This paper considers the sparse recovery with shuffled labels, i.e., $\by = \bPitrue \bX \bbetatrue + \bw$, where $\by \in \RR^n$, $\bPi\in \RR^{n\times n}$, $\bX\in \RR^{n\times p}$, $\bbetatrue\in \RR^p$, $\bw \in \RR^n$ denote the sensing result, the unknown permutation matrix, the design matrix, the sparse signal, and the additive noise, respectively. Our goal is to reconstruct both the permutation matrix $\bPitrue$ and the sparse signal $\bbetatrue$. We investigate this problem from both the statistical and computational aspects. From the statistical aspect, we first establish the minimax lower bounds on the sample number $n$ and the \emph{signal-to-noise ratio} ($\snr$) for the correct recovery of permutation matrix $\bPitrue$ and the support set $\supp(\bbetatrue)$, to be more specific, $n \gtrsim k\log p$ and $\log\snr \gtrsim \log n + \frac{k\log p}{n}$. Then, we confirm the tightness of these minimax lower bounds by presenting an exhaustive-search based estimator whose performance matches the lower bounds thereof up to some multiplicative constants. From the computational aspect, we impose a parsimonious assumption on the number of permuted rows and propose a computationally-efficient estimator accordingly. Moreover, we show that our proposed estimator can obtain the ground-truth $(\bPitrue, \supp(\bbetatrue))$ under mild conditions. Furthermore, we provide numerical experiments to corroborate our claims.

翻译：本文研究了标签洗牌下的稀疏恢复问题，即 $\by = \bPitrue \bX \bbetatrue + \bw$，其中 $\by \in \RR^n$、$\bPi\in \RR^{n\times n}$、$\bX\in \RR^{n\times p}$、$\bbetatrue\in \RR^p$、$\bw \in \RR^n$ 分别表示感知结果、未知置换矩阵、设计矩阵、稀疏信号和加性噪声。我们的目标是同时重构置换矩阵 $\bPitrue$ 和稀疏信号 $\bbetatrue$。我们从统计和计算两个角度研究该问题。在统计层面，我们首先建立了正确恢复置换矩阵 $\bPitrue$ 和支持集 $\supp(\bbetatrue)$ 所需样本数 $n$ 与信噪比 ($\snr$) 的极小化下界，具体而言：$n \gtrsim k\log p$ 且 $\log\snr \gtrsim \log n + \frac{k\log p}{n}$。随后，我们通过提出一种基于穷举搜索的估计器验证了这些极小化下界的紧致性——该估计器的性能与下界仅相差若干乘法常数。在计算层面，我们对置换行数施加简约性假设，并据此提出一种计算高效的估计器。我们证明，该估计器能在温和条件下获取真实参数 $(\bPitrue, \supp(\bbetatrue))$。最后，我们通过数值实验验证了理论结论。