geman-mcclure loss

**Geman-McClure Loss** is a **robust loss function that strongly discounts the influence of outliers** — using the form $L(r) = frac{r^2}{2(1 + r^2/c^2)}$ which saturates for large residuals, providing strong robustness to outliers in regression problems. **Geman-McClure Properties** - **Form**: $L(r) = frac{r^2}{2(1 + r^2/c^2)}$ — maximal loss is $c^2/2$ for any residual. - **Influence Function**: $psi(r) = frac{r}{(1 + r^2/c^2)^2}$ — re-descending, meaning very large residuals have near-zero influence. - **Re-Descending**: Unlike Huber (which has constant influence for outliers), Geman-McClure completely eliminates outlier influence. - **Non-Convex**: The nonconvexity means multiple local minima — requires good initialization. **Why It Matters** - **Strong Robustness**: Outliers are completely ignored — the re-descending influence function drives their gradient toward zero. - **Computer Vision**: Widely used in motion estimation, optical flow, and 3D reconstruction. - **Trade-Off**: Non-convexity makes optimization harder, but provides stronger outlier rejection than convex alternatives. **Geman-McClure** is **the outlier eraser** — a re-descending robust loss that drives the influence of extreme outliers to zero.