Tight space-noise tradeoffs in computing the ergodic measure

TL;DR

This paper establishes tight bounds on the space complexity of approximating the ergodic measure of low-dimensional noisy dynamical systems, revealing fundamental tradeoffs and a novel matrix exponentiation technique in space-bounded computation.

Contribution

It provides the first tight bounds on space complexity for computing ergodic measures in noisy systems and introduces a space-efficient matrix exponentiation method.

Findings

Space complexity is polynomial in log(1/ε) and log log(1/δ).

The bounds are tight up to polynomial factors.

A new space-efficient matrix exponentiation technique is proven.

Abstract

In this note we obtain tight bounds on the space-complexity of computing the ergodic measure of a low-dimensional discrete-time dynamical system affected by Gaussian noise. If the scale of the noise is $\varepsilon$, and the function describing the evolution of the system is not by itself a source of computational complexity, then the density function of the ergodic measure can be approximated within precision $\delta$ in space polynomial in $\log 1/\varepsilon+\log\log 1/\delta$. We also show that this bound is tight up to polynomial factors. In the course of showing the above, we prove a result of independent interest in space-bounded computation: that it is possible to exponentiate an $n$ by $n$ matrix to an exponentially large power in space polylogarithmic in $n$.