高维低秩矩阵的估计
Estimation of high-dimensional low-rank matrices
摘要：
假设我们观察到被噪声破坏的未知$m\times T$ -矩阵$A$的元素，或者更一般地，观察到元素的线性组合。我们特别感兴趣的是在高维设置中，未知条目的数量$mT$可以远远大于样本大小$N$。受几个应用的启发，我们考虑估计矩阵$A$的假设下，它有小的秩。这可以被视为降维或稀疏性假设。为了缩小到一个低秩表示，我们研究了惩罚最小二乘估计与Schatten-$p$准范数惩罚项，$p\leq1$。我们在两种可能的假设下研究了这些估计--“主要结果被陈述为预测风险和估计量的Schatten-$q$风险的非渐近上界，其中$q\in[p,2]$。我们获得的预测风险率的形式为$rm/N$（对于$m=T$），直至对数因子，其中$r$是$A$的秩。给出了多任务学习和矩阵完成的具体例子。证据是基于从经验过程理论的工具。作为一个副产品，我们得到的准凸Schatten类嵌入$S_p^M\hookrightarrow S_2^M$，$p<1$，这是独立的兴趣的$k$ th熵数的界限。
英文摘要：
Suppose that we observe entries or, more generally, linear combinations of entries of an unknown $m\times T$-matrix $A$ corrupted by noise. We are particularly interested in the high-dimensional setting where the number $mT$ of unknown entries can be much larger than the sample size $N$. Motivated by several applications, we consider estimation of matrix $A$ under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink toward a low-rank representation, we investigate penalized least squares estimators with a Schatten-$p$ quasi-norm penalty term, $p\leq1$. We study these estimators under two possible assumptions---a modified version of the restricted isometry condition and a uniform bound on the ratio "empirical norm induced by the sampling operator/Frobenius norm." The main results are stated as nonasymptotic upper bounds on the prediction risk and on the Schatten-$q$ risk of the estimators, where $q\in[p,2]$. The rates that we obtain for the prediction risk are of the form $rm/N$ (for $m=T$), up to logarithmic factors, where $r$ is the rank of $A$. The particular examples of multi-task learning and matrix completion are worked out in detail. The proofs are based on tools from the theory of empirical processes. As a by-product, we derive bounds for the $k$th entropy numbers of the quasi-convex Schatten class embeddings $S_p^M\hookrightarrow S_2^M$, $p<1$, which are of independent interest.
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