arXiv subjects

Robin Armstrong

Publications and source records attributed to Robin Armstrong.

4 recordsLinked to original sources

arXiv2026

Estimating Hierarchically Rank Structured Covariance Matrices

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

stat.CO↗

arXiv2025

Data Assimilation With An Integral-Form Ensemble Square-Root Filter

Geoscientific applications of ensemble Kalman filters face several computational challenges arising from the high dimensionality of the forecast covariance matrix, particularly when this matrix incorporates localization. For square-root filters, updating the perturbations of the ensemble members from their mean is an especially challenging step, one which generally requires approximations that introduce a trade-off between accuracy and computational cost. This paper describes an ensemble square-root filter which achieves a favorable trade-off between these factors by discretizing an integral representation of the Kalman filter update equations, and in doing so, avoids a direct evaluation of the matrix square-root in the perturbation update stage. This algorithm, which we call InFo-ESRF ("Integral-Form Ensemble Square-Root Filter"), is parallelizable and uses a preconditioned Krylov method to update perturbations to a high degree of accuracy. Through numerical experiments with both a Gaussian forecast model and a multi-layer Lorenz-type system, we demonstrate that InFo-ESRF is competitive or superior to several existing localized square-root filters in terms of accuracy and cost.

physics.comp-ph↗

arXiv2025

Collect, Commit, Expand: Efficient CPQR-Based Column Selection for Extremely Wide Matrices

Column-pivoted QR (CPQR) factorization is a computational primitive used in numerous applications that require selecting a small set of ``representative'' columns from a much larger matrix. These include applications in spectral clustering, model-order reduction, low-rank approximation, and computational quantum chemistry, where the matrix being factorized has a moderate number of rows but an extremely large number of columns. We describe a modification of the Golub-Businger algorithm which, for many matrices of this type, can perform CPQR-based column selection much more efficiently. This algorithm, which we call CCEQR, is based on a three-step ``collect, commit, expand'' strategy that limits the number of columns being manipulated, while also transferring more computational effort from level-2 BLAS to level-3. Unlike most CPQR algorithms that exploit level-3 BLAS, CCEQR is deterministic, and provably recovers a column permutation equivalent to the one computed by the Golub-Businger algorithm. Tests on spectral clustering and Wannier basis localization problems demonstrate that on appropriately structured problems, CCEQR can significantly outperform GEQP3.

math.NA↗

arXiv2023

Structure-Aware Analyses and Algorithms for Interpolative Decompositions

Low-rank approximation is a task of critical importance in modern science, engineering, and statistics. Many low-rank approximation algorithms, such as the randomized singular value decomposition (RSVD), project their input matrix into a subspace approximating the span of its leading singular vectors. Other algorithms compress their input into a small subset of representative rows or columns, leading to a so-called interpolative decomposition. This paper investigates how the accuracy of interpolative decompositions is affected by the structural properties of the input matrix being operated on, including how these properties affect the performance comparison between interpolative decompositions and RSVD. We also introduce a novel method of interpolative decomposition in the form of the randomized Golub-Klema-Stewart (RGKS) algorithm, which combines RSVD with a pivoting strategy for column subset selection. Through numerical experiments, we find that matrix structures including singular subspace geometry and singular spectrum decay play a significant role in determining the performance comparison between these different algorithms. We also prove inequalities which bound the error of a general interpolative decomposition in terms of these matrix structures. Lastly, we develop forms of these bounds specialized to RGKS while considering how randomization affects the approximation error of this algorithm.

math.NA↗