  This paper proposed a new regression model called $l_1$-regularized outlier
isolation and regression (LOIRE) and a fast algorithm based on block coordinate
descent to solve this model. Besides, assuming outliers are gross errors
following a Bernoulli process, this paper also presented a Bernoulli estimate
model which, in theory, should be very accurate and robust due to its complete
elimination of affections caused by outliers. Though this Bernoulli estimate is
hard to solve, it could be approximately achieved through a process which takes
LOIRE as an important intermediate step. As a result, the approximate Bernoulli
estimate is a good combination of Bernoulli estimate's accuracy and LOIRE
regression's efficiency with several simulations conducted to strongly verify
this point. Moreover, LOIRE can be further extended to realize robust rank
factorization which is powerful in recovering low-rank component from massive
corruptions. Extensive experimental results showed that the proposed method
outperforms state-of-the-art methods like RPCA and GoDec in the aspect of
computation speed with a competitive performance.
