相关论文: Revisiting CUR Perturbation Analysis: A Local Tang…

The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large-scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends…

This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix…

In data analysis, there continues to be a need for interpretable dimensionality reduction methods whereby instrinic meaning associated with the data is retained in the reduced space. Standard approaches such as Principal Component Analysis…

By exploiting the random sampling techniques, this paper derives an efficient randomized algorithm for computing a generalized CUR decomposition, which provides low-rank approximations of both matrices simultaneously in terms of some of…

High-dimensional Schr\"odinger systems arising from tensor-product discretizations suffer from exponential state growth, making direct controller synthesis and real-time closed-loop simulation computationally challenging. Hierarchical…

This paper introduces the concept of generalized interlacing families of polynomials, which extends the classical interlacing polynomial method to handle polynomials of varying degrees. We establish a fundamental property for these…

The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix…

A low-rank approximation of a parameter-dependent matrix $A(t)$ is an important task in the computational sciences appearing for example in dynamical systems and compression of a series of images. In this work, we introduce AdaCUR, an…

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained…

It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the…

We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$…

Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality…

Tensors are often studied by introducing preorders such as restriction and degeneration: the former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local…

Random sampling is a fundamental tool in modern machine learning and numerical linear algebra for reducing the computational cost of large-scale matrix problems. Existing analyses, however, rely primarily on subspace embedding guarantees,…

Truncated Singular Value Decomposition (SVD) calculates the closest rank-$k$ approximation of a given input matrix. Selecting the appropriate rank $k$ defines a critical model order choice in most applications of SVD. To obtain a principled…

In computed tomography (CT), data truncation is a common problem. Images reconstructed by the standard filtered back-projection algorithm from truncated data suffer from cupping artifacts inside the field-of-view (FOV), while anatomical…

Robust principal component analysis (RPCA) is a widely used tool for dimension reduction. In this work, we propose a novel non-convex algorithm, coined Iterated Robust CUR (IRCUR), for solving RPCA problems, which dramatically improves the…

CUR and low-rank approximations are among most fundamental subjects of numerical linear algebra, with a wide range of applications to a variety of highly important areas of modern computing, which range from the machine learning theory and…

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality…

Interpolative and CUR decompositions involve "natural bases" of row and column subsets, or skeletons, of a given matrix that approximately span its row and column spaces. These low-rank decompositions preserve properties such as sparsity or…