On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

TL;DR

This paper investigates the surprising effectiveness of a simple first-order matrix exponential derivative approximation, providing theoretical bounds and demonstrating its application to high-dimensional CTMC models in SARS-CoV-2 spread analysis.

Contribution

It offers rigorous error bounds for the naive approximation and applies it successfully to high-dimensional CTMC models in epidemiology.

Findings

The naive approximation performs well even in high-dimensional settings.

Theoretical bounds confirm the approximation's accuracy.

Application to SARS-CoV-2 shows practical utility in epidemiological modeling.

Abstract

The continuous-time Markov chain (CTMC) is the mathematical workhorse of evolutionary biology. Learning CTMC model parameters using modern, gradient-based methods requires the derivative of the matrix exponential evaluated at the CTMC's infinitesimal generator (rate) matrix. Motivated by the derivative's extreme computational complexity as a function of state space cardinality, recent work demonstrates the surprising effectiveness of a naive, first-order approximation for a host of problems in computational biology. In response to this empirical success, we obtain rigorous deterministic and probabilistic bounds for the error accrued by the naive approximation and establish a "blessing of dimensionality" result that is universal for a large class of rate matrices with random entries. Finally, we apply the first-order approximation within surrogate-trajectory Hamiltonian Monte Carlo for the analysis of the early spread of SARS-CoV-2 across 44 geographic regions that comprise a state space of unprecedented dimensionality for unstructured (flexible) CTMC models within evolutionary biology.