Related papers: The Generalized Matrix Chain Algorithm
Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in…
Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization, $\ell_1$ norm regularized optimization, and $\ell_0$ norm regularized…
We present a novel algorithm attaining excessively fast, the sought solution of linear systems of equations. The algorithm is short in its basic formulation and, by definition, vectorized, while the memory allocation demands are trivial,…
We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the…
Modern research in code generators for dense linear algebra computations has shown the ability to produce optimized code with a performance which compares and often exceeds the one of state-of-the-art implementations by domain experts.…
A generalized exponential matrix based on the construction of kernel operators for generalized summability is defined and analyzing its main properties, generalizing the classical exponential matrix and fractional exponential matrix. This…
Matrix--vector algorithms, particularly Krylov subspace methods, are widely viewed as the most effective algorithms for solving large systems of linear equations. This paper establishes lower bounds on the worst-case number of…
Matrix completion aims to predict missing elements in a partially observed data matrix which in typical applications, such as collaborative filtering, is large and extremely sparsely observed. A standard solution is matrix factorization,…
Large Language Models (LLMs) have already become quite proficient at solving simpler programming tasks like those in HumanEval or MBPP benchmarks. However, solving more complex and competitive programming tasks is still quite challenging…
Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer…
In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In…
Choosing the most adequate kernel is crucial in many Machine Learning applications. Gaussian Process is a state-of-the-art technique for regression and classification that heavily relies on a kernel function. However, in the Gaussian…
Many useful tasks in data science and machine learning applications can be written as simple variations of matrix multiplication. However, users have difficulty performing such tasks as existing matrix/vector libraries support only a…
The biggest cost of computing with large matrices in any modern computer is related to memory latency and bandwidth. The average latency of modern RAM reads is 150 times greater than a clock step of the processor. Throughput is a little…
A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the Boolean satisfiability problem (SAT), and…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it,…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
One of the greatest efforts of computational scientists is to translate the mathematical model describing a class of physical phenomena into large and complex codes. Many of these codes face the difficulty of implementing the mathematical…