Related papers: Reproducing Kernel Banach Spaces with the l1 Norm

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of…

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to…

Let $X$ be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_n:X\to X$ with the property that, for each $f\in X$, the functions $T_n(f)$ are…

This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with…

We characterize the reproducing kernel Hilbert spaces whose elements are $p$-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for $p=2$ we show that the spectral…

This note deals with the question: If T is a linear mapping between Banach spaces X and Y, and x belongs to X and has small norm, is x close to the kernel of T? It draws on notions of Z-stability and provides an affirmative constructive…

We develop a variational framework to understand the properties of functions learned by fitting deep neural networks with rectified linear unit activations to data. We propose a new function space, which is reminiscent of classical bounded…

Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice…

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of $L^p(\Rd), 1\le p\le \infty$, associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity.…

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…

In this paper, we will see that the Cartesian product of two 2-Banach spaces is also 2-Banach space and discuss some properties of closed linear operator in linear 2-normed space. We also describe the concept of different types of…

Motivated by the growing interest in representation learning approaches that uncover the latent structure of high-dimensional data, this work proposes new algorithms for reconstruction-based manifold learning within Reproducing-Kernel…

We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples…

We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in…

We propose inferential tools for functional linear quantile regression where the conditional quantile of a scalar response is assumed to be a linear functional of a functional covariate. In contrast to conventional approaches, we employ…

In continuation of the paper [3], we discuss various consequences of Hahn-Banach theorem for bounded b-linear functional in linear n-normed space and describe the notion of reflexivity of linear n-normed space with respect to bounded…

This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator,…

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by…

We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this…

Kernel theorems, in general, provide a convenient representation of bounded linear operators. For the operator acting on a concrete function space, this means that its action on any element of the space can be expressed as a generalised…