The inference of a large symmetric signal-matrix $\mathbf{S} \in \mathbb{R}^{N\times N}$ corrupted by additive Gaussian noise, is considered for two regimes of growth of the rank $M$ as a function of $N$. For sub-linear ranks $M=\Theta(N^\alpha)$ with $\alpha\in(0,1)$ the mutual information and minimum mean-square error (MMSE) are derived for two classes of signal-matrices: (a) $\mathbf{S}=\mathbf{X}\mathbf{X}^\intercal$ with entries of $\mathbf{X}\in\mathbb{R}^{N\times M}$ independent identically distributed; (b) $\mathbf{S}$ sampled from a rotationally invariant distribution. Surprisingly, the formulas match the rank-one case. Two efficient algorithms are explored and conjectured to saturate the MMSE when no statistical-to-computational gap is present: (1) Decimation Approximate Message Passing; (2) a spectral algorithm based on a Rotation Invariant Estimator. For linear ranks $M=\Theta(N)$ the mutual information is rigorously derived for signal-matrices from a rotationally invariant distribution. Close connections with scalar inference in free probability are uncovered, which allow to deduce a simple formula for the MMSE as an integral involving the limiting spectral measure of the data matrix only. An interesting issue is whether the known information theoretic phase transitions for rank-one, and hence also sub-linear-rank, still persist in linear-rank. Our analysis suggests that only a smoothed-out trace of the transitions persists. Furthermore, the change of behavior between low and truly high-rank regimes only happens at the linear scale $\alpha=1$.

翻译：本文考虑在加性高斯噪声干扰下，对大型对称信号矩阵 $\mathbf{S} \in \mathbb{R}^{N\times N}$ 的推断问题，重点关注秩 $M$ 随 $N$ 增长的两种机制。对于次线性秩 $M=\Theta(N^\alpha)$（其中 $\alpha\in(0,1)$），我们推导了两类信号矩阵的互信息和最小均方误差（MMSE）：(a) $\mathbf{S}=\mathbf{X}\mathbf{X}^\intercal$，其中 $\mathbf{X}\in\mathbb{R}^{N\times M}$ 的元素独立同分布；(b) $\mathbf{S}$ 从旋转不变分布中采样。令人惊讶的是，所得公式与秩为1的情况一致。本文探索了两种高效算法，并推测当不存在统计-计算差距时，它们能饱和MMSE：(1) 抽选近似消息传递（Decimation Approximate Message Passing）；(2) 基于旋转不变估计量的谱算法。对于线性秩 $M=\Theta(N)$，我们从旋转不变分布的信号矩阵出发，严格推导了互信息。研究揭示了与自由概率论中标量推断的密切联系，从而得到一个简单的MMSE公式，该公式仅涉及数据矩阵极限谱测度的积分。一个有趣的问题是：已知秩为1（进而次线性秩）的信息论相变在线性秩中是否仍然存在？我们的分析表明，仅存在相变的平滑迹。此外，低秩与真正高秩行为的变化仅发生在线性尺度 $\alpha=1$ 处。