Robust approximate linear regression without correspondence
This work addresses a domain-specific problem in computational neuroscience for matching neurons with outliers, representing an incremental advancement by expanding previous theory to include outlier cases.
The paper tackles the problem of estimating correspondence between point sets with outliers in both source and target, proposing algorithms for robust regression without correspondence. It demonstrates effectiveness in simulated data and a neuron matching application for C. elegans, showing concrete results in handling outliers.
We propose methods for estimating correspondence between two point sets under the presence of outliers in both the source and target sets. The proposed algorithms expand upon the theory of the regression without correspondence problem to estimate transformation coefficients using unordered multisets of covariates and responses. Previous theoretical analysis of the problem has been done in a setting where the responses are a complete permutation of the regressed covariates. This paper expands the problem setting by analyzing the cases where only a subset of the responses is a permutation of the regressed covariates in addition to some covariates being outliers. We term this problem \textit{robust regression without correspondence} and provide several algorithms based on random sample consensus for exact and approximate recovery in a noiseless and noisy one-dimensional setting as well as an approximation algorithm for multiple dimensions. The theoretical guarantees of the algorithms are verified in simulated data. We demonstrate an important computational neuroscience application of the proposed framework by demonstrating its effectiveness in a \textit{Caenorhabditis elegans} neuron matching problem where the presence of outliers in both the source and target nematodes is a natural tendency.