Supplement to “ Graph Sparsification Approaches for Laplacian Smoothing ”

This document contains proofs, supplementary details, and supplementary experiments for the paper " Graph Sparsification Approaches for Laplacian Smoothing ". All section numbers, equation numbers, and figure numbers in this supplementary document are preceded by the letter A, to distinguish them from those from the main paper. Part (a). By optimality ofˆθ for problem (5), y − ˆ θ 2 2 + λ ˆ θ T ˜ L ˆ θ ≤ y − ˆ β 2 2 + λ ˆ β T ˜ L ˆ β ≤ y − ˆ β 2 2 + λ(1 +) ˆ β T L ˆ β, where we have used the spectral similarity of L, ˜ L. Rearranging, we see thatˆθ 2 2 − ˆ β 2 2 ≤ 2y T (ˆ θ − ˆ β) + λ(1 +) ˆ β T L ˆ β − λ ˆ θ T ˜ L ˆ θ. Substituting y = ˆ β + y − ˆ β on the right-hand side, and again rearranging, ˆ θ − ˆ β 2 2 ≤ 2(y − ˆ β) T (ˆ θ − ˆ β) + λ(1 +) ˆ β T L ˆ β − λ ˆ θ T ˜ L ˆ θ. Using y − ˆ β = λLˆβ from the stationarity condition for (1), ˆ θ − ˆ β 2 2 ≤ 2λˆθ T L ˆ β − λ(1 −) ˆ β T L ˆ β − λ ˆ θ T ˜ L ˆ θ ≤ 2λL 1/2 ˆ θ 2 L 1/2 ˆ β 2 − λ(1 −) ˆ β T L ˆ β − λ ˆ θ T ˜ L ˆ θ ≤ 2λ √ 1 + ˜ L 1/2 ˆ θ 2 L 1/2 ˆ β 2 − λ(1 −) ˆ β T L ˆ β − λ ˆ θ T ˜ L ˆ θ, where we have again used the spectral similarity of L, ˜ L. Now we examine two cases for last line above. If √ 1 + ˜ L 1/2 ˆ θ 2 ≤ L 1/2 ˆ β 2 , thenˆθ − ˆ β 2 2 ≤ λ(1 +) ˆ β T L ˆ β − λ ˆ θ T ˜ L ˆ θ. If √ 1 + ˜ L 1/2 ˆ θ 2 > L 1/2 ˆ β 2 , thenˆθ − ˆ β 2 2 ≤ λ(1 + 2) ˆ …