Nonparametric Matrix Estimation with One-Sided Covariates

TL;DR

This paper introduces a nonparametric matrix estimation method leveraging observed column covariates and unobserved row covariates, achieving minimax optimal rates and outperforming baselines in simulations.

Contribution

It proposes a novel algorithm for matrix estimation with unobserved row covariates and observed column covariates, achieving optimal nonparametric rates.

Findings

Algorithm outperforms naive methods in low data regimes.

Achieves minimax optimal nonparametric rates.

Demonstrates effectiveness through simulated experiments.

Abstract

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 $\mathbb{E}[X_{ui}] = f(\alpha_u, \beta_i)$ for some Holder smooth function $f$. We consider the setting where the row covariates $\alpha$ are unobserved yet the column covariates $\beta$ 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.