Clayton Scott

EECS and StatisticsUniversity of Michigan

Robust Kernel Density Estimation

Problem Statement

Kernel Density Estimate

Gaussian RKHS

KDE = mean in RKHS

Robust Kernel Density Estimate

Example

Outline

Robust Multivariate Mean

Majorization / Minimization

Iterative Re-Weighted Least Squares

IRWLS Computation

Kernel IRWLS

Kernel IRWLS Convergence

Representer Theorem

Robustness Interpretation # 1

Connection to Data Depth

Influence Function

Influence Function

Influence Function

Example

Robustness Interpretation # 2

Asymptotics: σ fixed

Asymptotics: σ fixed

Robustness Interpretation # 3

Asymptotics: σ → 0

Hampel Loss

Example: Hampel Loss

Anomaly Detection

Anomaly Detection

Anomaly Detection: AUC versus ε

Anomaly Detection: AUC versus ε

Anomaly Detection: AUC versus ε

Anomaly Detection: AUC versus ε

Anomaly Detection: AUC versus ε

Anomaly Detection: AUC versus ε

Explanation

Robustness Interpretation # 4:

Conclusions

Extensions / Open Questions

Acknowledgments

Parameter Tuning

Proof Idea

Anomaly Detection: Average Ranks

Spatial Depth