Spanning-tree polynomial lower bounds to improve Gaussian χ^2 approximation

Derive nontrivial lower bounds on the spanning-tree polynomial \sum_{T} \prod_{(i,j)\in T} A_{ij} of the doubly-stochastic PSD matrix A associated with the Gaussian location family (where A_{ij} = (1/n) ∫ (dP_i dP_j)/d\overline{P}) to obtain improved upper bounds on χ^2(\mathbb{P}_n \| \mathbb{Q}_n) beyond those achievable using only trace and spectral gap information.

Background

The authors’ permanent-based bounds imply χ^2(\mathbb{P}_n | \mathbb{Q}n)+1 ≤ \prod{i=2}^n (1−λ_i)^{−1}, which equals a spanning-tree polynomial via the weighted matrix tree theorem applied to I−A.

They note that exploiting structural properties of A beyond trace and spectral gap could tighten bounds for specific families, notably Gaussian mixtures. They explicitly leave this exploitation as an open question and conjecture that lower bounding the spanning-tree polynomial would yield improved χ^2 bounds in the Gaussian family.