Sharp Limits and Efficient Recovery of Latent Memberships in Random-Design Regression

Mixed regression models capture population heterogeneity through distinct regression relationships across latent groups. When group memberships are scientifically or practically meaningful, recovering them becomes a fundamental inferential problem. Yet the statistical limits of membership recovery remain much less understood than those of estimating the regression components. For the two-component Gaussian random-design model with $p\lesssim n$, we show that the minimax mis-clustering risk is of order $\min (1,r)$ for noise-to-signal ratio $r$, while exact recovery is possible if and only if $nr\to 0$. These limits arise from an intrinsic ambiguity induced by the random regression margin, whose mass near zero determines the mis-clustering scale, while the minimum realized margin across the sample governs exact recovery. We further develop an efficient procedure attaining both limits, in which a bounded-residual spectral direction is combined with the pooled residual to form an initial partition, followed by hard least-squares iterations that geometrically contract a margin-weighted Hamming loss.