流形正则化的大间隔 $\ell_p$-SVDD 用于多维时间序列异常检测
Signature-Kernel One-Class Learning with Manifold Regularisation for Multivariate Time Series Anomaly Detection
AI总结：
本文通过流形正则化推广大间隔 $\ell_p$-SVDD 方法，利用数据几何结构进行时间序列异常检测，理论分析泛化性能并实验验证其有效性。
摘要：
我们将最近提出的大间隔 $\ell_p$-SVDD 方法进行推广，通过流形正则化来利用数据分布的几何结构，用于时间序列异常检测。具体而言，我们构建了 $\ell_p$-SVDD 方法的一个流形正则化变体，以鼓励底层流形上的标签平滑性，从而捕获结构信息以提高检测性能。借鉴现有的表示定理，我们为所提出的方法提供了一种有效的优化技术。我们使用 Rademacher 复杂度对所提出的方法进行理论研究，分析其泛化性能，并在各种数据集上对所提出的方法进行实验评估，以将其性能与其他方法进行比较。
英文摘要：
One-class learning is an effective methodology for time series anomaly detection, yet kernel one-class approaches often compare sequences in a pointwise fashion and ignore the geometry of normal data. In this work, we present a manifold-regularised, large-margin $\ell_p$-support vector data description ($\ell_p$-SVDD) method that analyses sequences as paths via the signature kernel. By incorporating a a graph-Laplacian penalty into the $\ell_p$-SVDD objective function and utilising the representer theorem for manifold regularisation, we illustrate that the dual task retains the functional form of the original one with a modified kernel matrix. As such, existing Frank--Wolfe solvers can be applied unchanged. We show that this new modified kernel is a valid kernel matrix and under a generic norm bound on the discriminant, the proposed model yields a smaller Rademacher-complexity bound than the baseline. Furthermore, the signature kernel which is computed through a Goursat partial differential equation and can handle irregularly sampled data and is invariant to reparametrisation has not previously been used for one-class time series anomaly detection to the best of our knowledge. Our experiments on five widely used benchmarks, evaluated without point adjustment, demonstrate that the proposed approach compares very favourably with the state of the art approaches.
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