# Dimensionless Anomaly Detection on Multivariate Streams with Variance Norm and Path Signature

arXiv: https://arxiv.org/abs/2006.03487 · HTML: https://mlsanity.com/paper/2006.03487/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2020.

## Built on by (most cited first, 8 of 8)

- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Applications of Signature Methods to Market Anomaly Detection](https://mlsanity.com/paper/2201.02441/) (2022)
- [Novelty Detection on Radio Astronomy Data using Signatures](https://mlsanity.com/paper/2402.14892/) (2024)
- [Non-parametric online market regime detection and regime clustering for multidimensional and path-dependent data structures](https://mlsanity.com/paper/2306.15835/) (2023)
- [Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detection](https://mlsanity.com/paper/2407.11873/) (2024)
- [2DSig-Detect: a semi-supervised framework for anomaly detection on image data using 2D-signatures](https://mlsanity.com/paper/2409.04982/) (2024)
- [Path Signatures Enable Model-Free Mapping of RNA Modifications](https://mlsanity.com/paper/2511.08855/) (2025)
- [Novelty detection on path space](https://mlsanity.com/paper/2512.03243/) (2025)

## Abstract (arXiv)

In this paper, we propose a dimensionless anomaly detection method for multivariate streams. Our method is independent of the unit of measurement for the different stream channels, therefore dimensionless. We first propose the variance norm, a generalisation of Mahalanobis distance to handle infinite-dimensional feature space and singular empirical covariance matrix rigorously. We then combine the variance norm with the path signature, an infinite collection of iterated integrals that provide global features of streams, to propose SigMahaKNN, a method for anomaly detection on (multivariate) streams. We show that SigMahaKNN is invariant to stream reparametrisation, stream concatenation and has a graded discrimination power depending on the truncation level of the path signature. We implement SigMahaKNN as an open-source software, and perform extensive numerical experiments, showing significantly improved anomaly detection on streams compared to isolation forest and local outlier factors in applications ranging from language analysis, hand-writing analysis, ship movement paths analysis and univariate time-series analysis.