相关论文: Recurrence relations and fast algorithms

Recent studies show that the reasoning capabilities of Large Language Models (LLMs) can be improved by applying Reinforcement Learning (RL) to question-answering (QA) tasks in areas such as math and coding. With a long context length, LLMs…

Algorithms are the engine for reproducible problem-solving. We present a framework automating algorithm discovery by conceptualizing them as sequences of operations, represented as tokens. These computational tokens are chained using a…

Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and…

Inference algorithms based on evolving interactions between replicated solutions are introduced and analyzed on a prototypical NP-hard problem - the capacity of the binary Ising perceptron. The efficiency of the algorithm is examined…

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…

Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…

Large-scale sequential data is often exposed to some degree of inhomogeneity in the form of sudden changes in the parameters of the data-generating process. We consider the problem of detecting such structural changes in a high-dimensional…

The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres…

The computational cost required to calculate nuclear correlation functions grows factorially in the number of quarks, making the study of large nuclei inaccessible to ab initio study using lattice QCD at the present time. However, the…

Many applications that use empirically estimated functions face a curse of dimensionality, because the integrals over most function classes must be approximated by sampling. This paper introduces a novel regression-algorithm that learns…

We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to…

Replication of experimental results has been a challenge faced by many scientific disciplines, including the field of machine learning. Recent work on the theory of machine learning has formalized replicability as the demand that an…

Despite their tremendous success and versatility, Deep Neural Networks (DNNs) such as Large Language Models (LLMs) suffer from inference inefficiency and rely on advanced computational infrastructure. To address these challenges and make…

Recent efforts to improve the performance of neural network (NN) accelerators that meet today's application requirements have given rise to a new trend of logic-based NN inference relying on fixed-function combinational logic (FFCL). This…

This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…

We present a new algorithm for iterating over all permutations of a sequence. The algorithm leverages elementary~$O(1)$ operations on recursive lists. As a result, no new nodes are allocated during the computation. Instead, all elements are…

Linear algebra algorithms often require some sort of iteration or recursion as is illustrated by standard algorithms for Gaussian elimination, matrix inversion, and transitive closure. A key characteristic shared by these algorithms is that…

The number of n-gram features grows exponentially in n, making it computationally demanding to compute the most frequent n-grams even for n as small as 3. Motivated by our production machine learning system built on n-gram features, we ask:…

Different features have different relevance to a particular learning problem. Some features are less relevant; while some very important. Instead of selecting the most relevant features using feature selection, an algorithm can be given…

We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…