  Low-rank matrix recovery problems involving high-dimensional and
heterogeneous data appear in applications throughout statistics and machine
learning. The contribution of this paper is to establish the fundamental limits
of recovery for a broad class of these problems. In particular, we study the
problem of estimating a rank-one matrix from Gaussian observations where
different blocks of the matrix are observed under different noise levels. In
the setting where the number of blocks is fixed while the number of variables
tends to infinity, we prove asymptotically exact formulas for the minimum
mean-squared error in estimating both the matrix and underlying factors. These
results are based on a novel reduction from the low-rank matrix tensor product
model (with homogeneous noise) to a rank-one model with heteroskedastic noise.
  As an application of our main result, we show that recently proposed methods
based on applying principal component analysis (PCA) to weighted combinations
of the data are optimal in some settings but sub-optimal in others. We also
provide numerical results comparing our asymptotic formulas with the
performance of methods based on weighted PCA, gradient descent, and approximate
message passing.
