arXiv subjects

Jonathan Leake

Publications and source records attributed to Jonathan Leake.

At least 19 recordsLinked to original sources

arXiv2026

Unique Minimizers for Permanents, Mixed Discriminants, and Log-concave Polynomials

The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of $\frac{n!}{n^n}$ are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.

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arXiv2026

New Bounds for Integer Flows and Verma Modules, via Denormalized Lorentzian Laurent Series

The theory of log concave polynomials has recently been developed to study objects and problems in combinatorics and other subfields in mathematics. Particular classes of log concave polynomials called Lorentzian polynomials and denormalized and dually Lorentzian polynomials have been used to prove log concavity statements for various combinatorial sequences. This includes the strongest form of Mason's log concavity conjecture on the independent sets of matroids and the log concavity of sequences of Kostka numbers. In this paper, we develop an analogous class of power series called denormalized Lorentzian (DL) Laurent series. This class is the natural generalization of DL polynomials to homogeneous power series with the benefit of capturing a number of combinatorial generating series including the Kostant partition function for integer flows of directed graphs. We then analyze specific DL Laurent series to obtain new bounds for integral flows on general directed acyclic graphs and new bounds for the dimensions of weight spaces of parabolic $\mathfrak{sl}_{n+1}(\mathbb{C})$ Verma modules.

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arXiv2026

Trickle-down Theorems via C-Lorentzian Polynomials II: Pairwise Spectral Influence and Improved Dobrushin's Condition

Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $μ$ on maximal faces. For any pair of vertices $u,v\in V$, define the pairwise spectral influence $\mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_w\in S_w$ for every $w\in V\setminus\{u,v\}$, and construct a matrix in $\mathbb{R}^{(S_u\cup S_v)\times (S_u\cup S_v)}$ where for any $s_u\in S_u, s_v\in S_v$, the $(us_u,vs_v)$-entry is the probability that $s_v$ is the spin of $v$ conditioned on $s_u$ being the spin of $u$ and on $σ$. Then $\mathcal{I}_{u,v}$ is the maximal second eigenvalue of this matrix, over all choices of spins for all $w\in V\setminus\{u,v\}$. Equivalently, $\mathcal{I}_{u,v}$ is the maximum local spectral expansion of links of codimension $2$ that include a spin for every $w \in V \setminus \{u,v\}$. We show that if the largest eigenvalue of the pairwise spectral influence matrix with entries $\mathcal{I}_{u,v}$ is bounded away from 1, i.e. $λ_{\max}(\mathcal{I})\leq 1-ε$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $μ$. This improves/generalizes the classical Dobrushin's influence matrix as the $\mathcal{I}_{u,v}$ lower-bounds the classical influence of $u\to v$. As an application, we prove that the Glauber dynamics mixes rapidly up to (approximately) the phase transition for the multi-state hardcore model--a widely studied model in telecommunication networks and statistical physics (generalizing the hardcore model) introduced by Mazel and Suhov. As a by-product of our results, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. Our proof builds on the trickle-down theorems via $\mathcal{C}$-Lorentzian polynomials machinery recently developed by the authors and Lindberg.

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arXiv2025

Lorentzian polynomials on cones

Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies. We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan. Several different characterizations of Lorentzian polynomials on cones are provided.

math.CO↗

arXiv2025

Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences

Let $X$ be a $d$-partite $d$-dimensional simplicial complex with parts $T_1,\dots,T_d$ and let $μ$ be a distribution on the facets of $X$. Informally, we say $(X,μ)$ is a path complex if for any $i<j<k$ and $F \in T_i,G \in T_j, K\in T_k$, we have $\mathbb{P}_μ[F,K | G]=\mathbb{P}_μ[F|G]\cdot\mathbb{P}_μ[K|G].$ We develop a new machinery with $\mathcal{C}$-Lorentzian polynomials to show that if all links of $X$ of co-dimension 2 have spectral expansion at most $1/2$, then $X$ is a $1/2$-local spectral expander. We then prove that one can derive fast-mixing results and log-concavity statements for top-link spectral expanders. We use our machinery to prove fast mixing results for sampling maximal flags of flats of distributive lattices (a.k.a. linear extensions of posets) subject to external fields, and to sample maximal flags of flats of "typical" modular lattices. We also use it to re-prove the Heron-Rota-Welsh conjecture and to prove a conjecture of Chan and Pak which gives a generalization of Stanley's log-concavity theorem. Lastly, we use it to prove near optimal trickle-down theorems for "sparse complexes" such as constructions by Lubotzky-Samuels-Vishne, Kaufman-Oppenheim, and O'Donnell-Pratt.

