Novelty detection on path space
This addresses novelty detection for path data, which is incremental by extending existing methods to path space with theoretical guarantees.
The paper tackles novelty detection on path space by framing it as a hypothesis testing problem with signature-based test statistics, obtaining tail bounds for false positive rates and deriving exact formulae for smooth CVaR surrogates. It results in new one-class SVM algorithms and provides numerical evaluation on synthetic and real-world data.
We frame novelty detection on path space as a hypothesis testing problem with signature-based test statistics. Using transportation-cost inequalities of Gasteratos and Jacquier (2023), we obtain tail bounds for false positive rates that extend beyond Gaussian measures to laws of RDE solutions with smooth bounded vector fields, yielding estimates of quantiles and p-values. Exploiting the shuffle product, we derive exact formulae for smooth surrogates of conditional value-at-risk (CVaR) in terms of expected signatures, leading to new one-class SVM algorithms optimising smooth CVaR objectives. We then establish lower bounds on type-$\mathrm{II}$ error for alternatives with finite first moment, giving general power bounds when the reference measure and the alternative are absolutely continuous with respect to each other. Finally, we evaluate numerically the type-$\mathrm{I}$ error and statistical power of signature-based test statistic, using synthetic anomalous diffusion data and real-world molecular biology data.