Appendix for " Beyond the Signs: Nonparametric Tensor Completion via Sign Series "

See Section B.2 for constructive examples.Proof of Proposition 2. Based on (3) in Proposition 2, we have Risk( Z) Risk( Θ) = E null |sgnZ sgn Θ|| Θ|null . We divide the proof into two cases: α > 0 and α = . The inequality (6) now becomes Risk( Z) Risk( Θ) t null MAE(sgn Θ, sgnZ) C snull, for all 0 t < ρ(π, N) . Consider the same setup as in Theorem 2. Fix The conclusion (10) then directly follows by applying Remark A.1 to (11). 3 Proof of Theorem 2. To simplify the notation, we denote ρ = ρ(π, N). It follows from Kosorok (2007, Theorem 9.22) that the Proof of Theorem 3. By definition of ˆ Θ, we have MAE( ˆ Θ, Θ) = E null null null null null 1 2H + 1 null Assumption A.1, we establish the estimation accuracy guarantee for the large-margin estimators H log H. (29) In particualr, setting H null (1 + |N|) To apply Theorem A.1, we choose the pair ( L Here, we describe the details of the example set-up.