Faster dynamic programming for tridiagonal maximum-entropy sampling

2026-10-06

AI Summary

This study addresses the high computational complexity of maximum entropy sampling for tridiagonal covariance matrices by proposing an efficient dynamic programming algorithm grounded in Ornstein–Uhlenbeck process modeling. By exploiting prefix dependencies within the recursive structure and leveraging Monge matrix properties to optimize state transitions, the proposed method significantly reduces the computational complexity from O(n⁵) to O(ns²) and further to O(ns), while also identifying permuted tridiagonal cases. Moreover, this work presents acceleration schemes for optimal value computation, derives closed-form solutions for first-order autoregressive models, and establishes complexity lower bounds for spider graph structures. Collectively, these contributions provide an efficient and theoretically guaranteed framework for sampling from high-dimensional structured covariances.

Abstract

The maximum-entropy sampling problem (MESP) seeks, for an order-$n$ covariance matrix $C$, a principal submatrix of order $s$ with maximum log-determinant. Mostly for convenience, we assume that $C$ is nonsingular. Al-Thani and Lee (2023) solved MESP in $O(n^5)$ time when $C$ or $C^{-1}$ is tridiagonal. We show that the inner maximization of their recursion depends only on a prefix of the index set and that no piece of a solution is longer than $s$; this gives an $O(ns^2)$-time algorithm that returns the optimal value for every budget $t\le s$. When $C^{-1}$ is tridiagonal, $C$ is, up to scaling, the covariance matrix of an Ornstein--Uhlenbeck process observed at unevenly spaced times, and MESP becomes choosing points on a line under a concave gap function with the Monge property; this gives an $O(ns)$-time algorithm and, in the first-order autoregressive case, a closed-form solution. Given only $C$, we solve MESP in $O(n^2)$ time whenever $C$ or $C^{-1}$ is tridiagonal, up to a symmetric permutation, and we recognize these cases within the same bound. For spiders, we make explicit, and sharpen, the dependence on the number of legs, and, drawing on a hardness result of Ohsaka for stars, we observe that, unless $\mathrm{P}=\mathrm{NP}$, the exponent of the running time must grow with the number of legs.

Problem

• Maximum-entropy sampling problem
• Tridiagonal matrix
• Log-determinant maximization
• Dynamic programming
• Ornstein-Uhlenbeck process

Innovation

• Maximum-entropy sampling
• Dynamic programming
• Tridiagonal matrix
• Monge property
• Ornstein-Uhlenbeck process