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We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a conformal deformation of the metric. We derive some existence results using a variational approach, either by…

Geodesic convexity (g-convexity) is a natural generalization of convexity to Riemannian manifolds. However, g-convexity lacks many desirable properties satisfied by Euclidean convexity. For instance, the natural notions of half-spaces and…

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural…

There is increasing interest in the problem of nonparametric regression with high-dimensional predictors. When the number of predictors $D$ is large, one encounters a daunting problem in attempting to estimate a $D$-dimensional surface…

This is the first paper of a series of two devoted to develop a practical method to describe the growth history of bound virialized objects in the gravitational instability scenario without resorting to $N$-body simulations. Here we present…

Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally…

We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebro-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and…

We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal…

Several mean-field theories predict that Hessian matrices of amorphous solids can be written by using the random matrix in the limit of the large spatial dimensions $d\to\infty$. Motivated by these results, we here propose a way to map a…

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence--a natural extension relevant to shapes extracted…

We revisit the problem of bounding the expected supremum of a canonical Gaussian process indexed by a convex set $T \subset \mathbf{R}^d$. We develop two decompositions for the Gaussian width, based on the geometry of the index set. The…

Gaussian processes are used in many machine learning applications that rely on uncertainty quantification. Recently, computational tools for working with these models in geometric settings, such as when inputs lie on a Riemannian manifold,…

We discuss the method of bounding suprema of canonical processes based on the inclusion of their index set into a convex hull of a well-controlled set of points. While the upper bound is immediate, the reverse estimate was established to…

We propose an algorithm to approximate solutions of global optimization problems in Sobolev spaces that follows the spirit of Consensus-based algorithms in finite dimensions. The main ingredient are Gaussian processes. In fact, we exploit…

The simplicial volume is a homotopy invariant of oriented closed connected manifolds measuring the efficiency of representing the fundamental class by singular chains with real coefficients. Despite of its topological nature, the simplicial…

We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their…

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…

This paper models the increase of density of a virgin loose granular sample submitted to a progressive axisymmetric compression (either isotropic or anisotropic) as an irreversible process which destroys the larger voids; a statistical…

Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample…

Markov processes on the lattices with arbitrary dimension are omnipresent in statistical mechanics; however their algebraic description is complete only in dimension 1, for which linear algebra provides many tools complementary to the…