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In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower bounds for this problem. It is NP-hard, and does not have a PTAS unless NP has subexponential time algorithms. On the other(More)
Quadratic Programming (QP) is the well-studied problem of maximizing over {−1, 1} values the quadratic form i =j a ij x i x j. QP captures many known combinatorial optimization problems, and assuming the unique games conjecture, semidefinite programming techniques give optimal approximation algorithms. We extend this body of work by initiating the study of(More)
Low rank decomposition of tensors is a powerful tool for learning generative models. The uniqueness results that hold for tensors give them a significant advantage over matrices. However, tensors pose serious algorithmic challenges; in particular, much of the matrix algebra toolkit fails to generalize to tensors. Efficient decomposition in the overcomplete(More)
Probabilistic modeling of ranking data is an extensively studied problem with applications ranging from understanding user preferences in electoral systems and social choice theory, to more modern learning tasks in online web search, crowd-sourcing and recommendation systems. This work concerns learning the Mallows model – one of the most popular(More)
We show that for any odd k and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a 1 2 + Ω(1/ √ D) fraction of 's constraints, where D is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent(More)
The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness(More)
We consider the problem of computing the q → p norm of a matrix A, which is defined for p, q ≥ 1, as Aq →p = max x = 0 Axp xq. This is in general a non-convex optimization problem, and is a natural generalization of the well-studied question of computing singular values (this corresponds to p = q = 2). Different settings of parameters give rise to a variety(More)
We give a robust version of the celebrated result of Kruskal on the uniqueness of tensor decompositions: we prove that given a tensor whose decomposition satisfies a robust form of Kruskal's rank condition, it is possible to approximately recover the decomposition if the tensor is known up to a sufficiently small (inverse polynomial) error. Kruskal's(More)
The Stochastic Block Model or the Planted Partition Model is the most widely used probabilistic model for community detection and clustering graphs in various fields, including machine learning, statistics, and social sciences. Many existing algorithms successfully learn the communities or clusters when the data is drawn exactly according to the model, but(More)
In this paper, we propose and study a new semi-random model for graph partitioning problems. We believe that it captures many properties of real-world instances. The model is more flexible than the semi-random model of Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and Sipser. We develop a general framework for solving semi-random(More)