Wasserstein mixing of a systematic-scan random rotation sampler

Abstract

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.