The Heavy Tail

GPS in cities is unreliable. Buildings reflect and block satellite signals, producing measurement errors that are nothing like the Gaussian noise the positioning algorithms assume. The errors have heavy tails — occasional large outliers from multipath and non-line-of-sight reception that the Gaussian model assigns near-zero probability.

Li et al. replace the Gaussian with a logistic distribution. The logistic has heavier tails — it assigns more probability to large errors without treating them as impossible. The resulting estimator (Least Quasi-Log-Cosh) reduces 3D positioning error by 11-31% compared to standard least-squares across real urban datasets.

The through-claim: the positioning error wasn’t algorithmic — it was distributional. The standard algorithm (weighted least squares) is mathematically optimal under its assumed error model. The problem is the assumed model is wrong. Heavy-tailed errors violate the Gaussian assumption, and the optimal estimator for a Gaussian distribution is a poor estimator for a logistic distribution. Changing the error model is the entire fix.

The logistic distribution is a specific choice with a specific property: its influence function is bounded, meaning outliers have limited effect on the solution. A Gaussian-optimal estimator gives outliers unlimited influence — a single reflected signal can drag the position estimate by meters. The logistic estimator caps this influence.

The satellite signals were fine. The error model was wrong. Fix the model, fix the position.