We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in article amsmath,amsfonts empty . We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x − y|−s+o(1) when |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp r1 Δ+o(1) in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210-227, 2011 (Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.)
For @math , the present paper states @math . Here we note that @math as @math which, formally, is in agreement with . For @math we in turn have @math and so, at @math , a polylogarithmic growth is no longer sustainable. Instead, for the case of the decay @math one expects that where @math varies through @math as @math sweeps through @math . This claim is supported by upper and lower bounds in somewhat restricted one-dimensional cases (Benjamini and Berger @cite_13 , Coppersmith, Gamarnik and Sviridenko @cite_12 ). However, even the existence of a sharp exponent @math has been elusive so far.
@cite_13 @cite_12
1968939650 2088668624
Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math Q @math $ . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. For any real number @math , we nearly completely characterize the space complexity of estimating @math for @math matrices @math in which each row and each column has @math non-zero entries and whose entries are presented one at a time in a data stream model. Here the @math are the singular values of @math , and when @math , @math is the @math -th power of the Schatten @math -norm. We show that when @math is not an even integer, to obtain a @math -approximation to @math with constant probability, any @math -pass algorithm requires @math bits of space, where @math as @math and @math is a constant independent of @math . However, when @math is an even integer, we give an upper bound of @math bits of space, which holds even in the turnstile data stream model. The latter is optimal up to @math factors. Our results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices @math : the previous best lower bound was @math for @math , @math for @math and @math for @math . We note for @math , while our lower bound for even integers is the same, for other @math in this range our lower bound is @math , which is considerably stronger than the previous @math for small enough constant @math . We obtain similar near-linear lower bounds for Ky-Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to our work. We prove lower bounds on the complexity of finding @math -stationary points (points @math such that @math ) of smooth, high-dimensional, and potentially non-convex functions @math . We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of @math at a query point @math . We show that for any (potentially randomized) algorithm @math , there exists a function @math with Lipschitz @math th order derivatives such that @math requires at least @math queries to find an @math -stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton's method, and generalized @math th order regularization are worst-case optimal within their natural function classes. In the distributed all-pairs shortest paths problem (APSP), every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds (typically called time complexity ). The problem admits @math -approximation @math -time algorithm and a nearly-tight @math lower bound [Nanongkai, STOC'14; Lenzen and Patt-Shamir PODC'15] @math , @math and @math hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios LenzenP_podc13,HolzerW12,PelegRT12,Nanongkai-STOC14 . . For the exact case, Elkin [STOC'17] presented an @math time bound, which was later improved to @math [Huang, Nanongkai, Saranurak FOCS'17]. It was shown that any super-linear lower bound (in @math ) requires a new technique [Censor-Hillel, Khoury, Paz, DISC'17], but otherwise it remained widely open whether there exists a @math -time algorithm for the exact case, which would match the best possible approximation algorithm. This paper resolves this question positively: we present a randomized (Las Vegas) @math -time algorithm, matching the lower bound up to polylogarithmic factors. Like the previous @math bound, our result works for directed graphs with zero (and even negative) edge weights. In addition to the improved running time, our algorithm works in a more general setting than that required by the previous @math bound; in our setting (i) the communication is only along edge directions (as opposed to bidirectional), and (ii) edge weights are arbitrary (as opposed to integers in 1, 2, ... poly(n) ). ... Given an undirected @math -node unweighted graph @math , a spanner with stretch function @math is a subgraph @math such that, if two nodes are at distance @math in @math , then they are at distance at most @math in @math . Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the @math - @math distance only for pairs @math in a given set @math . Such @math -spanners were studied before [Coppersmith,Elkin'05] only in the special case that @math is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same @math ) and of the best known spanners (with the same @math ). In more detail, for arbitrary @math , we show that there exists a @math -spanner of size @math with @math . Alternatively, for any @math , there exists a @math -spanner of size @math with @math . We also consider the relevant special case that there is a critical set of nodes @math , and we wish to approximate either the distances within nodes in @math or from nodes in @math to any other node. We show that there exists an @math -spanner of size @math with @math , and an @math -spanner of size @math with @math . All the mentioned pairwise spanners can be constructed in polynomial time. Let @math be the maximal value such that the product of an @math matrix by an @math matrix can be computed with @math arithmetic operations. In this paper we show that @math , which improves the previous record @math by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an @math matrix by an @math matrix, for any value @math . The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for @math ), we recover exactly the complexity of the algorithm by Coppersmith and Wino grad (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a @math -time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, where @math denotes the number of vertices, and also improve the time complexity of sparse square matrix multiplication. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math . We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of @math nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level @math . