Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains

TL;DR

This paper introduces a new spectral gap decomposition method for Markov chains, enabling polynomial bounds on the spectral gap for complex biased permutation chains with many classes.

Contribution

It develops a novel Complementary Decomposition theorem that iterates efficiently for orthogonal chains, improving spectral gap bounds without analyzing the projection chain.

Findings

Provides a $1/n$-orthogonal decomposition for biased monotone permutations.

Proves the first polynomial spectral gap bound for chains with $k = \Theta(n/\log n)$ classes.

Allows iterative application of the decomposition theorem $n$ times with only constant loss.

Abstract

The spectral gap of a Markov chain can be bounded by the spectral gaps of constituent "restriction" chains and a "projection" chain, and the strength of such a bound is the content of various decomposition theorems. In this paper, we introduce a new parameter that allows us to improve upon these bounds. We further define a notion of orthogonality between the restriction chains and "complementary" restriction chains. This leads to a new Complementary Decomposition theorem, which does not require analyzing the projection chain. For $\epsilon$-orthogonal chains, this theorem may be iterated $O(1/\epsilon)$ times while only giving away a constant multiplicative factor on the overall spectral gap. As an application, we provide a $1/n$-orthogonal decomposition of the nearest neighbor Markov chain over $k$-class biased monotone permutations on [$n$], as long as the number of particles in each class is at least $C\log n$. This allows us to apply the Complementary Decomposition theorem iteratively $n$ times to prove the first polynomial bound on the spectral gap when $k$ is as large as $\Theta(n/\log n)$. The previous best known bound assumed $k$ was at most a constant.