Maximum Consensus by Weighted Influences of Monotone Boolean Functions

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.