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This paper considers the task of linear regression with shuffled labels,i.e., $\mathbf Y = \mathbf \Pi \mathbf X \mathbf B + \mathbf W$, where $\mathbfY \in \mathbb R^{n\times m}, \mathbf Pi \in \mathbb R^{n\times n}, \mathbf X\in\mathbb R^{n\times p}, \mathbf B \in \mathbb R^{p\times m}$, and $\mathbf W\in\mathbb R^{n\times m}$, respectively, represent the sensing results, (unknownor missing) corresponding information, sensing matrix, signal of interest, andadditive sensing noise. Given the observation $\mathbf Y$ and sensing matrix$\mathbf X$, we propose a one-step estimator to reconstruct $(\mathbf \Pi,\mathbf B)$. From the computational perspective, our estimator's complexity is$O(n^3 + np^2m)$, which is no greater than the maximum complexity of a linearassignment algorithm (e.g., $O(n^3)$) and a least square algorithm (e.g.,$O(np^2 m)$). From the statistical perspective, we divide the minimum $snr$requirement into four regimes, e.g., unknown, hard, medium, and easy regimes;and present sufficient conditions for the correct permutation recovery undereach regime: $(i)$ $snr \geq \Omega(1)$ in the easy regime; $(ii)$ $snr \geq\Omega(\log n)$ in the medium regime; and $(iii)$ $snr \geq \Omega((\logn)^{c_0}\cdot n^{{c_1}/{srank(\mathbf B)}})$ in the hard regime ($c_0, c_1$ aresome positive constants and $srank(\mathbf B)$ denotes the stable rank of$\mathbf B$). In the end, we also provide numerical experiments to confirm theabove claims.

Optimal Estimator for Linear Regression with Shuffled Labels