Proceedings of the forty-seventh annual ACM symposium on Theory of Computing最新文献_第8页

Bypassing KLS: Gaussian Cooling and an O^*(n3) Volume Algorithm 绕过KLS:高斯冷却和O^*(n3)体积算法

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-09-21 DOI: 10.1145/2746539.2746563

Benjamin R. Cousins, S. Vempala

引用次数: 54

Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms 超图马尔可夫算子，特征值和近似算法

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-08-11 DOI: 10.1145/2746539.2746555

Anand Louis

{"title":"Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms","authors":"Anand Louis","doi":"10.1145/2746539.2746555","DOIUrl":"https://doi.org/10.1145/2746539.2746555","url":null,"abstract":"The celebrated Cheeger's Inequality [AM85,a86] establishes a bound on the expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. Our operator can be viewed as the gradient operator applied to a certain natural quadratic form for hypergraphs. We show that various hypergraph parameters (for e.g. expansion, diameter, etc) can be bounded using this operator's eigenvalues. We study the heat diffusion process associated with this Laplacian operator, and bound its parameters in terms of its spectra. All our results are generalizations of the corresponding results for graphs. We show that there can be no linear operator for hypergraphs whose spectra captures hypergraph expansion in a Cheeger-like manner. Our Laplacian operator is non-linear, and thus computing its eigenvalues exactly is intractable. For any k, we give a polynomial time algorithm to compute an approximation to the kth smallest eigenvalue of the operator. We show that this approximation factor is optimal under the SSE hypothesis (introduced by [RS10]) for constant values of k. Finally, using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88692151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 61

Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices Vandermonde矩阵的超分辨率、极值函数和条件数

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-08-07 DOI: 10.1145/2746539.2746561

Ankur Moitra

{"title":"Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices","authors":"Ankur Moitra","doi":"10.1145/2746539.2746561","DOIUrl":"https://doi.org/10.1145/2746539.2746561","url":null,"abstract":"Super-resolution is a fundamental task in imaging, where the goal is to extract fine-grained structure from coarse-grained measurements. Here we are interested in a popular mathematical abstraction of this problem that has been widely studied in the statistics, signal processing and machine learning communities. We exactly resolve the threshold at which noisy super-resolution is possible. In particular, we establish a sharp phase transition for the relationship between the cutoff frequency (m) and the separation (Δ). If m > 1/Δ + 1, our estimator converges to the true values at an inverse polynomial rate in terms of the magnitude of the noise. And when m < (1-ε) /Δ no estimator can distinguish between a particular pair of Δ-separated signals even if the magnitude of the noise is exponentially small. Our results involve making novel connections between extremal functions and the spectral properties of Vandermonde matrices. We establish a sharp phase transition for their condition number which in turn allows us to give the first noise tolerance bounds for the matrix pencil method. Moreover we show that our methods can be interpreted as giving preconditioners for Vandermonde matrices, and we use this observation to design faster algorithms for super-resolution. We believe that these ideas may have other applications in designing faster algorithms for other basic tasks in signal processing.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73245694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 157

2-Server PIR with Sub-Polynomial Communication 基于次多项式通信的2-Server PIR

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-07-24 DOI: 10.1145/2746539.2746546

Zeev Dvir, Sivakanth Gopi

引用次数: 75

Time Lower Bounds for Nonadaptive Turnstile Streaming Algorithms 非自适应转门流算法的时间下界

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-07-08 DOI: 10.1145/2746539.2746542

Kasper Green Larsen, Jelani Nelson, Huy L. Nguyen

引用次数: 16

Consistency Thresholds for the Planted Bisection Model 种植平分模型的一致性阈值

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-07-07 DOI: 10.1145/2746539.2746603

Elchanan Mossel, Joe Neeman, A. Sly

引用次数: 188

Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method 基于平方和方法的字典学习和张量分解

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-07-06 DOI: 10.1145/2746539.2746605

B. Barak, Jonathan A. Kelner, David Steurer

{"title":"Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method","authors":"B. Barak, Jonathan A. Kelner, David Steurer","doi":"10.1145/2746539.2746605","DOIUrl":"https://doi.org/10.1145/2746539.2746605","url":null,"abstract":"We give a new approach to the dictionary learning (also known as \"sparse coding\") problem of recovering an unknown n x m matrix A (for m ≥ n) from examples of the form [y = Ax + e,] where x is a random vector in Rm with at most τ m nonzero coordinates, and e is a random noise vector in Rn with bounded magnitude. For the case m=O(n), our algorithm recovers every column of A within arbitrarily good constant accuracy in time mO(log m/log(τ-1)), in particular achieving polynomial time if τ = m-δ for any δ>0, and time mO(log m) if τ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most √n nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser x. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor T, given access to a tensor T' that is τ-close to T in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of T and T' have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82368776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 178

Inapproximability of Nash Equilibrium 纳什均衡的不可逼近性

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-05-13 DOI: 10.1145/2746539.2746578

A. Rubinstein

引用次数: 106

Hardness of Graph Pricing Through Generalized Max-Dicut 基于广义极大分割的图定价的硬度

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-05-04 DOI: 10.1145/2746539.2746549

Euiwoong Lee

{"title":"Hardness of Graph Pricing Through Generalized Max-Dicut","authors":"Euiwoong Lee","doi":"10.1145/2746539.2746549","DOIUrl":"https://doi.org/10.1145/2746539.2746549","url":null,"abstract":"The Graph Pricing problem is among the fundamental problems whose approximability is not well-understood. While there is a simple combinatorial 1/4-approximation algorithm, the best hardness result remains at 1/2 assuming the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate within a factor better than 1/4 under the UGC, so that the simple combinatorial algorithm might be the best possible. We also prove that for any ε > 0, there exists δ > 0 such that the integrality gap of nδ-rounds of the Sherali-Adams hierarchy of linear programming for Graph Pricing is at most 1/4 + ε. This work is based on the effort to view the Graph Pricing problem as a Constraint Satisfaction Problem (CSP) simpler than the standard and complicated formulation. We propose the problem called Generalized Max-Dicut(T), which has a domain size T + 1 for every T ≥ 1. Generalized Max-Dicut(1) is well-known Max-Dicut. There is an approximation preserving reduction from Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and both our results are achieved through this reduction. Besides its connection to Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own right since in most arity two CSPs studied in the literature, SDP-based algorithms perform better than LP-based or combinatorial algorithms --- for this arity two CSP, a simple combinatorial algorithm does the best.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77361423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 10

Tight Bounds for Learning a Mixture of Two Gaussians 学习两个高斯函数混合的紧界

Proceedings of the forty-seventh annual ACM symposium on Theory of Computing Pub Date : 2014-04-19 DOI: 10.1145/2746539.2746579

Moritz Hardt, Eric Price

{"title":"Tight Bounds for Learning a Mixture of Two Gaussians","authors":"Moritz Hardt, Eric Price","doi":"10.1145/2746539.2746579","DOIUrl":"https://doi.org/10.1145/2746539.2746579","url":null,"abstract":"We consider the problem of identifying the parameters of an unknown mixture of two arbitrary d-dimensional gaussians from a sequence of independent random samples. Our main results are upper and lower bounds giving a computationally efficient moment-based estimator with an optimal convergence rate, thus resolving a problem introduced by Pearson (1894). Denoting by σ2 the variance of the unknown mixture, we prove that Θ(σ12) samples are necessary and sufficient to estimate each parameter up to constant additive error when d=1. Our upper bound extends to arbitrary dimension d>1 up to a (provably necessary) logarithmic loss in d using a novel---yet simple---dimensionality reduction technique. We further identify several interesting special cases where the sample complexity is notably smaller than our optimal worst-case bound. For instance, if the means of the two components are separated by Ω(σ) the sample complexity reduces to O(σ2) and this is again optimal. Our results also apply to learning each component of the mixture up to small error in total variation distance, where our algorithm gives strong improvements in sample complexity over previous work.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82529188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 99