Related papers: Computational Improvements to Matrix Operations

We propose a general technique for improving alternating optimization (AO) of nonconvex functions. Starting from the solution given by AO, we conduct another sequence of searches over subspaces that are both meaningful to the optimization…

It was recently shown that quantum annealing can be used as an effective, fast subroutine in certain types of matrix factorization algorithms. The quantum annealing algorithm performed best for quick, approximate answers, but performance…

Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…

The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is…

The achievable data rates of current fiber-optic wavelength-division-multiplexing (WDM) systems are limited by nonlinear interactions between different subchannels. Recently, it was thus proposed to replace the conventional Fourier…

The classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method…

The second order method as Newton Step is a suitable technique in Online Learning to guarantee regret bound. The large data is a challenge in Newton method to store second order matrices as hessian. In this paper, we have proposed an…

We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [OSV12], we establish an equivalence between matrix inversion and…

We assume the permutation $\pi$ is given by an $n$-element array in which the $i$-th element denotes the value $\pi(i)$. Constructing its inverse in-place (i.e. using $O(\log{n})$ bits of additional memory) can be achieved in linear time…

General-purpose multiprocessors (as, in our case, Intel IvyBridge and Intel Haswell) increasingly add GPU computing power to the former multicore architectures. When used for embedded applications (for us, Synthetic aperture radar) with…

The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of…

In recent years, with the rapid development of electro-optic modulators, optical computing has become a potential excellent candidate for various computing tasks. New structures and devices for optical computing are emerging one after…

Some techniques for the use of bitwise operations are described in the article. As an example, an open problem of isomorphism-free generations of combinatorial objects is discussed. An equivalence relation on the set of square binary…

Parker and L\^e introduced random butterfly transforms (RBTs) as a preprocessing technique to replace pivoting in dense LU factorization. Unfortunately, their FFT-like recursive structure restricts the dimensions of the matrix. Furthermore,…

A tight lower bound for required I/O when computing an ordinary matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of segments needed…

We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the…

This paper analyzes the convergence of fixed-point iterations of the form u = f(u) and the properties of the inverse of the related pentadiagonal matrices, associated with the fourth-order nonlinear beam equation. This nonlinear problem is…

Matrix multiplication is a fundamental classical computing operation whose efficiency becomes a major challenge at scale, especially for machine learning applications. Quantum computing, with its inherent parallelism and exponential storage…

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…

We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors…