Related papers: Entrywise error bounds for low-rank approximations…
The matrix rank minimization problem has applications in many fields such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization…
In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two…
Subset selection for the rank $k$ approximation of an $n\times d$ matrix $A$ offers improvements in the interpretability of matrices, as well as a variety of computational savings. This problem is well-understood when the error measure is…
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density…
Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of ``components.'' Typically, these components are linear combinations of the rows and columns of the matrix, and are thus…
We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces,…
We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries…
Many algorithms in scientific computing and data science take advantage of low-rank approximation of matrices and kernels, and understanding why nearly-low-rank structure occurs is essential for their analysis and further development. This…
The pivoted QLP decomposition is computed through two consecutive pivoted QR decompositions, and provides an approximation to the singular value decomposition. This work is concerned with a partial QLP decomposition of low-rank matrices…
Images are a substantial portion of the internet, making efficient compression important for reducing storage and bandwidth demands. This study investigates the use of Singular Value Decomposition and low-rank matrix approximations for…
CUR matrix decomposition is a randomized algorithm that can efficiently compute the low rank approximation for a given rectangle matrix. One limitation with the existing CUR algorithms is that they require an access to the full matrix A for…
Dual complex numbers can represent rigid body motion in 2D spaces. Dual complex matrices are linked with screw theory, and have potential applications in various areas. In this paper, we study low rank approximation of dual complex…
In this paper, we focus on developing randomized algorithms for the computation of low multilinear rank approximations of tensors based on the random projection and the singular value decomposition. Following the theory of the singular…
Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation…
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…
This paper studies inference in linear models with a high-dimensional parameter matrix that can be well-approximated by a ``spiked low-rank matrix.'' A spiked low-rank matrix has rank that grows slowly compared to its dimensions and nonzero…
Recent advances in representation learning reveal that widely used objectives, such as contrastive and non-contrastive, implicitly perform spectral decomposition of a contextual kernel, induced by the relationship between inputs and their…
Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is…
Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address…
The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known…