TY - JOUR

T1 - Ranking recovery from limited pairwise comparisons using low-rank matrix completion

AU - Levy, Tal

AU - Vahid, Alireza

AU - Giryes, Raja

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/9

Y1 - 2021/9

N2 - This paper proposes a new methodology for solving the well-known rank aggregation problem from pairwise comparisons using low-rank matrix completion. Partial and noisy data of pairwise comparisons is first transformed into a matrix form. We then use tools from matrix completion, which has served as a major component in the low-rank based completion solution for the Netflix challenge, to construct the preference of different objects. In our approach, the data from multiple comparisons is used to create an estimate of the probability of object i winning (or be chosen) over object j, where only a partial set of comparisons between the N objects is known. These probabilities can be transformed to take the form of a rank-one matrix. An alternating minimization algorithm, in which the target matrix takes a bilinear form, is used in combination with maximum likelihood estimation for both factors. The reconstructed matrix is used to obtain the true underlying preference intensity. We start by exploring the asymptotic case of an infinitely large number of comparisons (“noiseless case”). We then extend our solution to the case of a finite number of comparisons for a subset of pairs (“noisy case”). This work demonstrates the improvement of our proposed algorithm over the state-of-the-art techniques in both simulated scenarios and real data.

AB - This paper proposes a new methodology for solving the well-known rank aggregation problem from pairwise comparisons using low-rank matrix completion. Partial and noisy data of pairwise comparisons is first transformed into a matrix form. We then use tools from matrix completion, which has served as a major component in the low-rank based completion solution for the Netflix challenge, to construct the preference of different objects. In our approach, the data from multiple comparisons is used to create an estimate of the probability of object i winning (or be chosen) over object j, where only a partial set of comparisons between the N objects is known. These probabilities can be transformed to take the form of a rank-one matrix. An alternating minimization algorithm, in which the target matrix takes a bilinear form, is used in combination with maximum likelihood estimation for both factors. The reconstructed matrix is used to obtain the true underlying preference intensity. We start by exploring the asymptotic case of an infinitely large number of comparisons (“noiseless case”). We then extend our solution to the case of a finite number of comparisons for a subset of pairs (“noisy case”). This work demonstrates the improvement of our proposed algorithm over the state-of-the-art techniques in both simulated scenarios and real data.

KW - Low-rank

KW - Matrix completion

KW - Ranking

UR - http://www.scopus.com/inward/record.url?scp=85103381632&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2021.03.004

DO - 10.1016/j.acha.2021.03.004

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AN - SCOPUS:85103381632

SN - 1063-5203

VL - 54

SP - 227

EP - 249

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

ER -