Information theoretic properties of Markov Random Fields, and their algorithmic applications

Markov random fields are a popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for provably learning them relied on exhaustive search, correlation decay or various incoherence assumptions. Bresler [1] gave an algorithm for learning general Ising models on bounded degree graphs. His approach was based on a structural result about mutual information in Ising models. Here we take a more conceptual approach to proving lower bounds on the mutual information. Our proof generalizes well beyond Ising models, to arbitrary Markov random fields with higher order interactions. As an application, we obtain algorithms for learning Markov random fields on bounded degree graphs on n nodes with r-order interactions in n r time and log n sample complexity. Our algorithms also extend to various partial observation models

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies

Simplifying superstring and D-brane actions in AdS(4) x CP(3) superbackground

By making an appropriate choice for gauge fixing kappa-symmetry we obtain a relatively simple form of the actions for a D=11 superparticle in AdS(4) x S(7)/Z_k, and for a D0-brane, fundamental string and D2-branes in the AdS(4) x CP(3) superbackground. They can be used to study various problems of string theory and the AdS4/CFT3 correspondence, especially in regions of the theory which are not reachable by the OSp(6|4)/U(3) x SO(1,3) supercoset sigma-model. In particular, we present a simple form of the gauge-fixed superstring action in AdS(4) x CP(3) and briefly discuss issues of its T-dualization.Comment: 1+36 pages, v2,v3 clarifications and references adde

Applications and limits of convex optimization

Every algorithmic learning problem becomes vastly more tractable when reduced to a convex program, yet few can be simplified this way. At the heart of this thesis are two hard problems with unexpected convex reformulations. The Paulsen problem, a longstanding open problem in operator theory, was recently resolved by Kwok et al [40]. We use a convex program due to Barthe to present a dramatically simpler proof with an accompanying efficient algorithm that also achieves a better bound. Next, we examine the related operator scaling problem, whose fastest known algorithm uses convex optimization in non-Euclidean space. We expose a fundamental obstruction to such techniques by proving that, under realistic noise conditions, hyperbolic space admits no analogue of Nesterov’s accelerated gradient descent. Finally, we generalize Bresler’s structure learning algorithm from Ising models to arbitrary graphical models. We compare our results to a recent convex programming reformulation of the same problem. Notably, in variants of the problem where one only receives partial samples, our combinatorial algorithm is almost unaffected, whereas the convex approach fails to get off the ground.Ph.D

The Paulsen Problem Made Simple

© Linus Hamilton and Ankur Moitra. The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every -nearly equal norm Parseval frame in d dimensions is within squared distance O(?d13/2) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(ϵd2).NSF (Awards CCF-1453261, CCF-1565235

Anarchy Is Free in Network Creation

<p>The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game theoretic study of network formation, and our paper considers one of the most-well studied models, originally proposed by Fabrikant et al. In it, each of <em>n</em> agents corresponds to a vertex, which can create edges to other vertices at a cost of <em>α</em> each, for some parameter <em>α</em>. Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent’s cost function is further augmented by the sum total of all (graph theoretic) distances to all other vertices.</p> <p>Previous research proved that for many regimes of the (<em>α</em>,<em>n</em>) parameter space, the total social cost (sum of all agents’ costs) of every Nash equilibrium is bounded by at most a constant multiple of the optimal social cost. In algorithmic game theoretic nomenclature, this approximation ratio is called the price of anarchy. In our paper, we significantly sharpen some of those results, proving that for all constant non-integral <em>α</em> > 2, the price of anarchy is in fact 1 + <em>o</em>(1), i.e., not only is it bounded by a constant, but it tends to 1 as <em>n</em> → ∞. For constant integral <em>α</em> ≥ 2, we show that the price of anarchy is bounded away from 1. We provide quantitative estimates on the rates of convergence for both results.</p