Outlier-Resistant 𝐿_𝟏 Orthogonal Regression via the Reformulation-Linearization Technique

Figure 1

Two-dimensional illustration of different methods for incorporating the 𝐿₁ norm into orthogonal regression. In traditional orthogonal regression, the sum of distances of points (𝑥,𝑦) to 𝐛^𝑇(𝑥,𝑦)𝐛 is maximized, the sum of distances of points (𝑥,𝑦) to 𝐚^𝑇(𝑥,𝑦)𝐚 is minimized, and the sum of the magnitudes of 𝐚^𝑇(𝑥,𝑦)𝐚 is maximized. As noted in the text, each of these distance measures can be modified to incorporate the 𝐿₁ norm to derive different results. In this paper, the approach is to maximize the sum of 𝐿₁ distances of points (𝑥,𝑦) to 𝐛^𝑇(𝑥,𝑦)𝐛 which is illustrated by 𝑑₁+𝑑₂.