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Algorithmic and Optimization Aspects of Brascamp-Lieb Inequalities, Via Operator Scaling

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Algorithmic and Optimization Aspects of Brascamp-Lieb Inequalities, Via Operator Scaling Algorithmic and Optimization Aspects of Brascamp-Lieb Inequalities, via Operator Scaling Ankit Garg Leonid Gurvits Microsoft Research New England City College of New York Cambridge, Massachusetts, USA New York City, New York, USA [email protected] [email protected] Rafael Oliveira Avi Wigderson Princeton University Institute for Advanced Studies Princeton, New Jersey, USA Princeton, New Jersey, USA [email protected] [email protected] ABSTRACT On a higher level, our application of operator scaling algorithm The celebrated Brascamp-Lieb (BL) inequalities and their reverse to BL-inequalities further connects analysis and optimization with form of Barthe are an important mathematical tool, unifying and the diverse mathematical areas used so far to motivate and solve generalizing numerous inequalities in analysis, convex geometry the operator scaling problem, which include commutative invariant and information theory, with many used in computer science. While theory, non-commutative algebra, computational complexity and their structural theory is very well understood, far less is known quantum information theory. about computing their main parameters (which we later dene be- low). Prior to this work, the best known algorithms for any of these CCS CONCEPTS optimization tasks required at least exponential time. In this work, • Theory of computation → Mathematical optimization; we give polynomial time algorithms to compute: (1) Feasibility of BL-datum, (2) Optimal BL-constant, (3) Weak separation oracle for BL-polytopes. KEYWORDS What is particularly exciting about this progress, beyond the Optimization, Brascamp-Lieb, Operator Scaling, Inequalities. better understanding of BL-inequalities, is that the objects above naturally encode rich families of optimization problems which had ACM Reference format: no prior ecient algorithms. In particular, the BL-constants (which Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson. 2017. Al- we eciently compute) are solutions to non-convex optimization gorithmic and Optimization Aspects of Brascamp-Lieb Inequalities, via problems, and the BL-polytopes (for which we provide ecient Operator Scaling. In Proceedings of 49th Annual ACM SIGACT Symposium on membership and separation oracles) are linear programs with ex- the Theory of Computing, Montreal, Canada, June 2017 (STOC’17), 13 pages. ponentially many facets. Thus we hope that new combinatorial DOI: 10.1145/3055399.3055458 optimization problems can be solved via reductions to the ones above, and make modest initial steps in exploring this possibility. Our algorithms are obtained by a simple ecient reduction of a given BL-datum to an instance of the Operator Scaling problem de- ned by Gurvits. To obtain the results above, we utilize the two (very 1 INTRODUCTION recent and dierent) algorithms for the operator scaling problem This long introduction summarizes the main contributions of the (by Garg et al. and Ivanyos et al.). Our reduction implies algorithmic paper and the intuitions behind many of the denitions, results and versions of many of the known structural results on BL-inequalities, proofs. We start with the main object of this paper: introducing and in some cases provide proofs that are dierent or simpler than the Brascamp-Lieb inequalities, and surveying known structural existing ones. Further, the analytic properties of the algorithm by results and our new algorithmic results. We then describe how to Garg et al. provide new, eective bounds on the magnitude and instantiate the operator scaling algorithm of [29] directly in the continuity of BL-constants, with applications to non-linear ver- context of BL-inequalities, in a way that does not require previous sions of BL-inequalities; prior work relied on compactness, and familiarity with it. The properties of this algorithm lead to further thus provided no bounds. structural results, as well as highlight it as a provably ecient in- stance of the general alternate minimization 1 heuristic. Finally, we discuss several known and potential applications of Brascamp-Lieb Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed inequalities in computer science and optimization, which further for prot or commercial advantage and that copies bear this notice and the full citation motivate this and future studies. on the rst page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). STOC’17, Montreal, Canada © 2017 Copyright held by the owner/author(s). 978-1-4503-4528-6/17/06...$15.00 DOI: 10.1145/3055399.3055458 1sometimes also called alternating minimization. 