significantly heterogeneous regions characterized as "remote regions" at the poles and next to the medulla, while a large portion of the network promotes uniform exploration by T cells.

Restoring images degraded by spatially varying blur is a problem encountered in many disciplines such as astrophysics, computer vision, and biomedical imaging. One of the main challenges in performing this task is to design efficient numerical algorithms to approximate integral operators. We introduce a new method based on a sparse approximation of the blurring operator in the wavelet domain. This method requires $\mathcal{O}(N \epsilon^{-d/M})$ operations to provide $\epsilon$-approximations, where $N$ is the number of pixels of a $d$-dimensional image and $M\geq 1$ is a scalar describing the regularity of the blur kernel. In addition, we propose original methods to define sparsity patterns when only the operator regularity is known. Numerical experiments reveal that our algorithm provides a significant improvement compared to standard methods based on windowed convolutions.