[Paper Review] On the Further Structure of the Finite Free Convolutions

Jonathan Leake, Nick Ryder|arXiv (Cornell University)|Nov 14, 2018

Random Matrices and Applications4 references4 citations

TL;DR

This paper generalizes Marcus, Spielman, and Srivastava's root bound framework for finite free convolutions to all differential operators preserving real-rootedness, extending their largest root bounds beyond specific operators. It introduces new root bounds for interior roots using hyperbolic polynomials and proposes conjectures on submodular majorization and multivariate extensions, while providing a counterexample to a natural multivariate generalization of their bound.

ABSTRACT

Since the celebrated resolution of Kadison-Singer (via the Paving Conjecture) by Marcus, Spielman, and Srivastava, much study has been devoted to further understanding and generalizing the techniques of their proof. Specifically, their barrier method was crucial to achieving the required polynomial root bounds on the finite free convolution. But unfortunately this method required individual analysis for each usage, and the existence of a larger encapsulating framework is an important open question. In this paper, we make steps toward such a framework by generalizing their root bound to all differential operators. We further conjecture a large class of root bounds, the resolution of which would require for more robust techniques. We further give an important counterexample to a very natural multivariate version of their bound, which if true would have implied tight bounds for the Paving Conjecture.

Motivation & Objective

• To generalize Marcus, Spielman, and Srivastava's largest root bound for finite free convolutions to all differential operators preserving real-rootedness.
• To develop new bounds on interior roots (not just the largest root) using the theory of hyperbolic polynomials.
• To formulate and investigate a class of conjectures on submodular majorization of root sequences under finite free convolution.
• To explore the limitations of multivariate extensions of the finite free convolution framework, particularly by providing a counterexample to a natural multivariate generalization of the root bound.
• To lay foundational groundwork for a unified framework to generalize the barrier method beyond ad hoc applications in problems like the Paving Conjecture and Kadison-Singer.

Proposed method

• Generalizes the largest root bound from specific operators (e.g., $I - \alpha D$) to all differential operators preserving real-rootedness via the finite free convolution $p \boxplus^n q$.
• Uses the theory of hyperbolic polynomials to derive bounds on the movement of interior roots, not just the largest root.
• Applies induction and extremal polynomial arguments to prove submodular-like inequalities on root sequences, leading to a contradiction if a stronger inequality were to hold.
• Employs continuity and compactness arguments to extend results from degree-$n$ polynomials to limits involving roots tending to $-\infty$, ensuring stability of inequalities.
• Introduces a counterexample to a natural multivariate extension of the root bound, demonstrating that such a generalization fails in the multivariate setting.
• Proposes a framework for future work on $b$-additive convolutions using finite differences, which generalize the standard derivative-based convolution.

Experimental results

Research questions

• RQ1 Can the largest root bound for finite free convolutions be extended to all differential operators that preserve real-rootedness, not just specific forms like $I - \alpha D$?
• RQ2 What are the structural constraints on the movement of interior roots (other than the largest) under finite free convolution, and can they be bounded using hyperbolic polynomial theory?
• RQ3 Is there a submodular majorization structure in the root sequences of finite free convolutions, and if so, can it be proven?
• RQ4 Does a natural multivariate generalization of the Marcus-Srivastava root bound hold, and if not, what are the obstructions?
• RQ5 Can the framework of finite free convolutions be extended to $b$-additive convolutions using finite differences, and what advantages might this offer over derivative-based methods?

Key findings

• The largest root bound of Marcus, Spielman, and Srivastava is generalized to all differential operators preserving real-rootedness via the finite free convolution $p \boxplus^n q$.
• A counterexample is constructed showing that a natural multivariate generalization of the largest root bound does not hold, disproving a potential strengthening of their result.
• For polynomials of degree at least 1, the inequality $\lambda_1(p \boxplus^n q \boxplus^n r) + \lambda_1(r) \leq \lambda_1(p \boxplus^n r) + \lambda_1(q \boxplus^n r)$ holds, establishing a submodular-type relation on the largest roots.
• The proof of the submodular inequality relies on an extremal polynomial argument and a contradiction derived from assuming a maximal polynomial $p$ with $\beta(p) > 0$, leading to a violation of maximality.
• The authors show that the triangle inequality $\lambda_1(p \boxplus^n q) \leq \lambda_1(p) + \lambda_1(q)$ holds for the finite free convolution, consistent with known properties.
• The paper suggests that $b$-additive convolutions (replacing derivatives with finite differences) may provide a more structured setting for studying root behavior, with potential for stronger results than in the derivative case.