397 STOC’17, June 2017, Montreal, Canada Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson 1.1 The Brascamp-Lieb Inequalities: Basic turn to the computational complexity of the existing algorithms Notions and our new ones. The rst, regarding feasibility, is from [11], and the second, on the optimal constant, is from [40]. Before doing so, Many important inequalities, including Hölder’s inequality, Loomis- we make two comments, one about the “reverse BL-inequalities” Whitney inequality, Young’s convolution inequality, hypercontrac- and the other regarding input size when analyzing algorithms. tivity inequalities and many others are all special cases of the ex- tremely general Brascamp-Lieb inequalities, introduced by these Reverse BL inequalities. This informal comment simply claries authors in [14, 40]. Yet many others, including Prekopa-Lindler that all results in this paper stated for BL-inequalities hold for inequality, versions of Brunn-Minkowski inequality and others are their reverse form. In [6], Barthe introduced a reverse form of special cases of Barthe’s reverse form of Brascamp-Lieb [6]. We the Brascamp-Lieb inequalities, sometimes called RBL inequalities, discuss below only the original form, and only for Euclidean spaces, which turn out to generalize some known inequalities not captured for simplicity of exposition. The notation we use is taken mostly by the the original BL inequalities. These RBL inequalities take from [11], an excellent source both for background on this topic, the same data as standard BL inequalities, and have an optimal as well as the state-of-art on the basic questions in this eld. As is RBL constant associated with each. All questions raised above for common, we often use BL as abbreviation for Brascamp-Lieb. the BL inequalities, like feasibility, computation of the constant A Brascamp-Lieb datum is specied by twom-tuples: one of linear and its continuity properties are relevant in this reverse setting. n nj transformations B = ¹B1; B2;:::; Bmº, with Bj : R ! R (along However, they are actually equivalent to the original ones for a with ¹n;n1;:::;nmº, the vector of dimensions), and another of non- simple reason: Barthe [6] proved that for any feasible datum the negative reals p = ¹p1;p2;:::;pmº (which is a vector of exponents). optimal BL constant and the optimal RBL constant multiply to This combined information is denoted by ¹B; pº.A Brascamp-Lieb 1, and when the BL datum is infeasible the RBL constant is 0. In 2 inequality with the datum above states that for every tuple of m short, these two optimal constants determine each other. Thus all non-negative functions, f = ¹f1; f2;:::; fmº, which are integrable structural results above translate to the reverse setting, and so do according to the Lebesgue measure, we have the following inequal- all our algorithmic results. ity. ¹ m m ¹ !pj Input size conventions. Before we start, let us formalize the input Ö p Ö ¹fj ¹Bjxºº j dx ≤ C fj ¹xj ºdxj conventions to all algorithms, and the parameters we use to measure n nj x 2R j=1 j=1 xj 2R their complexity. The input to all algorithms will be a BL datum ¹B; pº, and the “size" of each part will be measured dierently. The where C 2 ¹0; 1¼ is independent of the functions f . Of course, we entries of the matrices B will be rational numbers, given in binary. really get an interesting inequality if C is nite. In that case, we call j We will let b = b¹Bº denote their total binary length (note that the datum ¹B; pº feasible, and we denote smallest C for which this in particular b ≥ nm, and so upper bounds the “combinatorial inequality holds by BL¹B; pº, called the BL-constant. size” of the input). The vector p will be given as a sequence of Many familiar special cases of this inequality are listed in [11]; rationals with a common denominator, namely p = c /d with we describe only a very simple one here, which hopefully makes j j c ;d integers. We use this convention for two reasons. First, our concrete and intuitive the complicated formal expression above and j algorithms will use this integer representation, and their complexity its constituents. A special case of the Loomis-Whitney inequality will depend on d = d¹pº (while in many cases d is only polynomial asserts that the volume of every measurable set S in the unit cube in the dimensions n;m of the problem, it can denitely be as large is at most the square root of of the product of the areas of its pro- as exponential in other cases). Second, in the context of quiver jections on the three coordinate planes. In this theorem, the linear representations, which is very relevant to this study, it is natural to transformations B : R3 ! R2 are dened by the simple projec- j use these integer “weights” c and d. tions B ¹x;y; zº = ¹y; zº, B ¹x;y; zº = ¹x; zº and B ¹x;y; zº = ¹x;yº, j 1 2 3 Summarizing, the two size parameters for a BL datum ¹B; pº will the functions fj are the indicator functions of these three projec- 1 1 1 be b and d as above.
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