Presented on October 2, 2017 at 11:00 a.m. in the Klaus Advanced Computing Building, room 1116E.
As for Swendsen--Wang, comparison estimates due to Ullrich @cite_2 @cite_39 imply that its inverse gap is at most that of Glauber dynamics on any graph and at any temperature (see Theorem ); thus for @math on @math it also has @math for all @math and for all @math thanks to duality, and similarly at @math it has @math .
@cite_39 @cite_2
1980404829 1982190141
A new bound (Theorem ) for the duration of the chip-firing game with @math chips on a @math -vertex graph is obtained, by a careful analysis of the pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is expressed in terms of the entries of the pseudo-inverse. It is shown (Section 5) to be always better than the classic bound due to Bj "o rner, Lov ' a sz and Shor. In some cases the improvement is dramatic. For instance: for strongly regular graphs the classic and the new bounds reduce to @math and @math , respectively. For dense regular graphs - @math - the classic and the new bounds reduce to @math and @math , respectively. This is a snapshot of a work in progress, so further results in this vein are in the works. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all @math . This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function. Taking the zero-temperature limit gives new results in extremal combinatorics: the number of @math -colorings of a @math -regular graph, for any @math , is maximized by a union of @math 's. This proves the @math case of a conjecture of Galvin and Tet ali. The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as @math , etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical @math -Coloring . A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the @math -level Hierarchical @math -Coloring problem is @math , for @math . The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized @math -time algorithm solving the LCL can be transformed into a deterministic @math -time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges @math --- @math or @math --- @math . 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in @math time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be @math and @math . Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. This paper makes two contributions to the areas of anti-concentration and non-asymptotic random matrix theory. First, we study the counting problem in inverse Littlewood-Offord theory for general random variables: for random variables @math which are i.i.d. copies of a random variable @math (satisfying some mild hypotheses), how many integer vectors @math in a prescribed box have large @math ? Building on recent work of Ferber, Jain, Luh, and Samotij (who treated the case when @math is a Rademacher random variable), we provide significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu, and Nguyen and Vu. Next, we study the non-asymptotic behavior of the least singular value @math of a random @math square matrix @math with i.i.d. entries, which are only assumed to be centered with variance @math . As an application of our counting theorem, and utilizing and developing a recent work by the author, we show that for all @math , @math , where @math is an absolute constant, thereby providing a considerably simpler proof of an approximate version of an essentially optimal result due to Rebrova and Tikhomirov. Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix @math , one may look at the spectrum of @math for a properly chosen @math . The issue is that the spectrum of @math might be contaminated by non-informational top eigenvalues, e.g., due to scale variations in the data, and the application of @math aims to remove these. Designing a good functional @math (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, @math where @math denotes the adjacency matrix and @math is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance @math , and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd o s-R 'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to massoulie-STOC , settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts massoulie-STOC,Mossel_SBM2,bordenave,colin3 . This is illustrated on a geometric block model that is dense in cliques. We introduce a Gibbs measure on nearest-neighbour paths of length @math in the Euclidean @math -dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature @math . We prove that, for all @math , the random walk condensates to a set of diameter @math in dimension @math , up to a multiplicative constant. In all dimensions @math , we also prove that the volume is bounded above by @math and the diameter is bounded below by @math . Similar results hold for a random walk conditioned to have local time greater than @math everywhere in its range when @math is larger than some explicit constant, which in dimension two is the logarithm of the connective constant. Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs. A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional @math @math models for @math (Ising) and @math ( @math ) at their critical temperatures, and for @math (Heisenberg) with correlation lengths around 10 and 20. On lattices up to @math no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities. Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process. We study the directed polymer model in dimension @math when the environment is heavy-tailed, with a decay exponent @math . We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature @math vanishes as the size of the system n goes to infinity. When @math , we show that all possible transversal fluctuations @math can be achieved by tuning properly @math , allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when @math , we show that there are only two regimes: the transversal fluctuations are either @math or @math . As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP]. Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast @math -coloring of trees requires random bits: Building on the recent lower bounds of , we prove that the randomized complexity of @math -coloring a tree with maximum degree @math is @math , whereas its deterministic complexity is @math for any @math . This also establishes a large separation between the deterministic complexity of @math -coloring and @math -coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in @math rounds can be transformed to run in @math rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires @math time deterministically. 3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size @math is at least its deterministic complexity on instances of size @math . This shows that a deterministic @math lower bound for any problem implies a randomized @math lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . We give the first improvement to the space approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size @math and, when queried about vertices at distance @math , returns a path of length @math . For weighted graphs with @math edges, our distance oracle has size @math and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space @math . For unweighted graphs, this implies a @math space lower bound to achieve approximation @math . We resolve several fundamental questions in the area of distributed functional monitoring, initiated by Cormode, Muthukrishnan, and Yi (SODA, 2008). In this model there are @math sites each tracking their input and communicating with a central coordinator that continuously maintain an approximate output to a function @math computed over the union of the inputs. The goal is to minimize the communication. We show the randomized communication complexity of estimating the number of distinct elements up to a @math factor is @math , improving the previous @math bound and matching known upper bounds up to a logarithmic factor. For the @math -th frequency moment @math , @math , we improve the previous @math communication bound to @math . We obtain similar improvements for heavy hitters, empirical entropy, and other problems. We also show that we can estimate @math , for any @math , using @math communication. This greatly improves upon the previous @math bound of Cormode, Muthukrishnan, and Yi for general @math , and their @math bound for @math . For @math , our bound resolves their main open question. Our lower bounds are based on new direct sum theorems for approximate majority, and yield significant improvements to problems in the data stream model, improving the bound for estimating @math in @math passes from @math to @math , giving the first bound for estimating @math in @math passes of @math bits of space that does not use the gap-hamming problem. We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures. A function @math is @math -resilient if all its Fourier coefficients of degree at most @math are zero, i.e., @math is uncorrelated with all low-degree parities. We study the notion of @math @math of Boolean functions, where we say that @math is @math -approximately @math -resilient if @math is @math -close to a @math -valued @math -resilient function in @math distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class @math over the uniform distribution. Roughly speaking, if all functions in a class @math are far from being @math -resilient then @math can be learned agnostically in time @math and conversely, if @math contains a function close to being @math -resilient then agnostic learning of @math in the statistical query (SQ) framework of Kearns has complexity of at least @math . This characterization is based on the duality between @math approximation by degree- @math polynomials and approximate @math -resilience that we establish. In particular, it implies that @math approximation by low-degree polynomials, known to be sufficient for agnostic learning over product distributions, is in fact necessary. Focusing on monotone Boolean functions, we exhibit the existence of near-optimal @math -approximately @math -resilient monotone functions for all @math . Prior to our work, it was conceivable even that every monotone function is @math -far from any @math -resilient function. Furthermore, we construct simple, explicit monotone functions based on @math and @math that are close to highly resilient functions. Our constructions are based on a fairly general resilience analysis and amplification. These structural results, together with the characterization, imply nearly optimal lower bounds for agnostic learning of monotone juntas. Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations. A family @math of graphs on a fixed set of @math vertices is called triangle-intersecting if for any @math , the intersection @math contains a triangle. More generally, for a fixed graph @math , a family @math is @math -intersecting if the intersection of any two graphs in @math contains a sub-graph isomorphic to @math . In [D. Ellis, Y. Filmus, and E. Friedgut, Triangle-intersecting families of graphs, J. Eur. Math. Soc. 14 (2012), pp. 841--885], Ellis, Filmus and Friedgut proved a 36-year old conjecture of Simonovits and Sos stating that the maximal size of a triangle-intersecting family is @math . Furthermore, they proved a @math -biased generalization, stating that for any @math , we have @math , where @math is the probability that the random graph @math belongs to @math . In the same paper, conjectured that the assertion of their biased theorem holds also for @math , and more generally, that for any non- @math -colorable graph @math and any @math -intersecting family @math , we have @math for all @math . In this note we construct, for any fixed @math and any @math , an @math -intersecting family @math of graphs such that @math , where @math depends only on @math and @math , thus disproving both conjectures.