Polynomial-Time MIMO Detection at the Maximum-Likelihood Threshold

2026-09-16

AI Summary

本文证明了在平方高斯二元MIMO模型中，通过舍入线性MMSE和最陡单比特下降法可以在多项式时间内达到最大似然检测的信噪比阈值。

Abstract

We prove that exact block recovery in the square Gaussian binary MIMO model can be achieved in polynomial time at the same first-order SNR threshold as exhaustive maximum-likelihood detection. Specifically, for $y=\sqrt{ρ/N}Hx^\star+w, \; x^\star\in\{\pm1\}^N,$ and independent standard Gaussian $H\in\mathbb R^{N\times N}$ and $w$, rounded linear MMSE followed by steepest single-bit descent recovers $x^\star$ with failure probability tending to zero, uniformly over every transmitted word and every $ρ\ge2\log N$, using $O(N^3)$ unit-cost exact-real arithmetic operations. The model is a special case of Gaussian random linear estimation, for which AMP state evolution and replica/MMSE formulas rigorously characterize fixed-parameter normalized performance. Those results predict the same $2\log N$ scale, but do not by themselves yield an all-coordinate guarantee in the dimension-dependent regime considered here. To the best of our knowledge, no prior work gives polynomial-time exact block recovery at the ML boundary for this setting; the closest prior square-system theorem, for the box relaxation, has first-order threshold $4\log N$. The proof places the rounded LMMSE estimate at sublinear Hamming distance from the truth, and then establishes, uniformly over every error set the local search can visit, that some wrong bit offers a quantified cost decrease while an objective barrier confines the search path. Conversely, if $0<ρ\le2\log N-\log\log N-s_N$ with $s_N\to\infty$ and $s_N=o(\log N)$, then a one-bit neighbor beats the transmitted word with probability tending to one, so even ML detection fails. Therefore, the statistical and polynomial-time exact-recovery thresholds coincide to first order in the stated arithmetic model.

Problem

• MIMO detection
• maximum-likelihood threshold
• polynomial-time
• Gaussian binary MIMO model

Innovation

• polynomial-time MIMO detection
• maximum-likelihood threshold
• rounded linear MMSE
• steepest single-bit descent