Temporal label recovery from noisy dynamical data

TL;DR

This paper introduces spectral algorithms based on manifold learning to recover temporal labels from noisy dynamical data, applicable to both periodic and aperiodic systems without requiring monotonic similarity assumptions.

Contribution

The work develops novel spectral methods that leverage Fiedler vectors for temporal label recovery, handling noise and complex dynamics without eigen-gap assumptions.

Findings

Outperforms existing spectral seriation algorithms in numerical tests.

Provides $ ext{l}_{ ext{infinity}}$ error bounds for label estimation.

Demonstrates effectiveness on synthetic biomolecule data.

Abstract

Analyzing dynamical data often requires information of the temporal labels, but such information is unavailable in many applications. Recovery of these temporal labels, closely related to the seriation or sequencing problem, becomes crucial in the study. However, challenges arise due to the nonlinear nature of the data and the complexity of the underlying dynamical system, which may be periodic or non-periodic. Additionally, noise within the feature space complicates the theoretical analysis. Our work develops spectral algorithms that leverage manifold learning concepts to recover temporal labels from noisy data. We first construct the graph Laplacian of the data, and then employ the second (and the third) Fiedler vectors to recover temporal labels. This method can be applied to both periodic and aperiodic cases. It also does not require monotone properties on the similarity matrix, which are commonly assumed in existing spectral seriation algorithms. We develop the $\ell_{\infty}$ error of our estimators for the temporal labels and ranking, without assumptions on the eigen-gap. In numerical analysis, our method outperforms spectral seriation algorithms based on a similarity matrix. The performance of our algorithms is further demonstrated on a synthetic biomolecule data example.