Robust Loss Functions

Outliers dominate the ordinary least-squares objective. Apex Solver ships 15 robust kernels — Huber, Cauchy, Tukey, Welsch, Fair, GemanMcClure, Andrews, Trimmed, L1, Lp, Barron variants and more — all in apex_solver::core::loss_functions.

Usage

#![allow(unused)]
fn main() {
use apex_solver::core::loss_functions::HuberLoss;

problem.add_residual_block(
    &[k_pose, k_landmark],
    Box::new(projection_factor),
    Some(Box::new(HuberLoss::new(1.345)?)),
);
}

What the solver actually computes

For a squared residual norm $s=\Vert \mathbf{r}{\Vert }^2$, the robust objective is

$$F(\mathbf{x})=\frac{1}{2}\sum _i{\rho }_i\left(s_i(\mathbf{x})\right)$$

Minimizing $F$ is equivalent to an iteratively reweighted least-squares problem whose residual and Jacobian are

$$\stackrel{=\rho \prime (s)r,J}{\mathbf{r}}=\sqrt{{\rho }^{\prime }(s)}\left(\mathbf{J}-\frac{{\rho }^{\prime \prime }(s)}{2{\rho }^{\prime }(s)}\alpha \mathbf{r}{\mathbf{r}}^{\top }\mathbf{J}\right),$$

with $\alpha$ from the Triggs correction. Apex Solver applies exactly this correction in-place during block evaluation, and reports the cost as $\frac{1}{2}\rho (s)$ — not as the norm of the corrected residual. The two are different functions; conflating them silently corrupts every reported cost and trust-region ratio on robust problems.