arXiv research

Introduces generalized almost statistical convergence and its properties.

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

Deep neural networks define suitable reproducing kernel Banach spaces.

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

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…

Paper characterizes embeddability of function spaces into $L_p$-type RKBS via metric entropy.

We endow projective (resp. direct) limits of Banach tensor structures with Fréchet (resp. convenient) structures and study adapted connections to $G$-structures in both frameworks. This situation is illustrated by a lot of examples.

Researchers develop neural networks for approximating functions in Banach spaces.

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

Suppose M be the projective limit of weak symplectic Banach manifolds \{(M_i,φ_{ij})\}_{i,j\in\mathbb N}, where M_i are modeled over reflexive Banach space and σis compatible with the inverse system(defined in the article). We associate to each point x\in M, a Fréchet space H_x(defined in section 3). We prove that if H…

Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.

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…

Extends Gaussian process theory to Banach spaces.

New neural architectures with multivariate nonlinearities are optimal in function space.

The paper uses Banach spaces to analyze neural networks.

We characterize the class of separable Banach spaces $X$ such that for every continuous function $f:X\to\mathbb{R}$ and for every continuous function $ε:X\to\mathbb(0,+\infty)$ there exists a $C^1$ smooth function $g:X\to\mathbb{R}$ for which $|f(x)-g(x)|\leqε(x)$ and $g'(x)\neq 0$ for all $x\in X$ (that is, $g$ has no…

New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous lin…

In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (nor…

Ambrose, Palais and Singer \cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath \cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is…

Develops a mathematical framework for causal fermion systems in infinite dimensions.

The paper constructs Morse homology for functionals involving the p-Laplacian in Banach spaces.

We define the notion of strong projective limit of Banach Lie algebroids. We study the associated structures of Fréchet bundles and the compatibility with the different morphisms. This kind of structure seems to be a convenient framework for various situations.

This paper presents the theory of non-smooth Lie group actions on chains of Banach manifolds. The rigorous functional analytic spaces are given to deal with quotients of such actions. A hydrodynamical example is studied in detail.

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem li…

Backtracking Gradient Descent method converges to critical points in Banach spaces.

Random feature models approximate functions in Banach spaces efficiently.

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic …

We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a sui…

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 …

Proves critical points of ADM mass correspond to specific initial data sets.

In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^k$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structu…

Study Cowen-Douglas operators from analytic function spaces.

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

Mathematical construction of Chern-Simons partition function using reflection positivity.

In this paper we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel-Rockafellar duality theory, we give explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving…

The paper defines a hypothesis space for deep learning using DNNs.

Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space $X$. Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds…

New RL method improves financial index tracking accuracy.

The concept of subdifferentiability is studied in the context of $C^1$ Finsler manifolds (modeled on a Banach space with a Lipschitz $C^1$ bump function). A class of Hamilton-Jacobi equations defined on $C^1$ Finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions…

Proves linear extension of isometries in smooth 2D Banach spaces.

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

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

Formalizes integral curves on Banach manifolds in Lean.

Represents neural networks as solutions to inverse problems in Banach spaces.

Anosov maps study with new Banach space and foliation method.