Stable Filtering for Efficient Dimensionality Reduction of Streaming Manifold Data

2026-01-13

AI Summary

This work addresses the challenge of efficiently processing high-dimensional streaming data, which is often massive in volume and underlain by an unknown low-dimensional manifold structure. Conventional dimensionality reduction methods typically rely on training procedures or fail to preserve intrinsic geometric properties. To overcome these limitations, the authors propose Randomized Filtering (RF), a novel approach that, for the first time, leverages random projection theory to create a training-free, online dimensionality reduction tool. RF operates without assumptions about the underlying data distribution and effectively preserves the nonlinear geometric structure of unknown attractor manifolds. Theoretical analysis and extensive experiments demonstrate that RF achieves substantial computational efficiency while rigorously maintaining manifold geometry, making it suitable for diverse scientific applications.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Data Stream Mining
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• dimensionality reduction
• streaming data
• manifold geometry
• data efficiency
• geometry preservation

Innovation

• Randomized Filtering
• Manifold Preservation
• Streaming Data
• Dimensionality Reduction
• Data-Independent Embedding