  Robust model fitting is a fundamental problem in computer vision: used to
pre-process raw data in the presence of outliers. Maximisation of Consensus
(MaxCon) is one of the most popular robust criteria and widely used. Recently
(Tennakoon et al. CVPR2021), a connection has been made between MaxCon and
estimation of influences of a Monotone Boolean function. Equipping the Boolean
cube with different measures and adopting different sampling strategies (two
sides of the same coin) can have differing effects: which leads to the current
study. This paper studies the concept of weighted influences for solving
MaxCon. In particular, we study endowing the Boolean cube with the Bernoulli
measure and performing biased (as opposed to uniform) sampling. Theoretically,
we prove the weighted influences, under this measure, of points belonging to
larger structures are smaller than those of points belonging to smaller
structures in general. We also consider another "natural" family of
sampling/weighting strategies, sampling with uniform measure concentrated on a
particular (Hamming) level of the cube.
  Based on weighted sampling, we modify the algorithm of Tennakoon et al., and
test on both synthetic and real datasets. This paper is not promoting a new
approach per se, but rather studying the issue of weighted sampling.
Accordingly, we are not claiming to have produced a superior algorithm: rather
we show some modest gains of Bernoulli sampling, and we illuminate some of the
interactions between structure in data and weighted sampling.
