[Paper Review] Moments of isotropic measures and optimal projective codes

Alexey Glazyrin|arXiv (Cornell University)|Apr 25, 2019

Mathematical Approximation and Integration70 references4 citations

TL;DR

This paper develops a linear programming framework to derive sharp upper bounds on moments of isotropic measures in real, complex, and quaternionic spaces, which are then used to establish new lower bounds for $p$-frame energy in projective codes. The key result resolves a conjecture by Bukh and Cox by proving that the $q$-th moment of an isotropic measure in $K^d$ is bounded by $\beta^q + \frac{1 - \beta^q}{M}$, with equality for maximal projective simplices at $d = 2,3,7,23$ ($K=\mathbb{R}$), $d=3$ ($K=\mathbb{H}$), and conjecturally all $d$ ($K=\mathbb{C}$).

ABSTRACT

In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtained energy bounds are sharp for several infinite families of codes.

Motivation & Objective

• To extend the linear programming approach to bound moments of isotropic measures in $\mathbb{R}^d$, $\mathbb{C}^d$, and $\mathbb{H}^d$.
• To derive new lower bounds for $p$-frame energy potentials in projective codes using these moment bounds.
• To resolve a conjecture by Bukh and Cox on the sharpness of first-moment bounds for isotropic measures.
• To construct infinite families of projective codes for which the derived energy bounds are tight.
• To generalize the duality between projective codes and tight frames using Gale transforms and Hölder's inequality.

Proposed method

• Adapt the Yudin-type linear programming method to isotropic measures in $K^d$ for $K = \mathbb{R}, \mathbb{C}, \mathbb{H}$.
• Derive a general linear programming bound for $q$-th moments of isotropic measures with $q \in [1,2]$.
• Use the bound $\beta^q + \frac{1 - \beta^q}{M}$ with $\beta = \sqrt{\frac{1}{d + 2(dim_{\mathbb{R}}K)^{-1}}}$ and $M = d + \frac{d^2 - d}{2} dim_{\mathbb{R}}K$.
• Apply the moment bounds to derive lower bounds for $p$-frame energy potentials on projective codes.
• Construct minimizing configurations via a one-parameter family of Gram matrices derived from Gale duality of tight frames.
• Verify sharpness of bounds by showing equality holds for uniform distributions over maximal projective simplices.

Experimental results

Research questions

• RQ1 What are the optimal upper bounds for $q$-th moments of isotropic measures in $K^d$ for $K = \mathbb{R}, \mathbb{C}, \mathbb{H}$?
• RQ2 Can the linear programming method be extended to derive sharp energy bounds for $p$-frame potentials on projective codes?
• RQ3 Is the conjecture by Bukh and Cox on the sharpness of the first-moment bound for isotropic measures true?
• RQ4 For which infinite families of projective codes are the derived energy bounds tight?
• RQ5 Can Gale duality be used to connect energies of projective codes and tight frames via functional inequalities?

Key findings

• The $q$-th moment of an isotropic measure in $K^d$ is bounded above by $\beta^q + \frac{1 - \beta^q}{M}$, where $\beta = \sqrt{\frac{1}{d + 2(dim_{\mathbb{R}}K)^{-1}}}$ and $M = d + \frac{d^2 - d}{2} dim_{\mathbb{R}}K$.
• This bound is sharp precisely for uniform distributions over maximal projective simplices, which exist for $d = 2,3,7,23$ when $K = \mathbb{R}$, $d = 3$ when $K = \mathbb{H}$, and conjecturally for all $d$ when $K = \mathbb{C}$.
• The paper resolves the conjecture of Bukh and Cox by showing their first-moment bound is a special case of this general result at $q = 1$.
• New lower bounds for $p$-frame energy potentials on projective codes in $\mathbb{R}^d$, $\mathbb{C}^d$, and $\mathbb{H}^d$ are derived using the moment bounds.
• Infinite families of projective codes are constructed where the derived energy bounds are tight, including a one-parameter family of minimizing Gram matrices for 6-point codes in $\mathbb{R}^4$.
• For the 6-point code in $\mathbb{R}^4$, the minimizing configurations form a line segment of matrices parameterized by $\alpha = 1 + \frac{1}{1 + 2^{\frac{p-2}{p-1}}}$, with $\alpha \in [\frac{4}{3}, \frac{3}{2}]$, yielding 32 distinct minimizing Gram matrices.