Outlier-robust learning with continuously differentiable least trimmed squares

Robust estimation is a fundamental task in statistical analysis, aimed at identifying models that can effectively eliminate the impact of noise, especially in the presence of outliers. The Least Trimmed Squares (LTS) estimation approach is widely recognized for its robustness in such scenarios. However, selecting a representative subset of samples for LTS estimation is computationally demanding, and the effectiveness of LTS is sensitive to the number of samples selected. In this study, we propose a novel approach, continuously differentiable LTS (CD-LTS), which employs a continuous function to approximate the original LTS. Due to its continuity and differentiability properties, CD-LTS can be used as a cost function for a range of learning models and avoids the need for additional sorting steps, thereby addressing the difficulty of applying traditional LTS directly. We utilize CD-LTS to develop four robust learning algorithms, including random vector functional link (RVFL), principal component analysis (PCA), iterative closest point (ICP), and orthogonal iterative (OI). The experimental results indicate that the proposed algorithms exhibit superior performance compared to existing methods.