Low-Rank Positive Semidefinite Matrix Recovery from Corrupted Rank-One Measurements

Yue Sun, Yuanxin Li, Yuejie Chi

Abstract:

We study the problem of estimating a low-rank
positive semidefinite (PSD) matrix from a set of rank-one
measurements using sensing vectors composed of i.i.d. standard
Gaussian entries, which are possibly corrupted by arbitrary
outliers. This problem arises from applications such as phase
retrieval, covariance sketching, quantum space tomography, and
power spectrum estimation. We first propose a convex optimization
algorithm that seeks the PSD matrix with the minimum ℓ1-
norm of the observation residual. The advantage of our algorithm
is that it is free of parameters, therefore eliminating the need
for tuning parameters and allowing easy implementations. We
establish that with high probability, a low-rank PSD matrix can
be exactly recovered as soon as the number of measurements
is large enough, even when a fraction of the measurements are
corrupted by outliers with arbitrary magnitudes. Moreover, the
recovery is also stable against bounded noise. With the additional
information of an upper bound of the rank of the PSD matrix,
we propose another non-convex algorithm based on subgradient
descent that demonstrates excellent empirical performance in
terms of computational efficiency and accuracy.