Flexible Modeling of Diversity with Strongly Log-Concave Distributions

Strongly log-concave (SLC) distributions are a rich class of discrete probability distributions over subsets of some ground set. They are strictly more general than strongly Rayleigh (SR) distributions such as the well-known determinantal point process. While SR distributions offer elegant models of diversity, they lack an easy control over how they express diversity. We propose SLC as the right extension of SR that enables easier, more intuitive control over diversity, illustrating this via examples of practical importance. We develop two fundamental tools needed to apply SLC distributions to learning and inference: sampling and mode finding. For sampling we develop an MCMC sampler and give theoretical mixing time bounds. For mode finding, we establish a weak log-submodularity property for SLC functions and derive optimization guarantees for a distorted greedy algorithm

Intersection theoretic inequalities via Lorentzian polynomials

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to $m$-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for $m$-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by $m$-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Pl\"{u}nnecke-Ruzsa inequalities) from additive combinatorics in our contexts.Comment: 27 pages; comments welcome

Amalgamation of real zero polynomials

With this article, we hope to launch the investigation of what we call the real zero amalgamation problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an extension of the first one. The real zero amalgamation problem asks when two (multivariate real) polynomials have a common extension (called amalgam) that is a real zero polynomial. We show that the obvious necessary conditions are not sufficient. Our counterexample is derived in several steps from a counterexample to amalgamation of matroids by Poljak and Turz\'ik. On the positive side, we show that even a degree-preserving amalgamation is possible in three very special cases with three completely different techniques. Finally, we conjecture that amalgamation is always possible in the case of two shared variables. The analogue in matroid theory is true by another work of Poljak and Turz\'ik. This would imply a very weak form of the Generalized Lax Conjecture.Comment: 24 page

Submodularity and Its Applications in Wireless Communications

This monograph studies the submodularity in wireless communications and how to use it to enhance or improve the design of the optimization algorithms. The work is done in three different systems. In a cross-layer adaptive modulation problem, we prove the submodularity of the dynamic programming (DP), which contributes to the monotonicity of the optimal transmission policy. The monotonicity is utilized in a policy iteration algorithm to relieve the curse of dimensionality of DP. In addition, we show that the monotonic optimal policy can be determined by a multivariate minimization problem, which can be solved by a discrete simultaneous perturbation stochastic approximation (DSPSA) algorithm. We show that the DSPSA is able to converge to the optimal policy in real time. For the adaptive modulation problem in a network-coded two-way relay channel, a two-player game model is proposed. We prove the supermodularity of this game, which ensures the existence of pure strategy Nash equilibria (PSNEs). We apply the Cournot tatonnement and show that it converges to the extremal, the largest and smallest, PSNEs within a finite number of iterations. We derive the sufficient conditions for the extremal PSNEs to be symmetric and monotonic in the channel signal-to-noise (SNR) ratio. Based on the submodularity of the entropy function, we study the communication for omniscience (CO) problem: how to let all users obtain all the information in a multiple random source via communications. In particular, we consider the minimum sum-rate problem: how to attain omniscience by the minimum total number of communications. The results cover both asymptotic and non-asymptotic models where the transmission rates are real and integral, respectively. We reveal the submodularity of the minimum sum-rate problem and propose polynomial time algorithms for solving it. We discuss the significance and applications of the fundamental partition, the one that gives rise to the minimum sum-rate in the asymptotic model. We also show how to achieve the omniscience in a successive manner

Approximation Algorithms for Independence Systems

In this thesis, we study three maximization problems over independence systems. • Chapter 2 – Weighted k-Set Packing is a fundamental combinatorial optimization problem that captures matching problems in graphs and hypergraphs. For over 20 years Berman’s algorithm stood as the state-of-the-art approximation algorithm for this problem, until Neuwohner’s recent improvements. Our focus is on the value k = 3 which is well motivated from theory and practice, and for which improvements are arguably the hardest. We largely improve upon her approximation, by giving an algorithm that yields state-of-the-art results. Our techniques are simple and naturally expand upon Berman’s analysis. Our analysis holds for any value of k with greater improvements over Berman’s result as k grows. • Chapter 3 – We continue the study of the weighted k-set packing problem. Building on Chapter 2, we reach the tightest approximation factor possible for k = 3, and k ≥ 7 using our techniques. As a consequence, we improve over all the results in Chapter 2. In particular, we obtain √3, and k/2 -approximation for k = 3 and k ≥ 7 respectively. Our result for k ≥ 7 is in fact analogous to that of Hurkens and Schrijver who obtained the same approximation factor for the unweighted problem. • Chapter 4 – We present improved multipass streaming algorithms for maximizing monotone and arbitrary submodular functions over independence systems. Our result demonstrates that the simple local-search algorithm for maximizing a monotone sub- modular function can be efficiently simulated using a few passes over the dataset. Our results improve the number of passes needed compared to the state-of-the-art. • Chapter 5 – We conclude the thesis by presenting improved approximation algorithms for Sparse Least-Square Estimation, Bayesian A-optimal Design, and Column Subset Selection over a matroid constraint. At the heart of this chapter is the demonstration of a new property that considered applications satisfy. We call it: β-weak submodularity. We leverage this property to derive new algorithms with strengthened guarantees. The notion of β-weak submodularity is of independent interest and we believe that it will have further use in machine learning and statistics

Online Dependent Rounding Schemes

We study the abstract problem of rounding fractional bipartite $b$-matchings online. The input to the problem is an unknown fractional bipartite $b$-matching, exposed node-by-node on one side. The objective is to maximize the \emph{rounding ratio} of the output matching $\mathcal{M}$, which is the minimum over all fractional $b$-matchings $\mathbf{x}$, and edges $e$, of the ratio $\Pr[e\in \mathcal{M}]/x_e$. In offline settings, many dependent rounding schemes achieving a ratio of one and strong negative correlation properties are known (e.g., Gandhi et al., J.ACM'06 and Chekuri et al., FOCS'10), and have found numerous applications. Motivated by online applications, we present \emph{online dependent-rounding schemes} (ODRSes) for $b$-matching. For the special case of uniform matroids (single offline node), we present a simple online algorithm with a rounding ratio of one. Interestingly, we show that our algorithm yields \emph{the same distribution} as its classic offline counterpart, pivotal sampling (Srinivasan, FOCS'01), and so inherits the latter's strong correlation properties. In arbitrary bipartite graphs, an online rounding ratio of one is impossible, and we show that a combination of our uniform matroid ODRS with repeated invocations of \emph{offline} contention resolution schemes (CRSes) yields a rounding ratio of $1-1/e\approx 0.632$. Our main technical contribution is an ODRS breaking this pervasive bound, yielding rounding ratios of $0.646$ and $0.652$ for $b$-matchings and simple matchings, respectively. We obtain these results by grouping nodes and using CRSes for negatively-correlated distributions, together with a new method we call \emph{group discount and individual markup}, analyzed using the theory of negative association. We present a number of applications of our ODRSes to online edge coloring, several stochastic optimization problems, and algorithmic fairness