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.