Results 1  10
of
145
Fast MonteCarlo algorithms for finding lowrank approximations
 IN PROCEEDINGS OF THE 39TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1998
"... We consider the problem of approximating a given m * n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the "best " such approximation. However, it takes time polynomial in m, n which is prohib ..."
Abstract

Cited by 237 (16 self)
 Add to MetaCart
We consider the problem of approximating a given m * n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the &quot;best &quot; such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this paper, we develop an algorithm which is qualitatively faster, provided we may sample the entries of the matrix according to a natural probability distribution. In many applications such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D * of rank at most k so that A D*2F < = min D,rank(D)<=k A D
Aggregating inconsistent information: ranking and clustering
, 2005
"... We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the number of disagreements with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc ..."
Abstract

Cited by 226 (17 self)
 Add to MetaCart
We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the number of disagreements with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a longstanding conjecture of BangJensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.
Limits of dense graph sequences
 J. Combin. Theory Ser. B
"... We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W: [0,1] 2 → [0, 1]. This limit object determines all the limits of subgraph ..."
Abstract

Cited by 207 (18 self)
 Add to MetaCart
(Show Context)
We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W: [0,1] 2 → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right. 1
Approximating the cutnorm via Grothendieck’s inequality
 Proc. of the 36 th ACM STOC
, 2004
"... ..."
(Show Context)
Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs
, 2004
"... ..."
(Show Context)
The counting lemma for regular kuniform hypergraphs
, 2004
"... Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of ..."
Abstract

Cited by 105 (14 self)
 Add to MetaCart
Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
Testing that distributions are close
 In IEEE Symposium on Foundations of Computer Science
, 2000
"... Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions ..."
Abstract

Cited by 98 (15 self)
 Add to MetaCart
(Show Context)
Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases ɛ when the distance between the distributions is small (less than max ( 2 32 3 √ n, ɛ 4 √)) or large (more n than ɛ) in L1distance. We also give an Ω(n 2/3 ɛ −2/3) lower bound. Our algorithm has applications to the problem of checking whether a given Markov process is rapidly mixing. We develop sublinear algorithms for this problem as well.
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
Abstract

Cited by 81 (4 self)
 Add to MetaCart
(Show Context)
Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
Property Testing
 Handbook of Randomized Computing, Vol. II
, 2000
"... this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well). ..."
Abstract

Cited by 75 (11 self)
 Add to MetaCart
(Show Context)
this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well).
Szemerédi’s lemma for the analyst
 Geom. Funct. Anal
"... Abstract Szemerédi's Regularity Lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi's Lemma can be thought of as a result in analysis ..."
Abstract

Cited by 71 (9 self)
 Add to MetaCart
(Show Context)
Abstract Szemerédi's Regularity Lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi's Lemma can be thought of as a result in analysis, and show some applications of analytic nature.