arXiv research

Introduces Hilbertian n-volume in metric spaces with inequalities.

The curvature-dimension condition implies a new weighted scalar curvature.

Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.

Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.

Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.

Paper proves weighted Riemannian manifolds are Hilbertian, embedding tangent modules.

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents a…

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvat…

Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite…

Abstract: Generalizes SGMs to infinite-dimensional Hilbertian setting.

New tensorization theorem for Sobolev spaces on product spaces.

We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Boch…

Study shows curvature bounds for CD and CAT spaces.

The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants …

Improved learning theory for kernel distribution regression with two-stage sampling.

New definition of metric current yields Finsler geometry volume densities.

We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian CD(K,N) spaces in the same form as the one available on smooth Riemannian manifolds.

Analyzes surfaces with bounded curvature, proving properties and continuity.

Analyzes surfaces with bounded curvature, proving properties and existence of heat kernels.

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g…

Piecewise flat extrinsic curvature approximations for simplicial manifolds.

The abstract discusses a new type of space and its properties.

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…

Study on entropy stability in product spaces of negatively curved symmetric spaces.

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric meas…

New algorithm for linear bandits tackles Optimal Transport problems.

The study proves a dichotomy for minimal hypersurfaces in certain thick manifolds.

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces …

Hilbert(ian) A-modules over finite von Neumann algebras A with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared, and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in…

We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard notions. A new feature of the twisted L^2-cohomology theory is that in addition to sat…

A new kernel for probability measures based on optimal transport.

This paper reviews the functional aspects of statistical learning theory. The main point under consideration is the nature of the hypothesis set when no prior information is available but data. Within this framework we first discuss about the hypothesis set: it is a vectorial space, it is a set of pointwise defined fun…

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …

For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In …

New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.

Gravity derived from thermodynamics via optimal transport.

Geometric methods solve sampling, optimisation, inference, and adaptive decision-making.

Study spectral and index properties of Hodge-Dirac operator on compact manifolds.

A new kernel-based CI test improves on existing methods.

Sharp gradient estimate for Green functions on non-parabolic RCD(0,N) spaces.

New framework for efficient PD averaging and clustering.

The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$ with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function $k:X\to \mathbb R$ we introduce the curvature-dimension condition $CD(k,\infty)$ which canonically ex…

Paper generalizes regularization methods for Banach spaces.

The paper shows how multi-task learning in neural networks is similar to kernel regression and Hilbert spaces.

The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.

Analysis of gradient descent on wide neural networks reveals strong generalization.

This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…