Recovering a planted vector $v$ in an $n$-dimensional random subspace of $\mathbb{R}^N$ is a generic task related to many problems in machine learning and statistics, such as dictionary learning, subspace recovery, and principal component analysis. In this work, we study computationally efficient estimation and detection of a planted vector $v$ whose $\ell_4$ norm differs from that of a Gaussian vector with the same $\ell_2$ norm. For instance, in the special case of an $N \rho$-sparse vector $v$ with Rademacher nonzero entries, our results include the following: (1) We give an improved analysis of (a slight variant of) the spectral method proposed by Hopkins, Schramm, Shi, and Steurer, showing that it approximately recovers $v$ with high probability in the regime $n \rho \ll \sqrt{N}$. In contrast, previous work required either $\rho \ll 1/\sqrt{n}$ or $n \sqrt{\rho} \lesssim \sqrt{N}$ for polynomial-time recovery. Our result subsumes both of these conditions (up to logarithmic factors) and also treats the dense case $\rho = 1$ which was not previously considered. (2) Akin to $\ell_\infty$ bounds for eigenvector perturbation, we establish an entrywise error bound for the spectral estimator via a leave-one-out analysis, from which it follows that thresholding recovers $v$ exactly. (3) We study the associated detection problem and show that in the regime $n \rho \gg \sqrt{N}$, any spectral method from a large class (and more generally, any low-degree polynomial of the input) fails to detect the planted vector. This establishes optimality of our upper bounds and offers evidence that no polynomial-time algorithm can succeed when $n \rho \gg \sqrt{N}$.

翻译：以 $\ ell_ gg 标准值 $@ 美元重现一个植入的矢量 $n 以维度随机亚空间 $\ mathb{R ⁇ N$ 是一个与机器学习和统计方面的许多问题有关的通用任务, 例如字典学习、 子空间恢复和主要组件分析。 在这项工作中, 我们研究一个植入的矢量 $v$的计算效率估算和检测, 其美元标准与高氏矢量的概率不同, 相同 $\ ell_ 2美元 标准。 例如, 在一个名为 N\ rho$ 的磁量矢量, 美元 美元与 Rademacher 的非零条目, 我们的结果包括:(1) 我们改进了对霍普金斯、 Schramm、 Shi 和 Steur 提议的光谱方法的分析, 显示它大约回收了$v$美元, $rholl\ sqr@ t} 标准。 当我们之前的光量值解算值 和 亚氏 解算值的數值分析结果中, 其结果可以由1\ rqrus rus rus 解到 解到 or oral 的结果, rmexmexmexmexm 。