Related papers: K-Dimensional Coding Schemes in Hilbert Spaces

The article explores an encoding and structural information processing approach using sparse bit vectors and fixed-length linear vectors. The following are presented: a discrete method of speculative stochastic dimensionality reduction of…

Hilbert space combines the properties of two fundamentally different types of mathematical spaces: vector space and metric space. While the vector-space aspects of Hilbert space, such as formation of linear combinations of state vectors,…

We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those…

A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing…

Encoding in a high-dimensional Hilbert space improves noise resilience in quantum information processing. This approach, however, may result in cross-mode coupling and detection complexities, thereby reducing quantum cryptography…

We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic…

This paper is concerned with the question of reconstructing a vector in a finite-dimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new…

We provide a characterization of the finite dimensionality of vector spaces in terms of the right-sided invertibility of linear operators on them.

This paper proposes a novel kernel approach to linear dimension reduction for supervised learning. The purpose of the dimension reduction is to find directions in the input space to explain the output as effectively as possible. The…

Reduced modeling in high-dimensional reproducing kernel Hilbert spaces offers the opportunity to approximate efficiently non-linear dynamics. In this work, we devise an algorithm based on low rank constraint optimization and kernel-based…

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer…

Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This…

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space R^n, Hilbert spaces admit various useful…

This article presents a simple characterization for entangled vectors in a finite dimensional Hilbert space $H$. The characterization is in terms of the coefficients of an expansion of the vector relative to an orthonormal basis for $H$.…

We study the typical behavior of bounded linear operators on infinite dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral…

Function encoders are a recent technique that learn neural network basis functions to form compact, adaptive representations of Hilbert spaces of functions. We show that function encoders provide a principled connection to feature learning…

We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…

We introduce the new concept of computation coding. Similar to how rate-distortion theory is concerned with the lossy compression of data, computation coding deals with the lossy computation of functions. Particularizing to linear…

Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which…