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Cauchy / Tukey / Geman–McClure robust misfits

All three losses are redescending M-estimators: their influence function saturates (Cauchy, Geman–McClure) or vanishes completely (Tukey) for large residuals, providing strong outlier resistance.

Definitions

Let \(r = d_{\mathrm{syn}} - d_{\mathrm{obs}}\) and \(c > 0\) the scale.

Cauchy / Lorentzian (Crase et al. 1990; Black & Anandan 1996):

\[ \rho^{\mathrm{C}}_c(r) = \tfrac{c^2}{2}\log\bigl(1 + (r/c)^2\bigr). \]

Tukey biweight (Beaton & Tukey 1974):

\[ \rho^{\mathrm{T}}_c(r) = \begin{cases} \tfrac{c^2}{6}\Bigl(1-\bigl[1-(r/c)^2\bigr]^3\Bigr) & |r| \le c,\\ \tfrac{c^2}{6} & |r| > c. \end{cases} \]

The gradient \(\rho_{\mathrm{T}}'(r)= r\,(1-(r/c)^2)^2 \mathbf 1_{|r|\le c}\) is identically zero for \(|r|>c\) — hence "outlier rejection".

Geman–McClure (Geman & McClure 1985):

\[ \rho^{\mathrm{GM}}_c(r) = c^2\,\frac{(r/c)^2}{1+(r/c)^2}. \]

All three reduce to a quadratic for \(|r|\ll c\).

When to use

  • Strong, isolated outliers (bad picks, mistraced gathers).
  • Tukey is the most aggressive and works best when you can already estimate a reasonable noise scale c.

API

from sweep_loss import CauchyLoss, TukeyLoss, GemanMcClureLoss
CauchyLoss(c=0.5)(syn, obs)
TukeyLoss(c=0.5)(syn, obs)
GemanMcClureLoss(c=0.5)(syn, obs)

Tests

tests/test_robust.py checks:

  • point-wise formulas on hand-picked residuals,
  • Cauchy log-growth for \(|r|\gg c\),
  • Tukey clamps to \(c^2/6\) for \(|r|>c\) and has zero gradient there,
  • Geman–McClure stays below \(c^2\) everywhere and is quadratic near zero,
  • leading-order agreement with L2 in the small-residual regime.

References

  • Beaton, A. E. & Tukey, J. W. (1974). The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16, 147-185. doi:10.1080/00401706.1974.10489171
  • Black, M. J. & Anandan, P. (1996). The robust estimation of multiple motions: parametric and piecewise-smooth flow fields. CVIU 63 (1), 75-104. doi:10.1006/cviu.1996.0006
  • Bube, K. P. & Nemeth, T. (2007). Fast line searches for the robust solution of linear systems in the hybrid l1/l2 and Huber norms. Geophysics 72 (2), A13-A17. doi:10.1190/1.2431639
  • Crase, E., Pica, A., Noble, M., McDonald, J. & Tarantola, A. (1990). Robust elastic nonlinear waveform inversion: application to real data. Geophysics 55 (5), 527-538. doi:10.1190/1.1442864
  • Geman, S. & McClure, D. E. (1985). Bayesian image analysis: an application to single photon emission tomography. Proc. Stat. Comp. Sect., ASA, 12-18. (no DOI; see permanent record)