Bounds on Squares of Two-Sets

For a finite group G, let pi(G) denote the proportion of (x,y) in GxG for which the set {x2,xy,yx,y2} has cardinality i. In this paper we develop estimates on the pi(G) for various i

Graphon Estimation in bipartite graphs with observable edge labels and unobservable node labels

Many real-world data sets can be presented in the form of a matrix whose entries correspond to the interaction between two entities of different natures (number of times a web user visits a web page, a student's grade in a subject, a patient's rating of a doctor, etc.). We assume in this paper that the mentioned interaction is determined by unobservable latent variables describing each entity. Our objective is to estimate the conditional expectation of the data matrix given the unobservable variables. This is presented as a problem of estimation of a bivariate function referred to as graphon. We study the cases of piecewise constant and H\"older-continuous graphons. We establish finite sample risk bounds for the least squares estimator and the exponentially weighted aggregate. These bounds highlight the dependence of the estimation error on the size of the data set, the maximum intensity of the interactions, and the level of noise. As the analyzed least-squares estimator is intractable, we propose an adaptation of Lloyd's alternating minimization algorithm to compute an approximation of the least-squares estimator. Finally, we present numerical experiments in order to illustrate the empirical performance of the graphon estimator on synthetic data sets

Upper bounds on the growth rates of hard squares and related models via corner transfer matrices

We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models - non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf's transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter's corner transfer matrix formalism, optimised with Nishino and Okunishi's corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude.Comment: Also submitted to FPSAC 2015 conferenc

The queen's domination problem

The queens graph Qn has the squares of then x n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A set D of squares of Qn is a dominating set for Qn if every square of Qn is either in D or adjacent to a square in D. If no two squares of a set I are adjacent then I is an independent set. Let 'J'(Qn) denote the minimum size of a dominating set of Qn and let i(Qn) denote the minimum size of an independent dominating set of Qn. The main purpose of this thesis is to determine new values for'!'( Qn). We begin by discussing the most important known lower bounds for 'J'(Qn) in Chapter 2. In Chapter 3 we state the hitherto known values of 'J'(Qn) and explain how they were determined. We briefly explain how to obtain all non-isomorphic minimum dominating sets for Q8 (listed in Appendix A). It is often useful to study these small dominating sets to look for patterns and possible generalisations. In Chapter 4 we determine new values for')' ( Q69 ) , ')' ( Q77 ), ')' ( Q30 ) and i (Q45 ) by considering asymmetric and symmetric dominating sets for the case n = 4k + 1 and in Chapter 5 we search for dominating sets for the case n = 4k + 3, thus determining the values of 'I' ( Q19) and 'I' (Q31 ). In Chapter 6 we prove the upper bound')' (Qn) :s; 1 8 5n + 0 (1), which is better than known bounds in the literature and in Chapter 7 we consider dominating sets on hexagonal boards. Finally, in Chapter 8 we determine the irredundance number for the hexagonal boards H5 and H7, as well as for Q5 and Q6Mathematical SciencesD.Phil. (Applied Mathematics

Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets

We consider the problem of computing the minimum value $f_{\min,K}$ of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can be reformulated as finding a probability measure $\nu$ on $K$ minimizing $\int_K f d\nu$. Lasserre showed that it suffices to consider such measures of the form $\nu = q\mu$, where $q$ is a sum-of-squares polynomial and $\mu$ is a given Borel measure supported on $K$. By bounding the degree of $q$ by $2r$ one gets a converging hierarchy of upper bounds $f^{(r)}$ for $f_{\min,K}$. When $K$ is the hypercube $[-1, 1]^n$, equipped with the Chebyshev measure, the parameters $f^{(r)}$ are known to converge to $f_{\min,K}$ at a rate in $O(1/r^2)$. We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in $O(\log r / r)$ when $K$ satisfies a minor geometrical condition, and in $O(\log^2 r / r^2)$ when $K$ is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $O(1 / \sqrt{r})$ and $O(1/r)$ for these two respective cases.Comment: 30 pages with 10 figures. Update notes for second version: Added a new section containing numerical examples that illustrate the theoretical results -- Fixed minor mistakes/typos -- Improved some notation -- Clarified certain explanations in the tex

Limitations of semidefinite programs for separable states and entangled games

Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $\omega(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published versio