相关论文: On the approximation of a matrix
Let $T$ be a square matrix with a real spectrum, and let $f$ be an analytic function. The problem of the approximate calculation of $f(T)$ is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that $T$…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
A sequence of approximations for the determinant and its logarithm of a complex matrixis derived, along with relative error bounds. The determinant approximations are derived from expansions of det(X)=exp(trace(log(X))), and they apply to…
Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater urgency recently, within the context of geometric…
This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
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…
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
Random matrices tend to be well conditioned, and we employ this well known property to advance matrix computations. We prove that our algorithms employing Gaussian random matrices are efficient, but in our tests the algorithms have…
Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices…
This work considers the low-rank approximation of a matrix $A(t)$ depending on a parameter $t$ in a compact set $D \subset \mathbb{R}^d$. Application areas that give rise to such problems include computational statistics and dynamical…
We present and analyze a simple numerical method that diagonalizes a complex normal matrix A by diagonalizing the Hermitian matrix obtained from a random linear combination of the Hermitian and skew-Hermitian parts of A.
In recent years, randomized algorithms have established themselves as fundamental tools in computational linear algebra, with applications in scientific computing, machine learning, and quantum information science. Many randomized matrix…
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are…
We present a method for randomizing formulas for bilinear computation of matrix products. We consider the implications of such randomization when there are two sources of error: One due to the formula itself only being approximately…
Computing accurate low rank approximations of large matrices is a fundamental data mining task. In many applications however the matrix contains sensitive information about individuals. In such case we would like to release a low rank…
In this paper, we study the general problem of optimizing a convex function $F(L)$ over the set of $p \times p$ matrices, subject to rank constraints on $L$. However, existing first-order methods for solving such problems either are too…
Given a matrix $A$, a matrix nearness problem seeks an $X$ that most closely approximates $A$ in the sense of minimizing $\lVert A - X\rVert$ under a variety of constraints on $X$. A generalized matrix nearness problem seeks the same but…
Motivated by recent progress in quantum technologies and in particular quantum software, research and industrial communities have been trying to discover new applications of quantum algorithms such as quantum optimization and machine…
We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle…