arXiv research

Global implicit function theorem for Fréchet spaces, solving derivative loss problems.

Recent advances in deep representation learning on Riemannian manifolds extend classical deep learning operations to better capture the geometry of the manifold. One possible extension is the Fréchet mean, the generalization of the Euclidean mean; however, it has been difficult to apply because it lacks a closed form w…

A theorem proves integrability of Fréchet tangent distributions.

Study on the smoothness of solutions to a specific type of stochastic differential equation.

The paper develops predictors for functional data on manifolds.

A new algorithm solves signed Fréchet regression on manifolds with bounded curvature.

Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for the…

Lean 4 formalizes Stokes' theorem for smooth singular cubes.

We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly sympl…

We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of …

Study shows zero probability of cut locus for Fréchet mean on Riemannian manifolds.

Estimates means in metric spaces using quantization.

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to …

GEORCE-FM algorithm optimizes Fréchet means and distances efficiently.

Paper develops statistical tests for covariance matrix regression on manifold.

Paper develops methods for semi-supervised Fréchet regression.

New framework for higher-order singular-value derivatives of rectangular matrices.

Paper introduces a medoid-based approach for efficient Fréchet regression.

The paper proves critical point results for Frechet manifolds.

Study shows continuous evolution of curves in Fréchet distance.

Paper studies weighted Fermat-Frechet problem for simplex edge lengths.

The consistency of Fréchet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fréchet medians are estimated. It…

The study finds the minimum number of critical points for functionals on Frechet spaces and Finsler manifolds.

In this paper we develop the geometry of bounded Fréchet manifolds. We prove that a bounded Fréchet tangent bundle admits a vector bundle structure. But the second order tangent bundle $T^2M$ of a bounded Fréchet manifold $M$, becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As a…

Sharp threshold found for Frechet mean of inhomogeneous graphs.

Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finite-dimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Frechet supermanifold. In addition, we derive results about the structure of…

Study on Frechet distance properties for paths and graphs.

This work analyzes Fréchet regression using comparison geometry, providing theoretical and practical insights.

New graph properties inherited by Frechet mean and median.

Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.

Proposes a new random forest weighted local Fréchet regression method.

Improved method for computing Fréchet means on SPD matrices.

The aim of this article is to present the category of bounded Frechet manifolds in respect to which we will review the geometry of Frechet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some applications to Riemannian geo…

Deep single-index Fréchet regression for metric space-valued outputs

We compute an approximate Fréchet mean for sets of sparse graphs.

New method uses CNNs to estimate graph means.

A new mechanism for differentially private Fréchet mean on SPD matrices.

This paper explores the impact of metric choice on Fréchet regression.

We give sufficient conditions for a $C^1_c$-local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a functio…

Improved Fréchet regression tackles noise and multicollinearity.

The paper computes an approximation to the sample Frechet mean of graph sets using spectral information.

Test partial effects in Frechet regression on Bures-Wasserstein manifolds.

We derive bounds on the distribution function, therefore also on the Value-at-Risk, of $\varphi(\mathbf X)$ where $\varphi$ is an aggregation function and $\mathbf X = (X_1,\dots,X_d)$ is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three type…

Paper optimizes FTPL for adversarial and stochastic bandits with specific tail distributions.

The abstract discusses transversality for infinite dimensional manifolds.

We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.

In this paper we prove a variation of the theorem in title, for equations with periodic coefficients, in Frechet spaces. The main result gives equivalent conditions ensuring the reduction of such an equation to one with constant coefficient. In the particular case of $C^{\infty}$, we obtain the exact analogue of the cl…

A method for reducing dimensions in Fréchet regression models.