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arXiv2024

Capacity bounds on integral flows and the Kostant partition function

The type $A$ Kostant partition function is an important combinatorial object with various applications: it counts integer flows on the complete directed graph, computes Hilbert series of spaces of diagonal harmonics, and can be used to compute weight and tensor product multiplicities of representations. In this paper we study asymptotics of the Kostant partition function, improving on various previously known lower bounds and settling conjectures of O'Neill and Yip. Our methods build upon recent results and techniques of Brändén-Leake-Pak, who used Lorentzian polynomials and Gurvits' capacity method to bound the number of lattice points of transportation and flow polytopes. Finally, we also give new two-sided bounds using the Lidskii formulas from subdivisions of flow polytopes.

math.CO↗

arXiv2024

Inequalities characterizing distinguished unipotent orbits

In this paper we prove a new characterization of the distinguished unipotent orbits of a connected reductive group over an algebraically closed field of characteristic 0. For classical groups we prove the characterization by a combinatorial computation, and for exceptional groups we check it with a computer. This characterization is needed in the theory of cuspidal sheaves on the stack of L-parameters in forthcoming work of the first two named authors.

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arXiv2024

Compatibility of Real-Rooted Polynomials with Mixed Signs

We characterize compatible families of real-rooted polynomials, allowing both positive and negative leading coefficients. Our characterization naturally generalizes the same-sign characterization used by Chudnovsky and Seymour in their famous 2007 paper proving the real-rootedness of independence polynomials of claw-free graphs, thus fully settling a question left open in their paper. Our methods are generally speaking elementary, utilizing mainly linear algebra and the established theory of interlacing polynomials, with a bit of invariant theory.

math.CO↗

arXiv2024

From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.

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arXiv2022

Deterministic Approximation Algorithms for Volumes of Spectrahedra

We give a method for computing asymptotic formulas and approximations for the volumes of spectrahedra, based on the maximum-entropy principle from statistical physics. The method gives an approximate volume formula based on a single convex optimization problem of minimizing $-\log \det P$ over the spectrahedron. Spectrahedra can be described as affine slices of the convex cone of positive semi-definite (PSD) matrices, and the method yields efficient deterministic approximation algorithms and asymptotic formulas whenever the number of affine constraints is sufficiently dominated by the dimension of the PSD cone. Our approach is inspired by the work of Barvinok and Hartigan who used an analogous framework for approximately computing volumes of polytopes. Spectrahedra, however, possess a remarkable feature not shared by polytopes, a new fact that we also prove: central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. We give two main applications of this method. First, we apply this method to what we call the "multi-way Birkhoff spectrahedron" and obtain an explicit asymptotic formula for its volume. This spectrahedron is the set of quantum states with maximal entanglement (i.e., the quantum states having univariant quantum marginals equal to the identity matrix) and is the quantum analog of the multi-way Birkhoff polytope. Second, we apply this method to explicitly compute the asymptotic volume of central sections of the set of density matrices.

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arXiv2022

Capacity Lower Bounds via Productization

We give a sharp lower bound on the capacity of a real stable polynomial, depending only on the value of its gradient at $x = 1$. This result implies a sharp improvement to a similar inequality proved by Linial-Samorodnitsky-Wigderson in 2000, which was crucial to the analysis of their permanent approximation algorithm. Such inequalities have played an important role in the recent work on operator scaling and its generalizations and applications, and in fact we use our bound to construct a new scaling algorithm for real stable polynomials. In addition, we give a strong improvement on previous lower bounds of the capacity of a non-homogeneous real stable polynomial, depending only on the value of its gradient at $x = 1$. Crucially, this new bound is independent of the degree of the polynomial, and has singly exponential dependence on the number of variables. This compares favorably to the bounds used recently in the fantastic work of Karlin-Klein-Oveis Gharan to give an improved approximation factor for metric TSP, where this dependence is doubly exponential. Such bounds were conjectured to exist by the authors, and thus our new bound should imply further improvement to the approximation factor for metric TSP. The new technique we develop to prove this bound is productization, which says that any real stable polynomial can be approximated at any point in the positive orthant by a product of linear forms. Beyond the results of this paper, our main hope is that this new technique will allow us to avoid "frightening technicalities", in the words of Laurent and Schrijver, that often accompany combinatorial lower bounds.

math.CO↗

arXiv2021

Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture

We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann relations of degree one for the Chow ring of a matroid.

math.CO↗

arXiv2021

Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory

In the last few years, the notion of symmetry has provided a powerful and essential lens to view several optimization or sampling problems that arise in areas such as theoretical computer science, statistics, machine learning, quantum inference, and privacy. Here, we present two examples of nonconvex problems in optimization and sampling where continuous symmetries play -- implicitly or explicitly -- a key role in the development of efficient algorithms. These examples rely on deep and hidden connections between nonconvex symmetric manifolds and convex polytopes, and are heavily generalizable. To formulate and understand these generalizations, we then present an introduction to Lie theory -- an indispensable mathematical toolkit for capturing and working with continuous symmetries. We first present the basics of Lie groups, Lie algebras, and the adjoint actions associated with them, and we also mention the classification theorem for Lie algebras. Subsequently, we present Kostant's convexity theorem and show how it allows us to reduce linear optimization problems over orbits of Lie groups to linear optimization problems over polytopes. Finally, we present the Harish-Chandra and the Harish-Chandra--Itzykson--Zuber (HCIZ) formulas, which convert partition functions (integrals) over Lie groups into sums over the corresponding (discrete) Weyl groups, enabling efficient sampling algorithms.

