Manifold-adaptive dimension estimation revisited

TL;DR

This paper improves the manifold-adaptive FSA dimension estimator, making it highly accurate and robust, and applies it to analyze neural dynamics during seizures, revealing low-dimensional brain activity sources.

Contribution

It introduces a median-based global intrinsic dimensionality estimator derived from the FSA method, with corrections for finite-sample effects, and demonstrates its superior performance over existing estimators.

Findings

The corrected-median-FSA estimator outperforms the ML estimator in synthetic benchmarks.

It matches DANCo's performance on standard benchmarks.

Applied to neural data, it identifies low-dimensional seizure onset zones.

Abstract

Data dimensionality informs us about data complexity and sets limit on the structure of successful signal processing pipelines. In this work we revisit and improve the manifold-adaptive Farahmand-Szepesv\'ari-Audibert (FSA) dimension estimator, making it one of the best nearest neighbor-based dimension estimators available. We compute the probability density function of local FSA estimates, if the local manifold density is uniform. Based on the probability density function, we propose to use the median of local estimates as a basic global measure of intrinsic dimensionality, and we demonstrate the advantages of this asymptotically unbiased estimator over the previously proposed statistics: the mode and the mean. Additionally, from the probability density function, we derive the maximum likelihood formula for global intrinsic dimensionality, if i.i.d. holds. We tackle edge and finite-sample effects with an exponential correction formula, calibrated on hypercube datasets. We compare the performance of the corrected-median-FSA estimator with kNN estimators: maximum likelihood (ML, Levina-Bickel) and two implementations of DANCo (R and matlab). We show that corrected-median-FSA estimator beats the ML estimator and it is on equal footing with DANCo for standard synthetic benchmarks according to mean percentage error and error rate metrics. With the median-FSA algorithm, we reveal diverse changes in the neural dynamics while resting state and during epileptic seizures. We identify brain areas with lower-dimensional dynamics that are possible causal sources and candidates for being seizure onset zones.