Nonparametric Matrix Estimation with One-Sided Covariates

Consider the task of matrix estimation, in which we desire to estimate a ground truth matrix given sparse and noisy observations. Each entry is observed independently with probability p, and additionally perturbed with additive observation noise. Assume the (u,i)-th entry of the ground truth matrix can be described by f(α
u
,β
i
) for some Holder smooth function f. We consider the setting where the row covariates α are unobserved yet the column covariates β are observed. We provide an algorithm and accompanying analysis which shows that our algorithm improves upon naively estimating each row separately when the number of rows is not too small. Furthermore when the matrix is moderately proportioned, our algorithm achieves the minimax optimal nonparametric rate of an oracle algorithm that knows the row covariates. In simulated experiments we show our algorithm outperforms other baselines in low data regimes.