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The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure
 Operations Research Letters
, 2015
"... Abstract We present a new polynomially solvable case of the Quadratic Assignment Problem in KoopmansBeckman form QAP(A, B), by showing that the identity permutation is optimal when A and B are respectively a Robinson similarity and dissimilarity matrix and one of A or B is a Toeplitz matrix. A Rob ..."
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Abstract We present a new polynomially solvable case of the Quadratic Assignment Problem in KoopmansBeckman form QAP(A, B), by showing that the identity permutation is optimal when A and B are respectively a Robinson similarity and dissimilarity matrix and one of A or B is a Toeplitz matrix. A Robinson (dis)similarity matrix is a symmetric matrix whose entries (increase) decrease monotonically along rows and columns when moving away from the diagonal, and such matrices arise in the classical seriation problem.
Serialrank: Spectral ranking using seriation
 in Advances in Neural Information Processing Systems
"... We describe a seriation algorithm for ranking a set of n items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriati ..."
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We describe a seriation algorithm for ranking a set of n items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder this matrix and construct a ranking. We first show that this spectral seriation algorithm recovers the true ranking when all pairwise comparisons are observed and consistent with a total order. We then show that ranking reconstruction is still exact even when some pairwise comparisons are corrupted or missing, and that seriation based spectral ranking is more robust to noise than other scoring methods. An additional benefit of the seriation formulation is that it allows us to solve semisupervised ranking problems. Experiments on both synthetic and real datasets demonstrate that seriation based spectral ranking achieves competitive and in some cases superior performance compared to classical ranking methods. 1
Beyond the Birkhoff polytope: Convex relaxations for vector permutation problems
 in Advances in Neural Information Processing Systems
, 2014
"... The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using Θ(n2) variables and constraints, significantly more than the n variables one could use to ..."
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The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using Θ(n2) variables and constraints, significantly more than the n variables one could use to represent a permutation as a vector. Using a recent construction of Goemans [1], we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to Θ(n logn) in theory and Θ(n log2 n) in practice. We modify the recent convex formulation of the 2SUM problem introduced by Fogel et al. [2] to use this polytope, and demonstrate how we can attain results of similar quality in significantly less computational time for large n. To our knowledge, this is the first usage of Goemans ’ compact formulation of the permutahedron in a convex optimization problem. We also introduce a simpler regularization scheme for this convex formulation of the 2SUM problem that yields good empirical results. 1
Similarity matrix Input Reconstructed
"... We’re given pairwise similarity information Aij on n variables. We suppose that the data has a serial structure, i.e. there is an underlying order π such that A π(i)π(j) decreases with i − j (Rmatrix) Can we recover π? ..."
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We’re given pairwise similarity information Aij on n variables. We suppose that the data has a serial structure, i.e. there is an underlying order π such that A π(i)π(j) decreases with i − j (Rmatrix) Can we recover π?
ADMCLE APPROACH FOR DETECTING SLOW VARIABLES IN CONTINUOUS TIME MARKOV CHAINS AND DYNAMIC DATA
, 2015
"... A method for detecting intrinsic slow variables in highdimensional stochastic chemical reaction networks is developed and analyzed. It combines anisotropic diffusion maps (ADM) with approximations based on the chemical Langevin equation (CLE). The resulting approach, called ADMCLE, has the poten ..."
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A method for detecting intrinsic slow variables in highdimensional stochastic chemical reaction networks is developed and analyzed. It combines anisotropic diffusion maps (ADM) with approximations based on the chemical Langevin equation (CLE). The resulting approach, called ADMCLE, has the potential of being more efficient than the ADM method for a large class of chemical reaction systems, because it replaces the computationally most expensive step of ADM (running local short bursts of simulations) by using an approximation based on the CLE. The ADMCLE approach can be used to estimate the stationary distribution of the detected slow variable, without any apriori knowledge of it. If the conditional distribution of the fast variables can be obtained analytically, then the resulting ADMCLE approach does not make any use of Monte Carlo simulations to estimate the distributions of both slow and fast variables.
SYNCRANK: ROBUST RANKING, CONSTRAINED RANKING AND RANK AGGREGATION VIA EIGENVECTOR AND SDP SYNCHRONIZATION
, 2015
"... Abstract. We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), ..."
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Abstract. We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine learning. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose usefulness has been demonstrated in numerous applications in recent years in computer vision and graphics, sensor network localization and structural biology. Its least squares solution can be approximated by either a spectral or a semidefinite programming (SDP) relaxation, followed by a rounding procedure. We show extensive numerical simulations on both synthetic and realworld data sets (Premier League soccer games, a Halo 2 game tournament and NCAA College Basketball games), which show that our proposed method compares favorably to other ranking methods from the recent literature. Existing theoretical guarantees on the group synchronization problem imply lower bounds on the largest amount of noise permissible in the data while still achieving an approximate recovery of the ground truth ranking. We propose a similar synchronizationbased algorithm for the rankaggregation problem, which integrates in a globally consistent ranking many pairwise rankoffsets or partial rankings, given by different rating systems on the same set of items, an approach which yields significantly more accurate results than other aggregation methods, including RankCentrality, a recent stateoftheart algorithm. Furthermore, we discuss the problem of semisupervised ranking when there is available information on the ground truth rank of a subset of players, and propose an algorithm based on SDP