Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2209.14869

Improved estimates for the number of non-negative integer matrices with given row and column sums

Maximilian Jerdee·Alec Kirkley·M. E. J. Newman
View original source ↗Read source document ↗Export metadata ↓

Abstract

The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We also use this new estimate as the starting point for an improved numerical method for either counting or sampling matrices using sequential importance sampling. Code implementing our methods is provided.

Explore related subjects

stat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maximilian Jerdee, Alec Kirkley, M. E. J. Newman. 2022-09-29. Improved estimates for the number of non-negative integer matrices with given row and column sums. https://doi.org/10.1098/rspa.2023.0470

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Wasserstein mixing of a systematic-scan random rotation sampler

We study the mixing time of a systematic-scan analogue of Kac's walk that was proposed as a fast surrogate for Haar-distributed orthogonal matrices in randomized high-dimensional algorithms and was conjectured to approach Haar measure after only logarithmically many sweeps. We show that this conjectured speed-up does not occur for convergence of the full matrix law to Haar measure in Frobenius Wasserstein distance. At fixed normalized accuracy, the mixing time lies between order $n/\log n$ and order $n$ sweeps; at fixed absolute Frobenius accuracy, the corresponding bounds are between order $n$ and order $n\log n$. More strongly, below the scale $n/\log n$, the normalized Wasserstein distance remains asymptotically at its extremal value. We also show that the output law is singular with respect to Haar measure for fewer than $n/2$ sweeps. Thus the sampler may provide effective application-specific randomization without exhibiting the much faster full-Haar mixing.

stat.CO↗
arXiv2026

Bayesian Calibration with Functional Outputs Using Elastic Partial Matching

Calibrating a simulation model involves estimating its parameters by comparing model outputs with experimental data, so that simulation results faithfully reproduce the experimental observations. When the outputs are functions of time, there are multiple ways to quantify the discrepancy between experimental and simulated curves. A recent approach based on elastic functional data analysis decomposes a functional output into two components: a function temporally aligned to a template, and the corresponding warping function. This decomposition splits the problem into two independent calibration tasks, thereby addressing functional misalignment. However, it assumes that experimental and simulated curves share the same temporal support, an assumption often violated in practice when initial or end times are themselves uncertain or depend on the calibration parameters. In this work, we reinterpret the decomposition step as an approximation to a more general Bayesian calibration problem that incorporates an error term on the time axis. This perspective allows us to naturally extend the framework to a broader family of time warpings with varying initial or end times, using partial elastic alignment. We illustrate the method on a synthetic test case, comparing it with existing Bayesian calibration methods and demonstrating improved surrogate performance and error modeling. We then apply the proposed approach to the calibration of an equation of state (a thermodynamic equation relating the state variables of a material).

stat.CO↗
arXiv2026

Delayed Acceptance Slice Sampling

Slice sampling is a well-established Markov chain Monte Carlo method for approximate sampling of target distributions which are only known up to a normalizing constant. The method is based on choosing a new state on a slice, i.e., a superlevel set of the given unnormalized target density (with respect to a reference measure). However, slice sampling algorithms usually require per step multiple evaluations of the target density, and thus can become computationally expensive. This is particularly the case for Bayesian inference with costly likelihoods. In this paper, we exploit deterministic approximations of the target density, which are relatively cheap to evaluate, and propose delayed acceptance versions of several common (hybrid) slice samplers. We show ergodicity of the resulting slice sampling methods, discuss the superiority of delayed acceptance (ideal) slice sampling over delayed acceptance Metropolis-Hastings algorithms, and illustrate the benefits of our novel approach in terms of improved computational efficiency in numerical experiments.

stat.CO↗