Nonparametric Matrix Estimation with One-Sided Covariates
This work addresses matrix estimation challenges in scenarios with partial covariate information, offering incremental improvements for applications like recommendation systems or data imputation.
The paper tackles the problem of matrix estimation with one-sided observed covariates, where only column covariates are known, and proposes an algorithm that improves upon naive row-wise estimation and achieves minimax optimal nonparametric rates under moderate matrix proportions.
Consider the task of matrix estimation in which a dataset $X \in \mathbb{R}^{n\times m}$ is observed with sparsity $p$, and we would like to estimate $\mathbb{E}[X]$, where $\mathbb{E}[X_{ui}] = f(α_u, β_i)$ 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.