  We study the problem of recovering an incomplete $m\times n$ matrix of rank
$r$ with columns arriving online over time. This is known as the problem of
life-long matrix completion, and is widely applied to recommendation system,
computer vision, system identification, etc. The challenge is to design
provable algorithms tolerant to a large amount of noises, with small sample
complexity. In this work, we give algorithms achieving strong guarantee under
two realistic noise models. In bounded deterministic noise, an adversary can
add any bounded yet unstructured noise to each column. For this problem, we
present an algorithm that returns a matrix of a small error, with sample
complexity almost as small as the best prior results in the noiseless case. For
sparse random noise, where the corrupted columns are sparse and drawn randomly,
we give an algorithm that exactly recovers an $\mu_0$-incoherent matrix by
probability at least $1-\delta$ with sample complexity as small as
$O\left(\mu_0rn\log (r/\delta)\right)$. This result advances the
state-of-the-art work and matches the lower bound in a worst case. We also
study the scenario where the hidden matrix lies on a mixture of subspaces and
show that the sample complexity can be even smaller. Our proposed algorithms
perform well experimentally in both synthetic and real-world datasets.
