Related papers: Singular Value Decomposition in Sobolev Spaces: Pa…
We propose definitions of SVD, spectral decomposition (for self-adjoint matrices) and Jordan decomposition which make sense for all rings. For many rings, these decompositions can be shown to exist. For some specific rings, these…
We study a semigroup of weighted composition operators on the Hardy space of the disk $H^2(\mathbb{D})$, and more generally on the Hardy space $H^2(U)$ attached to a simply connected domain $U$ with smooth boundary. Motivated by conformal…
Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An…
The Randomized Singular Value Decomposition (RSVD) is a widely used algorithm for efficiently computing low-rank approximations of large matrices, without the need to construct a full-blown SVD. Of interest, of course, is the approximation…
By using the vector-valued theory of singular integrals, we prove a Hardy--Littlewood--Sobolev inequality on product Hardy spaces $H^p_{\rm{prod}}$, which is a parallel result of the classical Hardy--Littlewood--Sobolev inequality. The same…
The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and $L^2$ forms. We further extend the Hodge decomposition to the Sobolev space $H^1$ for general…
The main goal of this paper is to introduce a new fractional anisotropic Sobolev space with variable exponent where the basic qualitative properties (completeness, separability, reflexivity, ...) are established, including the continuous…
Singular value decomposition is widely used in modal analysis, such as proper orthogonal decomposition and resolvent analysis, to extract key features from complex problems. SVD derivatives need to be computed efficiently to enable the…
This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional…
We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, named TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via…
Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value…
From linear classifiers to neural networks, image classification has been a widely explored topic in mathematics, and many algorithms have proven to be effective classifiers. However, the most accurate classifiers typically have…
Singular Value Decomposition (SVD) is a powerful tool for multivariate analysis. However, independent computation of the SVD for each sample taken from a bandlimited matrix random process will result in singular value sample paths whose…
The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fundamental but expensive tool for tensor computations. In the…
Efficient and fast computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial because of its many potential applications. The current/existing subspace randomized algorithms…
We present a new formulation of the hyperbolic singular value decomposition (HSVD) for an arbitrary complex (or real) matrix without hyperexchange matrices and redundant invariant parameters. In our formulation, we use only the concept of…
We characterize a weighted norm inequality which corresponds to the embedding of a class of absolutely continuous functions into the fractional order Sobolev space. The auxiliary result of the paper is of independent interest. It comprises…
The traditional method of computing singular value decomposition (SVD) of a data matrix is based on a least squares principle, thus, is very sensitive to the presence of outliers. Hence the resulting inferences across different applications…
In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor…
This paper describes the singular value decomposition (SVD) of the Poisson kernel for the Dirichlet problem for the Laplacian on bounded regions in R^N, N >=2. This operator is a compact linear transformation from L^2 of the boundary to L^2…