Matrix Inference and Estimation in Multi-Layer Models

We consider the problem of estimating the input and hidden variables of a stochastic multi-layer neural network from an observation of the output. The hidden variables in each layer are represented as matrices with statistical interactions along both rows as well as columns. This problem applies to matrix imputation, signal recovery via deep generative prior models, multi-task and mixed regression, and learning certain classes of two-layer neural networks. We extend a recently-developed algorithm -- Multi-Layer Vector Approximate Message Passing (ML-VAMP), for this matrix-valued inference problem. It is shown that the performance of the proposed Multi-Layer Matrix VAMP (ML-Mat-VAMP) algorithm can be exactly predicted in a certain random large-system limit, where the dimensions $N\times d$ of the unknown quantities grow as $N\rightarrow\infty$ with $d$ fixed. In the two-layer neural-network learning problem, this scaling corresponds to the case where the number of input features, as well as training samples, grow to infinity but the number of hidden nodes stays fixed. The analysis enables a precise prediction of the parameter and test error of the learning.