Related papers: A faster multipole Legendre-Chebyshev transform
We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…
A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking…
The inherent diversity of computation types within the deep neural network (DNN) models often requires a variety of specialized units in hardware processors, which limits computational efficiency, increasing both inference latency and power…
Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for…
As machine learning techniques become ubiquitous, the efficiency of neural network implementations is becoming correspondingly paramount. Frameworks, such as Halide and TVM, separate out the algorithmic representation of the network from…
Evaluation of the Voigt function, a convolution of a Lorentzian and a Gaussian profile, is essential in various fields such as spectroscopy, atmospheric science, and astrophysics. Efficient computation of the function is crucial, especially…
The rapid development of the Transformer-based Large Language Models (LLMs) in recent years has been closely linked to their ever-growing and already enormous sizes. Many LLMs contain hundreds of billions of parameters and require dedicated…
Despite their frequent slow convergence, proximal gradient schemes are widely used in large-scale optimization tasks due to their tremendous stability, scalability, and ease of computation. In this paper, we develop and investigate a…
We present a novel Bayesian inference tool that uses a neural network to parameterise efficient Markov Chain Monte-Carlo (MCMC) proposals. The target distribution is first transformed into a diagonal, unit variance Gaussian by a series of…
The Cholesky decomposition is a fundamental tool for solving linear systems with symmetric and positive definite matrices which are ubiquitous in linear algebra, optimization, and machine learning. Its numerical stability can be improved by…
This paper presents fast first-order methods for solving linear programs (LPs) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a…
The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that…
In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this…
The Simplex tableau has been broadly used and investigated in the industry and academia. With the advent of the big data era, ever larger problems are posed to be solved in ever larger machines whose architecture type did not exist in the…
Generalized linear models play an essential role in a wide variety of statistical applications. This paper discusses an approximation of the likelihood in these models that can greatly facilitate computation. The basic idea is to replace a…
While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to "factorize" them, or to decide whether they are "prime". The former is plain linear algebra, while the latter is heavy-duty…
We present a new tunably-accurate Laguerre Petrov-Galerkin spectral method for solving linear multi-term fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in…
Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these…
This work proposes an accelerated first-order algorithm we call the Robust Momentum Method for optimizing smooth strongly convex functions. The algorithm has a single scalar parameter that can be tuned to trade off robustness to gradient…
Efficient approximation lies at the heart of large-scale machine learning problems. In this paper, we propose a novel, robust maximum entropy algorithm, which is capable of dealing with hundreds of moments and allows for computationally…