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Regularization and semisupervised learning on large graphs
 In COLT
, 2004
"... Abstract. We consider the problem of labeling a partially labeled graph. This setting may arise in a number of situations from survey sampling to information retrieval to pattern recognition in manifold settings. It is also of potential practical importance, when the data is abundant, but labeling i ..."
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Cited by 114 (1 self)
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Abstract. We consider the problem of labeling a partially labeled graph. This setting may arise in a number of situations from survey sampling to information retrieval to pattern recognition in manifold settings. It is also of potential practical importance, when the data is abundant, but labeling is expensive or requires human assistance. Our approach develops a framework for regularization on such graphs. The algorithms are very simple and involve solving a single, usually sparse, system of linear equations. Using the notion of algorithmic stability, we derive bounds on the generalization error and relate it to structural invariants of the graph. Some experimental results testing the performance of the regularization algorithm and the usefulness of the generalization bound are presented. 1
A counterexample to strong parallel repetition
 In Proc. 49th FOCS. IEEE
, 2008
"... The parallel repetition theorem states that for any twoprover game, with value 1 − ɛ (for, say, ɛ ≤ 1/2), the value of the game repeated in parallel n times is at most (1 − ɛ c) Ω(n/s) , where s is the answers ’ length (of the original game) and c is a universal constant [R95]. Several researchers ..."
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Cited by 20 (2 self)
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The parallel repetition theorem states that for any twoprover game, with value 1 − ɛ (for, say, ɛ ≤ 1/2), the value of the game repeated in parallel n times is at most (1 − ɛ c) Ω(n/s) , where s is the answers ’ length (of the original game) and c is a universal constant [R95]. Several researchers asked wether this bound could be improved to (1 − ɛ) Ω(n/s) ; this question is usually referred to as the strong parallel repetition problem. We show that the answer for this question is negative. More precisely, we consider the odd cycle game of size m; a twoprover game with value 1 − 1/2m. We show that the value of the odd cycle game repeated in parallel n times is at least 1 − (1/m) · O ( √ n). This implies that for large enough n (say, n ≥ Ω(m 2)), the value of the odd cycle game repeated in parallel n times is at least (1 − 1/4m 2) O(n). Thus: 1. For parallel repetition of general games: the bounds of (1−ɛ c) Ω(n/s) given in [R95, Hol07] are of the right form, up to determining the exact value of the constant c ≥ 2. 2. For parallel repetition of XOR games, unique games and projection games: the bounds of (1 − ɛ 2) Ω(n) given in [FKO07] (for XOR games) and in [Rao07] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game: the bound of 1 − (1/m) · ˜ Ω ( √ n) given in [FKO07] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of MaxCut that are at least 1 − ɛ 2 satisfiable from instances that are at most 1 − (2/π) · ɛ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition. 1
Ranking Sports Teams and the Inverse Equal Paths Problem
, 2006
"... The problem of rank aggregation has been studied in contexts varying from sports, to multicriteria decision making, to machine learning, to academic citations, to ranking web pages, and to descriptive decision theory. Rank aggregation is the mapping of inputs that rank subsets of a set of objects ..."
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Cited by 2 (0 self)
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The problem of rank aggregation has been studied in contexts varying from sports, to multicriteria decision making, to machine learning, to academic citations, to ranking web pages, and to descriptive decision theory. Rank aggregation is the mapping of inputs that rank subsets of a set of objects into a consistent ranking that represents in some meaningful way the various inputs. In the ranking of sports competitors, or academic citations or ranking of web pages the inputs are in the form of pairwise comparisons. We present here a new paradigm using an optimization framework that addresses major shortcomings in current models of aggregate ranking. Ranking methods are often criticized for being subjective and ignoring some factors or emphasizing others. In the ranking scheme here subjective considerations can be easily incorporated while their contributions to the overall ranking are made explicit. The inverse equal paths problem is introduced here, and is shown to be tightly linked to the problem of aggregate ranking “optimally”. This framework is useful in making an optimization framework available and by introducing specific performance measures for the quality of the aggregate ranking as per its deviations from the input rankings provided. Presented as inverse equal paths problem we devise for the aggregate ranking problem polynomial time combinatorial algorithms for convex penalty functions of the deviations; and show the NPhardness of some forms of nonlinear penalty functions. Interestingly, the algorithmic setup of the problem is that of a network flow problem. We compare the equal paths scheme here to the eigenvector method, Google PageRank for ranking web sites, and the academic citation method for ranking academic papers.
Parallel Imaging Problem
"... Abstract. Metric Labeling problems have been introduced as a model for understanding noisy data with pairwise relations between the data points. One application of labeling problems with pairwise relations is image understanding, where the underlying assumption is that physically close pixels are ..."
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Abstract. Metric Labeling problems have been introduced as a model for understanding noisy data with pairwise relations between the data points. One application of labeling problems with pairwise relations is image understanding, where the underlying assumption is that physically close pixels are likely to belong to the same object. In this paper we consider a variant of this problem, we will call Parallel Imaging, where instead of directly observing the noisy data, the data undergoes a simple linear transformation first, such as adding different images. This class of problems arises in a wide range of imaging problems. Our study has been motivated by the Parallel Imaging problem in Magnetic Resonance Image (MRI) reconstruction. We give a constant factor approximation algorithm for the case of speedup of two with the truncated linear metric, motivated by the MRI reconstruction problem. Our method uses local search and graph cut techniques. 1
OPTIMAL ALLOCATION PROBLEM WITH QUADRATIC UTILITY FUNCTIONS AND ITS RELATIONSHIP WITH GRAPH CUT PROBLEM
, 2011
"... Abstract We discuss the optimal allocation problem in combinatorial auctions, where the items are allocated to bidders so that the sum of the bidders ’ utilities is maximized. In this paper, we consider the case where utility functions are given by quadratic functions; the class of such utility func ..."
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Abstract We discuss the optimal allocation problem in combinatorial auctions, where the items are allocated to bidders so that the sum of the bidders ’ utilities is maximized. In this paper, we consider the case where utility functions are given by quadratic functions; the class of such utility functions has a succinct representation but is sufficiently general. The main aim of this paper is to show the computational complexity of the optimal allocation problem with quadratic utility functions. We consider the cases where utility functions are submodular and supermodular, and show NPhardness and/or polynomialtime exact/approximation algorithms. These results are given by using the relationship with graph cut problems such as the min/max cut problem and the multiway cut problem.