Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons — Supplemental Materials —

This supplemental document presents details concerning analytical derivations that support the theorems made in the main text “Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons”, accepted to the 32th International Conference on Machine Learning (ICML 2015). One can find here the detailed proof of Theorems 2 4. 1 Main Theorems We repeat the main theorems as follows for convenience of presentation. Theorem 2 (Minimax Lower Bounds). Fix ∈ ( 0, 12 ) , and let G ∼ Gn,pobs . If L ≤ c (1− ) log n− 2 npobs∆K (1) holds for some absolute constant1 c > 0, then for any ranking scheme ψ, there exists a preference vector w with separation ∆K such that the probability of error Pe (ψ) ≥ . Theorem 3. Let c0, c1, c2, c3 > 0 be some sufficiently large constants. Suppose that L = O (poly (n)), the comparison graph G ∼ Gn,pobs with pobs > c0 log n/n, and assume that the separation measure satisfies ∆K > c1 √ log n npobsL . (2) Then with probability exceeding 1 − 1/n, Spectral MLE perfectly identifies the set of top-K ranked items, provided that the parameters obey T ≥ c2 log n and ξt := c3 { ξmin + 1 2t (ξmax − ξmin) } , (3) where ξmin := √ logn nLpobs and ξmax := √ logn pobsL . Theorem 4. Suppose that G ∼ Gn,pobs with pobs > c1 log n/n for some large constant c1, and that there exists a score ŵ ∈ [wmin, wmax] independent of G satisfying ∣∣ŵub i − wi∣∣ ≤ ξwmax, ∀1 ≤ i ≤ n; (4) ‖ŵ −w‖ ≤ δ ‖w‖ . (5) ∗Department of Statistics, Stanford University, CA 94305, U.S.A. †Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea 1More precisely, c = w4 min/(2w 4 max).