Stable Filtering for Efficient Dimensionality Reduction of Streaming Manifold Data

TL;DR

This paper introduces randomized filtering (RF), a data-independent, efficient dimensionality reduction method that preserves the geometry of streaming high-dimensional data on low-dimensional manifolds without requiring training.

Contribution

The paper presents RF, a novel randomized filtering technique that efficiently preserves non-linear manifold structures in streaming data, suitable for real-time scientific applications.

Findings

RF effectively preserves manifold geometry in streaming data.

RF outperforms traditional methods in computational efficiency.

Experimental results demonstrate RF's practical benefits across diverse datasets.

Abstract

Many areas in science and engineering now have access to technologies that enable the rapid collection of overwhelming data volumes. While these datasets are vital for understanding phenomena from physical to biological and social systems, the sheer magnitude of the data makes even simple storage, transmission, and basic processing highly challenging. To enable efficient and accurate execution of these data processing tasks, we require new dimensionality reduction tools that 1) do not need expensive, time-consuming training, and 2) preserve the underlying geometry of the data that has the information required to understand the measured system. Specifically, the geometry to be preserved is that induced by the fact that in many applications, streaming high-dimensional data evolves on a low-dimensional attractor manifold. Importantly, we may not know the exact structure of this manifold a priori. To solve these challenges, we present randomized filtering (RF), which leverages a specific instantiation of randomized dimensionality reduction to provably preserve non-linear manifold structure in the embedded space while remaining data-independent and computationally efficient. In this work we build on the rich theoretical promise of randomized dimensionality reduction to develop RF as a real, practical approach. We introduce novel methods, analysis, and experimental verification to illuminate the practicality of RF in diverse scientific applications, including several simulated and real-data examples that showcase the tangible benefits of RF.