We consider the twin problems of estimating the effective rank and the Schatten norms $\|{\bf A}\|_{s}$ of a rectangular $p\times q$ matrix ${\bf A}$ from noisy observations. When $s$ is an even integer, we introduce a polynomial-time estimator of $\|{\bf A}\|_s$ that achieves the minimax rate $(pq)^{1/4}$. Interestingly, this optimal rate does not depend on the underlying rank of the matrix. When $s$ is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves the rate $(q\wedge p)(pq)^{1/4}$, which turns out to be optimal up to a logarithmic multiplicative term. The tight minimax rate is achieved by a more involved polynomial approximation method. This allows us to build estimators for a class of effective rank indices. As a byproduct, we also characterize the minimax rate for estimating the sequence of singular values of a matrix.
翻译:我们考虑了估算有效等级的双重问题,以及从噪音观测中估算矩形 $p_time q$ qtrime $ qmission $ $ bf A} 和 schatten 规范$ $ {bf A}} 的双重问题。 当美元不是平均整数时, 最佳税率要慢得多。 单值的简单阈值估计值达到 $ (q\wedge p)(pq) {%1/4} 的汇率时, 就会达到一个对数的倍数术语。 最紧微量的负数率是通过一种更涉及的多数值近似法实现的。 这使我们能够为一个有效等级指数的类别建立估计值。 作为副产品, 我们还将估算一个矩阵单值序列的微值比率定性为最小值。