Nondominated sorting, also called Pareto depth analysis (PDA), is widely used in multiobjective optimization and has recently found important applications in multicriteria anomaly detection. Recently, a partial differential equation (PDE) continuum limit was discovered for nondominated sorting leading to a very fast approximate sorting algorithm called PDE-based ranking. We propose in this paper a fast real-time streaming version of the PDA algorithm for anomaly detection that exploits the computational advantages of PDE continuum limits. Furthermore, we derive new PDE continuum limits for sorting points within their nondominated layers and show how the new PDEs can be used to classify anomalies based on which criterion was more significantly violated. We also prove statistical convergence rates for PDE-based ranking, and present the results of numerical experiments with both synthetic and real data.
Bibliographical noteFunding Information:
∗Received by the editors March 16, 2017; accepted for publication (in revised form) December 1, 2017; published electronically March 27, 2018. http://www.siam.org/journals/siap/78-2/M112118.html Funding: The second author’s work was supported by NSF grant DMS-1500829. †Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 0B9, Canada (firstname.lastname@example.org, email@example.com). ‡School of Mathematics, University of Minnesota, Mineapolis, MN 55455 (firstname.lastname@example.org).
- Anomaly detection
- Continuum limits
- Longest chain problem
- Nondominated sorting
- Pareto depth analysis
- Partial differential equations
- Streaming data
- Upwind finite difference schemes
- Viscosity solutions