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.

翻译：考虑矩阵估测任务, 即用月度 $x $x $\ mathbb{E}[X]$, 美元= f( ALpha_ u, \beta_ i), 某些 Holder 光滑功能的 f( ) $f 美元。 我们考虑的是行COVeate $\ alpha$ 未观察到但列COVetarate $\ beta $ 的设置 。 我们提供算法和附带分析, 显示当行数不小时, 我们算法在天真的估算每行时会有所改进。 此外, 当矩阵是中等比例时, 我们的算法会达到了解行共变法的极算法的最小最大非参数率。 在模拟实验中, 我们显示我们的算法优于低数据系统中的其他基线 。