M-Estimator

Ordinary least squares is statistically optimal under Gaussian noise — and catastrophically fragile under outliers, because the squared loss grows without bound: a single gross outlier can drag the estimate arbitrarily far. M-estimators (maximum-likelihood-type estimators) fix this by replacing the squared loss with a robust kernel $\rho$ that grows more slowly for large residuals:

where $r_i$ is the $i$-th residual (e.g. a reprojection error) and $\sigma$ is a scale parameter that normalizes residuals to noise units.

Common robust kernels

• Huber: quadratic for small residuals, linear beyond a threshold $k$:

Convex, safe, mild — the default choice when unsure.

• Cauchy (Lorentzian): $\rho(r) = \frac{c^2}{2}\log\!\left(1 + r^2/c^2\right)$. Logarithmic growth, strongly down-weights large outliers, but non-convex.

• Tukey biweight: saturates completely — residuals beyond the threshold $c$ contribute a constant cost and exactly zero gradient, so confirmed outliers stop influencing the solution at all. Such redescending kernels reject outliers hardest but need a decent initialization, since faraway measurements cannot pull the estimate back.

The derivative $\psi(r) = \rho'(r)$ is called the influence function: it measures how much a datum at residual $r$ tugs on the estimate. For least squares $\psi(r) = r$ (unbounded influence); for Huber it is clipped at $\pm k$; for Tukey it redescends to zero.

Solving with IRLS

Setting the gradient of the robust cost to zero gives $\sum_i \psi(r_i)\,\partial r_i/\partial\theta = 0$. Defining a weight $w_i = \psi(r_i)/r_i$ turns this into the condition of a weighted least-squares problem, which suggests Iteratively Reweighted Least Squares (IRLS):

1. Compute residuals $r_i$ at the current estimate.
2. Compute weights $w_i = \rho'(r_i)/r_i$ (small residual → weight near 1; large residual → weight near 0).
3. Solve the weighted least-squares problem $\min_\theta \sum_i w_i\, r_i(\theta)^2$ (one Gauss-Newton/LM step).
4. Repeat until convergence.

In practice this integrates seamlessly into a Gauss-Newton or Levenberg-Marquardt loop: the robust kernel just rescales each residual block’s Jacobian and error. Ceres calls these LossFunctions, g2o calls them RobustKernels.

The scale $\sigma$ matters as much as the kernel: it defines what counts as “large”. A standard robust estimate is derived from the median absolute deviation, $\hat{\sigma} = 1.4826 \cdot \mathrm{median}_i\,|r_i - \mathrm{median}(r)|$, where the constant makes it consistent with the Gaussian standard deviation.

M-estimators vs. RANSAC

The two are complementary, and real pipelines use both:

• RANSAC makes a hard inlier/outlier decision and can recover from overwhelming outlier fractions, but its output is only as good as one minimal sample.
• M-estimators make soft, continuous decisions and refine all parameters jointly, but converge to the right basin only if started near it.

The standard recipe: RANSAC to find a coarse model and inlier set, then robust nonlinear refinement (Huber/Cauchy) over the inliers to absorb the remaining mismatches.

Why it matters for SLAM

Every residual in SLAM is occasionally wrong: mismatched features, moving objects, false loop closures. Feeding even a handful of these into a pure least-squares bundle adjustment corrupts the whole trajectory. Robust kernels are therefore ubiquitous — ORB-SLAM wraps reprojection errors in Huber kernels, pose-graph back-ends wrap loop-closure edges in Cauchy or switchable-constraint formulations, and modern globally robust methods (graduated non-convexity) are built directly on redescending M-estimators. Knowing which kernel a system uses, and at what threshold, tells you how it will behave the day the front-end lies to it.

Related

• RANSAC
• Non-linear optimization
• Bundle Adjustment
• Robust pose-graph optimization
• GNC