Pointwise permanent bounds for repeated-column eigenvector matrices

Establish pointwise upper bounds on Perm(U_{n−ℓ,ℓ2,…,ℓn}) for the matrices formed by repeating columns of the orthogonal eigenvector matrix U in the generalized Cauchy–Binet expansion of the permanent of A = U D U^T, where A_{ij} = (1/n) ∫ (dP_i dP_j)/d\overline{P} is the doubly-stochastic positive semidefinite matrix used to represent χ^2(\mathbb{P}_n \| \mathbb{Q}_n). Such bounds should exploit spectral information beyond trace and spectral gap to yield sharper control of χ^2(\mathbb{P}_n \| \mathbb{Q}_n).

Background

In Section 5 the authors express the χ^2 divergence between the permutation mixture \mathbb{P}n and its i.i.d. counterpart \mathbb{Q}_n as the permanent of a doubly-stochastic PSD matrix A, and then as a sum of homogeneous polynomials Sℓ in the eigenvalues of A via a generalized Cauchy–Binet formula.

The coefficients in this sum are permanents of matrices U_{n−ℓ,ℓ2,…,ℓn} constructed by repeating columns of the orthogonal matrix U of eigenvectors of A. Bounding these coefficients pointwise could provide refined bounds on χ^2 that exploit more of the spectrum than the trace and spectral gap. The authors explicitly state that they have not succeeded in this approach and identify it as an open direction.