Identifying Phase Transition Thresholds of Permuted Linear Regression via Message Passing

Hang Zhang, Ping Li

Abstract: This paper considers the permuted linear regression, i.e., ${\mathbf{Y}} = {\mathbf{\Pi}}^{\natural}{\mathbf{X}}{\mathbf{B}}^{\natural} + {\mathbf{W}}$, where ${\mathbf{Y}} \in \mathbb{R}^{n\times m}, {\mathbf{\Pi}}^{\natural}\in\mathbb{R}^{n\times n}, {\mathbf{X}} \in \mathbb{R}^{n\times p}, {\mathbf{B}}^{\natural}\in \mathbb{R}^{p\times m}$, and ${\mathbf{W}}\in \mathbb{R}^{n\times m}$ represent the observations, missing (or incomplete) information about ordering, sensing matrix, signal of interests, and additive sensing noise, respectively. As is shown in the previous work, there exists phase transition phenomena in terms of the \emph{signal-to-noise ratio} ($\mathsf{snr}$), number of permuted rows, etc. While all existing works only concern the convergence rates without specifying the associate constants in front of them, we give a precise identification of the phase transition thresholds via the message passing algorithm. Depending on whether the signal ${\mathbf{B}}^{\natural}$ is known or not, we separately identify the corresponding critical points around the phase transition regimes. Moreover, we provide numerical experiments and show the empirical phase transition points are well aligned with theoretical predictions.

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