50 research outputs found
A Brascamp-Lieb–Rary of Examples
This paper focuses on the Brascamp-Lieb inequality and its applications in analysis, fractal geometry, computer science, and more. It provides a beginner-level introduction to the Brascamp-Lieb inequality alongside re- lated inequalities in analysis and explores specific cases of extremizable, simple, and equivalent Brascamp-Lieb data. Connections to computer sci- ence and geometric measure theory are introduced and explained. Finally, the Brascamp-Lieb constant is calculated for a chosen family of linear maps
Operator scaling with specified marginals
The completely positive maps, a generalization of the nonnegative matrices,
are a well-studied class of maps from matrices to
matrices. The existence of the operator analogues of doubly stochastic scalings
of matrices is equivalent to a multitude of problems in computer science and
mathematics, such rational identity testing in non-commuting variables,
noncommutative rank of symbolic matrices, and a basic problem in invariant
theory (Garg, Gurvits, Oliveira and Wigderson, FOCS, 2016).
We study operator scaling with specified marginals, which is the operator
analogue of scaling matrices to specified row and column sums. We characterize
the operators which can be scaled to given marginals, much in the spirit of the
Gurvits' algorithmic characterization of the operators that can be scaled to
doubly stochastic (Gurvits, Journal of Computer and System Sciences, 2004). Our
algorithm produces approximate scalings in time poly(n,m) whenever scalings
exist. A central ingredient in our analysis is a reduction from the specified
marginals setting to the doubly stochastic setting.
Operator scaling with specified marginals arises in diverse areas of study
such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues
of sums of Hermitian matrices, and quantum information theory. Some of the
known theorems in these areas, several of which had no effective proof, are
straightforward consequences of our characterization theorem. For instance, we
obtain a simple algorithm to find, when they exist, a tuple of Hermitian
matrices with given spectra whose sum has a given spectrum. We also prove new
theorems such as a generalization of Forster's theorem (Forster, Journal of
Computer and System Sciences, 2002) concerning radial isotropic position.Comment: 34 pages, 3 page appendi
Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices
We address the computation of the degrees of minors of a noncommutative
symbolic matrix of form where are matrices over a
field , are noncommutative variables, are integer
weights, and is a commuting variable specifying the degree. This problem
extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can
formulate various combinatorial optimization problems. Extending the study by
Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and
polyhedral characterization for the maximum degrees of minors of of all
sizes, and develop a strongly polynomial-time algorithm for computing them.
This algorithm is viewed as a unified algebraization of the classical Hungarian
method for bipartite matching and the weight-splitting algorithm for linear
matroid intersection. As applications, we provide polynomial-time algorithms
for weighted fractional linear matroid matching and linear optimization over
rank-2 Brascamp-Lieb polytopes
Adjoint Brascamp-Lieb inequalities
The Brascamp-Lieb inequalities are a generalization of the H\"older,
Loomis-Whitney, Young, and Finner inequalities that have found many
applications in harmonic analysis and elsewhere. In this paper we introduce an
"adjoint" version of these inequalities, which can be viewed as an
version of the entropy Brascamp-Lieb inequalities of Carlen and
Cordero-Erausquin. As applications, we reprove a log-convexity property of the
Gowers uniformity norms, and establish some reverse inequalities for
various tomographic transforms. We conclude with some open questions.Comment: 43 page