  Recent work has demonstrated the effectiveness of gradient descent for
directly recovering the factors of low-rank matrices from random linear
measurements in a globally convergent manner when initialized properly.
However, the performance of existing algorithms is highly sensitive in the
presence of outliers that may take arbitrary values. In this paper, we propose
a truncated gradient descent algorithm to improve the robustness against
outliers, where the truncation is performed to rule out the contributions of
samples that deviate significantly from the {\em sample median} of measurement
residuals adaptively in each iteration. We demonstrate that, when initialized
in a basin of attraction close to the ground truth, the proposed algorithm
converges to the ground truth at a linear rate for the Gaussian measurement
model with a near-optimal number of measurements, even when a constant fraction
of the measurements are arbitrarily corrupted. In addition, we propose a new
truncated spectral method that ensures an initialization in the basin of
attraction at slightly higher requirements. We finally provide numerical
experiments to validate the superior performance of the proposed approach.
