# Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detection

arXiv: https://arxiv.org/abs/2407.11873 · HTML: https://mlsanity.com/paper/2407.11873/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 76 references that are in the map.
- First posted on arXiv 16 Jul 2024.

## Builds on (most cited first, 24 of 24)

- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Benchmarking Multivariate Time Series Classification Algorithms](https://mlsanity.com/paper/2007.13156/) (2020)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [Discrete-time signatures and randomness in reservoir computing](https://mlsanity.com/paper/2010.14615/) (2020)
- [PCF-GAN: generating sequential data via the characteristic function of measures on the path space](https://mlsanity.com/paper/2305.12511/) (2023)
- [Lecture notes on rough paths and applications to machine learning](https://mlsanity.com/paper/2404.06583/) (2024)
- [Dimension reduction in recurrent networks by canonicalization](https://mlsanity.com/paper/2007.12141/) (2020)
- [Signature Methods in Stochastic Portfolio Theory](https://mlsanity.com/paper/2310.02322/) (2023)
- [The Mahalanobis distance for functional data with applications to classification](https://mlsanity.com/paper/1304.4786/) (2013)
- [Applications of Signature Methods to Market Anomaly Detection](https://mlsanity.com/paper/2201.02441/) (2022)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Dimensionless Anomaly Detection on Multivariate Streams with Variance Norm and Path Signature](https://mlsanity.com/paper/2006.03487/) (2020)
- [Path Development Network with Finite-dimensional Lie Group Representation](https://mlsanity.com/paper/2204.00740/) (2022)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)
- [The Signature Kernel](https://mlsanity.com/paper/2305.04625/) (2023)
- [On Mahalanobis distance in functional settings](https://mlsanity.com/paper/1803.06550/) (2018)
- [G-Signatures: Global Graph Propagation With Randomized Signatures](https://mlsanity.com/paper/2302.08811/) (2023)
- [Randomized Signature Methods in Optimal Portfolio Selection](https://mlsanity.com/paper/2312.16448/) (2023)
- [Novelty Detection on Radio Astronomy Data using Signatures](https://mlsanity.com/paper/2402.14892/) (2024)
- [Universal randomised signatures for generative time series modelling](https://mlsanity.com/paper/2406.10214/) (2024)
- [On the effectiveness of Randomized Signatures as Reservoir for Learning Rough Dynamics](https://mlsanity.com/paper/2201.00384/) (2022)

## Built on by (most cited first, 1 of 1)

- [Generating Financial Time Series by Matching Random Convolutional Features](https://mlsanity.com/paper/2606.05138/) (2026)

## Abstract (arXiv)

The Mahalanobis distance is a classical tool used to measure the covariance-adjusted distance between points in bbRd. In this work, we extend the concept of Mahalanobis distance to separable Banach spaces by reinterpreting it as a Cameron-Martin norm associated with a probability measure. This approach leads to a basis-free, data-driven notion of anomaly distance through the so-called variance norm, which can naturally be estimated using empirical measures of a sample. Our framework generalizes the classical bbRd, functional (L2[0,1])d, and kernelized settings; importantly, it incorporates non-injective covariance operators. We prove that the variance norm is invariant under invertible bounded linear transformations of the data, extending previous results which are limited to unitary operators. In the Hilbert space setting, we connect the variance norm to the RKHS of the covariance operator, and establish consistency and convergence results for estimation using empirical measures with Tikhonov regularization. Using the variance norm, we introduce the notion of a kernelized nearest-neighbour Mahalanobis distance, and study some of its finite-sample concentration properties. In an empirical study on 12 real-world data sets, we demonstrate that the kernelized nearest-neighbour Mahalanobis distance outperforms the traditional kernelized Mahalanobis distance for multivariate time series novelty detection, using state-of-the-art time series kernels such as the signature, global alignment, and Volterra reservoir kernels.