# Optimal Spectral Recovery of a Planted Vector in a Subspace

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, principal component analysis, and non-Gaussian 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 where $v$ is an $N \rho$-sparse vector with Bernoulli-Gaussian or Bernoulli-Rademacher 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 (2016), showing that it approximately recovers $v$ with high probability in the regime $n \rho \ll \sqrt{N}$. This condition subsumes the conditions $\rho \ll 1/\sqrt{n}$ or $n \sqrt{\rho} \lesssim \sqrt{N}$ required by previous work up to polylogarithmic factors. We achieve $\ell_\infty$ error bounds for the spectral estimator via a leave-one-out analysis, from which it follows that a simple thresholding procedure exactly recovers $v$ with Bernoulli-Rademacher entries, even in the dense case $\rho = 1$. (2) 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 matches the condition for recovery and offers evidence that no polynomial-time algorithm can succeed in recovering a Bernoulli-Gaussian vector $v$ when $n \rho \gg \sqrt{N}$.

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