Related papers: Complex to Rational Fast Matrix Multiplication
Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods that obtain quantum speedups with respect to the…
Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant…
Although the matrix multiplication plays a vital role in computational linear algebra, there are few efficient solutions for matrix multiplication of the near-sparse matrices. The Sparse Approximate Matrix Multiply (SpAMM) is one of the…
We call matrix algorithms superfast if they use much fewer flops and memory cells than the input matrix has entries. Using such algorithms is indispensable for Big Data Mining and Analysis, where the input matrices are so immense that one…
We present an approximate algorithm for matrix multiplication based on matrix sketching techniques. First one of the matrix is chosen and sparsified using the online matrix sketching algorithm, and then the matrix product is calculated…
Computation of a signal's estimated covariance matrix is an important building block in signal processing, e.g., for spectral estimation. Each matrix element is a sum of products of elements in the input matrix taken over a sliding window.…
We study the practical performance of quantum-inspired algorithms for recommendation systems and linear systems of equations. These algorithms were shown to have an exponential asymptotic speedup compared to previously known classical…
The All-Pairs Shortest Paths (APSP) is a foundational problem in theoretical computer science. Approximating APSP in undirected unweighted graphs has been studied for many years, beginning with the work of Dor, Halperin and Zwick…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families…
This paper studies the problem of finding an $(1+\epsilon)$-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all…
Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random…
Rational and neural network based approximations are efficient tools in modern approximation. These approaches are able to produce accurate approximations to nonsmooth and non-Lipschitz functions, including multivariate domain functions. In…
We show how rational function approximations to the logarithm, such as $\log z \approx (z^2 - 1)/(z^2 + 6z + 1)$, can be turned into fast algorithms for approximating the determinant of a very large matrix. We empirically demonstrate that…
We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$…
It is shown how to compute quotients efficiently in non-commutative univariate polynomial rings. This extends earlier work where efficient generic quotients were studied with a primary focus on commutative domains. Fast algorithms are given…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation…
Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic…
We show that under some widely believed assumptions, there are no higher-order algorithms for basic tasks in computational mathematics such as: Computing integrals with neural network integrands, computing solutions of a Poisson equation…