High-Dimensional Multiple-Measurement-Vector Problem: Mutual Information and Message Passing Solution

In this paper, we study the high-dimensional multiple-measurement-vector (MMV) problem, which typically arises in massive machine-type communications (mMTC) that operates with massive multiple-input multiple-output (MIMO) in a grant-free manner. We derive an expression for the mutual information (MI) of the MMV channel, considering an input that has i.i.d. rows and a row-wise output that is randomly mapped from a linear combination of the input and the weighting. Our derivation follows from the replica method, a non-rigorous but very powerful approach prevailing in physics society, but unlike the classical treatment, the replica symmetry (RS) structure is extended here to comply with the MMV setting. This new structure requires an additional condition of simultaneously diagonalizability on the matrix parameters if the row-wise output mapping is non-Gaussian. The MI obtained suggests its first-order condition is pointing towards the fixed-point equation of some message passing algorithm. In the case of a correlated Gaussian channel, the MI further gives birth to a new identity on the celebrated MI-MMSE relation. To find an algorithm that matches the MI's first-order condition, we apply vector-form message passing, and circumvent the difficulty of non-commutable matrix products. We also derive its state evolution (SE), whose fixed-point equation clearly shows it perfectly agrees with the first-order condition of the MI derived.