cs.DS↗

arXiv2021

A Representation Theoretic Explanation of the Borcea-Brändén Characterization

In 2009, Borcea and Brändén characterize all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic interpretation of their findings, which generalizes and simplifies their result and leads to a conceptual unification of many related results in polynomial stability theory. At the heart of this unification is a generalized Grace's theorem which addresses polynomials whose roots are all contained in some real interval or ray. This generalization allows us to extend the Borcea-Brändén result to characterize a certain subclass of the linear operators which preserve such polynomials.

math.CV↗

arXiv2021

Sampling Matrices from Harish-Chandra-Itzykson-Zuber Densities with Applications to Quantum Inference and Differential Privacy

Given two $n \times n$ Hermitian matrices $Y$ and $Λ$, the Harish-Chandra-Itzykson-Zuber (HCIZ) distribution on the unitary group $\text{U}(n)$ is $e^{\text{tr}(UΛU^*Y)}dμ(U)$, where $μ$ is the Haar measure on $\text{U}(n)$. The density $e^{\text{tr}(UΛU^*Y)}$ is known as the HCIZ density. Random unitary matrices distributed according to the HCIZ density are important in various settings in physics and random matrix theory. However, the basic question of efficient sampling from the HCIZ distribution has remained open. We present two efficient algorithms to sample matrices from distributions that are close to the HCIZ distribution. The first algorithm outputs samples that are $ξ$-close in total variation distance and requires polynomially many arithmetic operations in $\log 1/ξ$ and the number of bits needed to encode $Y$ and $Λ$. The second algorithm comes with a stronger guarantee that the samples are $ξ$-close in infinity divergence, but the number of arithmetic operations depends polynomially on $1/ξ$, the number of bits needed to encode $Y$ and $Λ$, and the differences of the largest and the smallest eigenvalues of $Y$ and $Λ$. HCIZ densities can also be viewed as exponential densities on $\text{U}(n)$-orbits, and these densities have been studied in statistics, machine learning, and theoretical computer science. Thus our results have the following applications: 1) an efficient algorithm to sample from complex versions of matrix Langevin distributions studied in statistics, 2) an efficient algorithm to sample from continuous max-entropy distributions on unitary orbits, which implies an efficient algorithm to sample a pure quantum state from the entropy-maximizing ensemble representing a given density matrix, and 3) an efficient algorithm for differentially private rank-$k$ approximation, with improved utility bounds for $k>1$.

cs.DS↗

arXiv2021

Lower bounds for contingency tables via Lorentzian polynomials

We present a new lower bound on the number of contingency tables, improving upon and extending previous lower bounds by Barvinok and Gurvits. As an application, we obtain new lower bounds on the volumes of flow and transportation polytopes. Our proofs are based on recent results on Lorentzian polynomials.

math.CO↗

arXiv2020

On the Computability of Continuous Maximum Entropy Distributions: Adjoint Orbits of Lie Groups

Given a point $A$ in the convex hull of a given adjoint orbit $\mathcal{O}(F)$ of a compact Lie group $G$, we give a polynomial time algorithm to compute the probability density supported on $\mathcal{O}(F)$ whose expectation is $A$ and that minimizes the Kullback-Leibler divergence to the $G$-invariant measure on $\mathcal{O}(F)$. This significantly extends the recent work of the authors (STOC 2020) who presented such a result for the manifold of rank $k$-projections which is a specific adjoint orbit of the unitary group $\mathrm{U}(n)$. Our result relies on the ellipsoid method-based framework proposed in prior work; however, to apply it to the general setting of compact Lie groups, we need tools from Lie theory. For instance, properties of the adjoint representation are used to find the defining equalities of the minimal affine space containing the convex hull of $\mathcal{O}(F)$, and to establish a bound on the optimal dual solution. Also, the Harish-Chandra integral formula is used to obtain an evaluation oracle for the dual objective function. While the Harish-Chandra integral formula allows us to write certain integrals over the adjoint orbit of a Lie group as a sum of a small number of determinants, it is only defined for elements of a chosen Cartan subalgebra of the Lie algebra $\mathfrak{g}$ of $G.$ We show how it can be applied to our setting with the help of Kostant's convexity theorem. Further, the convex hull of an adjoint orbit is a type of orbitope, and the orbitopes studied in this paper are known to be spectrahedral. Thus our main result can be viewed as extending the maximum entropy framework to a class of spectrahedra.

math.OC↗

arXiv2020

Counting Matchings via Capacity Preserving Operators

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the Van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver's inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilized to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g., the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland's lower matching conjecture.

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