Mode-Seeking Kernel Losses for Robust Distribution System State Estimation: Contamination-Stability Theory and Cross-System Evidence

Learning-based distribution system state estimation (DSSE) predictors trained with the squared loss estimate the conditional mean of the state given the measurements. Under contaminated measurements, the conditional mean shifts, and the unbounded influence of the squared loss allows a few corrupted samples to dominate training. We replace the squared objective with the Gaussian kernel modal loss Lb(r)=1−exp(−r2/(2b2)), whose influence is bounded and redescending and whose population minimizer converges to the conditional mode as the bandwidth b→0. The mode-seeking formulation admits a learning-theoretic analysis: the population risk minimizer converges to the conditional mode as b→0; the score is bounded and redescending; a Rademacher bound of order O(1/n) controls generalization; the clean risk under Huber εc-contamination exceeds the optimal clean risk by at most 2εc+4Γn; and nonconvex SGD converges with bandwidth-controlled gradient variance. Experiments follow a corrected, seed-disciplined protocol (independent training runs, train-only normalization, paired corruption realizations, bootstrap confidence intervals) on four systems: CIGRE-14, IEEE-30, Oberrhein-70, and the independently generated external PanTaGruEl IEEE-14 benchmark. The modal-loss multilayer perceptron attains significantly lower voltage RMSE than its squared-loss counterpart at every tested condition on all four systems, with paired-bootstrap 95% confidence intervals excluding zero. In a comparison against conventional robust losses, no statistically detectable difference was found between the modal loss and the Tukey biweight under the tested configuration, while the modal loss outperformed MSE and Huber (δ=1.0) at the frozen parameter values; the modal objective replaces a finite residual cutoff with a smooth bandwidth parameter. For the graph neural estimator, the modal-loss advantage is significant on CIGRE-14 and Oberrhein-70 under mixture contamination but not on IEEE-30, indicating an architecture-dependent interaction. A multi-start analysis of the optimization-based WLS and Modal-WLS solvers shows that single-start results underestimate the achievable convergence of Modal-WLS under severe contamination (success rate 0.40 at K=1 vs. 0.80 at K=10), while estimation accuracy is largely unchanged.