arXiv subjects

Jonathan Leake

Publications and source records attributed to Jonathan Leake.

23 records · Page 2Linked to original sources

arXiv2020

On the computability of continuous maximum entropy distributions with applications

We initiate a study of the following problem: Given a continuous domain $Ω$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$ and a prior measure $μ$ on $Ω$, find the probability density over $Ω$ whose marginal is $A$ and that minimizes the KL-divergence to $μ$. This framework gives rise to several extremal distributions that arise in mathematics, quantum mechanics, statistics, and theoretical computer science. Our technical contributions include a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a "balance" property of the measure $μ$ on $Ω$, and exact algorithms for evaluating the dual and its gradient for several interesting settings of $Ω$ and $μ$. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL-divergence minimizing distributions in several cases. Applications of our results include: 1) an optimization characterization of the Goemans-Williamson measure that is used to round a positive semidefinite matrix to a vector, 2) the computability of the entropic barrier for polytopes studied by Bubeck and Eldan, and 3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy in the 1970s: this corresponds to the case when $Ω$ is the set of rank one projections matrices and $μ$ corresponds to the Haar measure on the unit sphere. Our techniques generalize to the setting of Hermitian rank $k$ projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicable even beyond, to adjoint orbits of compact Lie groups.

cs.DS↗

arXiv2019

Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.

math.CO↗

arXiv2018

On the Further Structure of the Finite Free Convolutions

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.

math.CO↗

arXiv2018

Mixed Determinants and the Kadison-Singer problem

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison-Singer problem to prove improved paving estimates. Working with Anderson's paving formulation of Kadison-Singer instead of Weaver's vector balancing version, we show that the machinery of interlacing polyomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic polynomials turn out to be related to the so called "mixed determinants" that have been carefully studied by Borcea and Branden. This technique allows us to show that any projection with diagonal entries strictly less than $\frac{1}{4}$ can be two paved, matching recent results of Bownik, Casazza, Marcus and Speegle, though our estimates are asymptotically weaker. We also show that any projection with diagonal entries at most $\frac{1}{2}$ can be four paved, yielding improvements over currently known estimates. We also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have some remarkable combinatorial properties.

math.FA↗

arXiv2017

Connecting the q-Multiplicative Convolution and the Finite Difference Convolution

In a recent paper, Brändén, Krasikov, and Shapiro consider root location preservation properties of finite difference operators. To this end, the authors describe a natural polynomial convolution operator and conjecture that it preserves root mesh properties. We prove this conjecture using two methods. The first develops a novel connection between the additive (Walsh) and multiplicative (Grace-Szegö) convolutions, which can be generically used to transfer results from multiplicative to additive. We then use this to transfer an analogous result, due to Lamprecht, which demonstrates logarithmic root mesh preservation properties of a certain $q$-multiplicative convolution operator. The second method proves the result directly using a modification of Lamprecht's proof of the logarithmic root mesh result. We present his original argument in a streamlined fashion and then make the appropriate alterations to apply it to the additive case.

math.CV↗