Effective Resistance in Fixed-Rank External-Field Measures and Constant-Stretch Correlated Sampling on the Hypersimplex

2026-07-15

AI Summary

This work investigates the pseudoinverse properties of indicator-vector covariance matrices under fixed-rank exterior field measures, aiming to establish a tight constant upper bound on effective resistances over hyper-simplices. By leveraging spectral analysis of covariance matrices, Moore–Penrose pseudoinverse estimation, and probabilistic bounds involving elementary symmetric polynomials, the study derives—for the first time—a constant upper bound on pairwise effective resistances without assuming the normalized covariance conjecture. Furthermore, it establishes a positive semidefinite lower bound for the covariance matrix. Building on these results, the authors present the first correlation sampling algorithm on the hyper-simplex with a constant stretch guarantee, improving upon prior logarithmic stretch bounds and resolving an open problem posed by Naor et al.

Abstract

We prove an effective-resistance bound for fixed-rank external-field measures. Let $d\ge2$ be an integer, let $m\in\{1,\ldots,d-1\}$. Let $w\in(0,+\infty)^d$, and let $\mathsf S$ be an $m$-element random subset of $[d]$ distributed according to the rank-$m$ external-field measure with weights $w$, i.e., \[\mathbb P(\mathsf S=S)=\frac{\prod_{i\in S}w_i}{e_m(w)},\qquad S\subseteq\{1,\dots,d\},\quad|S|=m,\] where \[e_m(w):=\sum_{\substack{T\subseteq\{1,\dots,d\}\\|T|=m}}\prod_{\ell\in T}w_\ell\] is the $m$th elementary symmetric polynomial in $w_1,\ldots,w_d$. Let $X:=(X_1,\dots,X_d)^\top$ be its indicator vector, i.e., \[X_i=\mathbb I\{i\in\mathsf S\},\qquad i\in\{1,\dots,d\}.\] Let $Σ:=\operatorname{Cov}(X)$, put $v_i:=Σ_{ii}$ for each $i\in\{1,\dots,d\}$, and let $\mathbf e_1,\ldots,\mathbf e_d$ denote the standard basis of $\mathbb R^d$. Our main result is that, for every $i\ne j$, \[(\mathbf e_i-\mathbf e_j)^\topΣ^\dagger(\mathbf e_i-\mathbf e_j)\le\frac1{v_i}+\frac1{v_j},\] where $Σ^\dagger$ is the Moore-Penrose pseudoinverse of $Σ$. As a consequence, if \[v:=(v_1,\ldots,v_d)^\top,\qquad D:=\operatorname{diag}(v),\qquad V:=\sum_{i=1}^dv_i,\] then, as a corollary, we obtain \[Σ\succeq\frac12\left(D-\frac{vv^\top}{V}\right),\] which establishes a factor-two relaxation of the normalized covariance bound conjectured by Anari, Haqi, and Ma. As a further corollary, combining our theorem with the recent framework of Anari, Haqi, and Ma yields a constant-stretch guarantee for correlated sampling on the hypersimplex without relying on the still-open normalized covariance conjecture assumed in their conditional result. Our result improves the logarithmic-in-$k$ stretch bound of Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc to a constant and resolves the open question posed in their work.

Problem

• effective resistance
• external-field measures
• correlated sampling
• hypersimplex
• covariance bound

Innovation

• effective resistance
• fixed-rank external-field measures
• constant-stretch correlated sampling
• hypersimplex
• covariance bound