arXiv research

Gradient flows for knot energies ensure long-term existence of knotted loops.

New Banach spaces for ReLU networks enable better function approximation and gradient dynamics analysis.

Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…

Formalizes integral curves on Banach manifolds in Lean.

Gradient flow of curve length on Sobolev metrics preserves convexity.

In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riem…

In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.

A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.

We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we defi…

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

This paper extends mirror descent to Banach spaces with reproducing kernels.

We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain …

Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.

New spectral theory for non-associative algebras with applications to Moufang dynamics.

Novel neural operator predicts complex spatiotemporal dynamics from partial observations.

We introduce several new black-box reductions that significantly improve the design of adaptive and parameter-free online learning algorithms by simplifying analysis, improving regret guarantees, and sometimes even improving runtime. We reduce parameter-free online learning to online exp-concave optimization, we reduce…

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasie…

Regulated curves on Banach manifolds with continuous projections and regulated derivatives are studied.

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

The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.

First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symme…

The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.

Constructs Poisson structure on Banach Lie algebroid predual.

Improved DeepONets for PDE solution operators with adaptive re-weighting and new architecture.

Researchers develop neural networks for approximating functions in Banach spaces.

Examines different approaches to Poisson structures in Banach spaces.

This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anc…

We prove that any isometry between the unit spheres of $C^2$-smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of absolutely smooth $2$-dimensional Banach spaces.

We study a few basic properties of Banach-Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach-Lie groupoid sheds fresh light on earlier research on some infinite-dimensional mani…

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump function $f$ such that $f'(x)\neq 0$ for every point $x$ in the interior of the …

Banach fibrations and Nijenhuis operators studied for vanishing torsion.

Differential structure on partial isometries over Grassmannian constructed.

This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. T…

A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold $M$ endowed with a multiplication map $μ\colon M \times M \to M$ such that each left multiplication map $μ_x := μ(x,\cdot)$ (with $x \in M$) is an involutive automorphism of $(M,μ)$ with the isolated fixed point $x$. We show that morphisms of …

Gradient descent in neural networks analyzed using RKBS for broader applicability.

New structure on unitary group of Hilbert space.

Extends Gaussian process theory to Banach spaces.

New algorithm solves saddle point problems in Banach spaces.

Optimizes shapes on non-standard manifolds.

The study integrates Banach manifolds into H-manifolds, integrating Lie algebras into H-groups.

Robust SVM optimization in Banach spaces tackles classification uncertainty.

The paper uses Banach spaces to analyze neural networks.

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, …

Improved sampling method using regularized Stein Variational Gradient Flow.

Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …