  This paper proposes the capped least squares regression with an adaptive
resistance parameter, hence the name, adaptive capped least squares regression.
The key observation is, by taking the resistant parameter to be data dependent,
the proposed estimator achieves full asymptotic efficiency without losing the
resistance property: it achieves the maximum breakdown point asymptotically.
Computationally, we formulate the proposed regression problem as a quadratic
mixed integer programming problem, which becomes computationally expensive when
the sample size gets large. The data-dependent resistant parameter, however,
makes the loss function more convex-like for larger-scale problems. This makes
a fast randomly initialized gradient descent algorithm possible for global
optimization. Numerical examples indicate the superiority of the proposed
estimator compared with classical methods. Three data applications to cancer
cell lines, stationary background recovery in video surveillance, and blind
image inpainting showcase its broad applicability.