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. We shall show that if the restricted isometry constant (RIC) @math of the measurement matrix @math satisfies @math then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every @math -sparse signal @math in @math iterations from @math . Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given @math , we shall construct a matrix @math with the RIC @math such that OMP may not recover some @math -sparse signal @math in @math iterations. Given an @math matrix @math , its Schatten- @math norm, @math , is defined as @math , where @math is the @math -th largest singular value of @math . These norms have been studied in functional analysis in the context of non-commutative @math -spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embeddability, are still open. Specifically, given a set of matrices @math , suppose we want to construct a linear map @math such that @math for each @math , where @math and @math , and further, @math for a given approximation factor @math and real number @math . Then how large do @math and @math need to be as a function of @math ? We nearly resolve this question for every @math , for the case where @math can be expressed as @math , where @math and @math are arbitrary matrices that are allowed to depend on @math , that is, @math can be implemented by left and right matrix multiplication. Namely, for every @math , we provide nearly matching upper and lower bounds on the size of @math and @math as a function of @math . Importantly, our upper bounds are oblivious , meaning that @math and @math do not depend on the @math , while our lower bounds hold even if @math and @math depend on the @math . As an application of our upper bounds, we answer a recent open question of about space-approximation trade-offs for the Schatten @math -norm, showing in a data stream it is possible to estimate the Schatten- @math norm up to a factor of @math using @math space. Given a set of @math points in @math , how many dimensions are needed to represent all pair wise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the @math norm, where @math dimensions suffice to achieve @math distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in @math . A recent result shows that distortion @math can be achieved with @math dimensions. On the other hand, the only lower bounds known are that distortion @math requires @math dimensions and that distortion @math requires @math dimensions. In this work, we show the first near linear lower bounds for dimension reduction in @math . In particular, we show that @math distortion requires at least @math dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of Brinkman-Charikar for lower bounds on dimension reduction in @math . In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. 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. For any integers @math and @math , we show the existence of a set of @math vectors @math such that any embedding @math satisfying @math must have @math This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of @math of interest, since there is always an isometric embedding into dimension @math (either the identity map, or projection onto @math ). Previously such a lower bound was only known to hold against linear maps @math , and not for such a wide range of parameters @math [LN16]. The best previously known lower bound for general @math was @math [Wel74, Lev83, Alo03], which is suboptimal for any @math . This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . In stochastic convex optimization the goal is to minimize a convex function @math over a convex set @math where @math is some unknown distribution and each @math in the support of @math is convex over @math . The optimization is commonly based on i.i.d. samples @math from @math . A standard approach to such problems is empirical risk minimization (ERM) that optimizes @math . Here we consider the question of how many samples are necessary for ERM to succeed and the closely related question of uniform convergence of @math to @math over @math . We demonstrate that in the standard @math setting of Lipschitz-bounded functions over a @math of bounded radius, ERM requires sample size that scales linearly with the dimension @math . This nearly matches standard upper bounds and improves on @math dependence proved for @math setting by Shalev- (2009). In stark contrast, these problems can be solved using dimension-independent number of samples for @math setting and @math dependence for @math setting using other approaches. We further show that our lower bound applies even if the functions in the support of @math are smooth and efficiently computable and even if an @math regularization term is added. Finally, we demonstrate that for a more general class of bounded-range (but not Lipschitz-bounded) stochastic convex programs an infinite gap appears already in dimension 2. We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on @math , for all @math . Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of @math . In more detail, we give an estimator that, for any @math and @math , draws @math samples from an unknown target log-concave density on @math , and outputs a hypothesis that (with high probability) is @math -close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of @math for this problem. For the set of graphs with a given degree sequence, consisting of any number of @math and @math , and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of @math -matchings. We find the expected value of the number of @math -matchings of @math -regular bipartite graphs on @math vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for @math -matchings in @math -regular bipartite graphs on @math vertices, and their asymptotic versions for infinite @math -regular bipartite graphs. We prove these conjectures for @math -regular bipartite graphs and for @math -matchings with @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. Let @math be an @math Fourier matrix over @math for some prime @math . We improve upon known lower bounds for the number of rows of @math that must be sampled so that the resulting matrix @math satisfies the restricted isometry property for @math -sparse vectors. This property states that @math is approximately @math for all @math -sparse vectors @math . In particular, if @math , we show that @math rows must be sampled to satisfy the restricted isometry property with constant probability.