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The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure
 Oper. Res. Lett
"... 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) ..."
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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.
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
ADMCLE APPROACH FOR DETECTING SLOW VARIABLES IN CONTINUOUS TIME MARKOV CHAINS AND DYNAMIC DATA∗
"... Abstract. 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 t ..."
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Abstract. 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.
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?