Nonparametric Detection of Geometric Structures over Networks

TL;DR

This paper introduces kernel-based nonparametric tests for detecting anomalous structures in networks, achieving order-level optimality and demonstrating effectiveness through theoretical analysis and numerical experiments.

Contribution

It proposes a novel kernel-based testing method for network anomaly detection that is order-level optimal and extends to various network structures.

Findings

Proposed tests are order-level optimal for line networks.

Necessary conditions for universal consistency are established.

Numerical results confirm the effectiveness of the tests.

Abstract

Nonparametric detection of existence of an anomalous structure over a network is investigated. Nodes corresponding to the anomalous structure (if one exists) receive samples generated by a distribution q, which is different from a distribution p generating samples for other nodes. If an anomalous structure does not exist, all nodes receive samples generated by p. It is assumed that the distributions p and q are arbitrary and unknown. The goal is to design statistically consistent tests with probability of errors converging to zero as the network size becomes asymptotically large. Kernel-based tests are proposed based on maximum mean discrepancy that measures the distance between mean embeddings of distributions into a reproducing kernel Hilbert space. Detection of an anomalous interval over a line network is first studied. Sufficient conditions on minimum and maximum sizes of candidate anomalous intervals are characterized in order to guarantee the proposed test to be consistent. It is also shown that certain necessary conditions must hold to guarantee any test to be universally consistent. Comparison of sufficient and necessary conditions yields that the proposed test is order-level optimal and nearly optimal respectively in terms of minimum and maximum sizes of candidate anomalous intervals. Generalization of the results to other networks is further developed. Numerical results are provided to demonstrate the performance of the proposed tests.