Online learning of quadratic manifolds from streaming data for nonlinear dimensionality reduction and nonlinear model reduction (2409.02703v1)

Abstract: This work introduces an online greedy method for constructing quadratic manifolds from streaming data, designed to enable in-situ analysis of numerical simulation data on the Petabyte scale. Unlike traditional batch methods, which require all data to be available upfront and take multiple passes over the data, the proposed online greedy method incrementally updates quadratic manifolds in one pass as data points are received, eliminating the need for expensive disk input/output operations as well as storing and loading data points once they have been processed. A range of numerical examples demonstrate that the online greedy method learns accurate quadratic manifold embeddings while being capable of processing data that far exceed common disk input/output capabilities and volumes as well as main-memory sizes.

• The paper presents an online greedy algorithm that incrementally updates a truncated SVD to build quadratic manifolds for nonlinear reduction.
• The paper demonstrates significant error reduction in complex simulations by effectively capturing nonlinear dynamics in Petabyte-scale datasets.
• The paper’s approach enables real-time, scalable in-situ analysis for applications in computational fluid dynamics and large-scale simulations.

Online Learning of Quadratic Manifolds from Streaming Data for Nonlinear Dimensionality Reduction and Nonlinear Model Reduction

The paper presents a novel online greedy algorithm designed to construct quadratic manifolds from streaming data, addressing challenges in nonlinear dimensionality reduction and nonlinear model reduction. Traditional approaches for manifold learning often operate in a batch setting, requiring all data upfront, which is not feasible for large datasets often exceeding memory capabilities, and are time-consuming due to multiple data passes. This online method instead processes data incrementally as they are streamed, offering a compelling alternative for in-situ analysis, especially well-suited for handling Petabyte-scale numerical simulation data.

Core Methodology and Contributions

Central to the method is the transformation of traditional greedy techniques to work directly with streaming data via an incremental singular value decomposition (SVD) updating procedure. The algorithm capitalizes on the SVD of the data matrix without needing to explicitly construct the entire data matrix, leveraging low-rank approximations and efficient linear algebra operations to maintain scalability and computational feasibility.

1. Incremental SVD: The technique relies on dynamically maintaining a truncated SVD of the data, updating it with each new data chunk. It aligns closely with methods for incremental learning of principal components and dominated subspaces.
2. Greedy Manifold Construction: The method extends previous greedy approaches by selecting basis vectors from this dynamically maintained rank-reduced representation, allowing adaptation to streaming settings without full data availability.
3. Quadratic Correction: By fitting a nonlinear correction term to a quadratic feature map, this approach manages to capture richer dynamics inherent in many physical phenomena without resorting to excessively high-dimensional linear models.

Numerical Results and Implications

Through comprehensive numerical experiments, including scenarios such as Hamiltonian wave equations and turbulent channel flows, the paper demonstrates this approach's effectiveness. Notably, it can handle Petabyte-scale data, showcasing its utility for contemporary and future high-fidelity simulations.

• Error Reduction: The method achieves significant decreases in approximation error over linear dimensionality reductions, particularly in scenarios where dynamics exhibit nonlinear characteristics.
• Scalability: Numerical experiments underscore the method's strength in processing large datasets in situ, a crucial consideration as simulations grow in complexity and quantity of generated data.

Implications and Future Prospects

This method's scalability and accuracy point to considerable potential applications in various fields, including computational fluid dynamics and other areas relying on large-scale simulations.

• Practical Deployment: On a practical level, the ability to process and extract meaningful low-dimensional representations from streaming data opens avenues for real-time data insights, sensor data analytics, and other on-the-fly computational tasks.
• Theoretical Implications: From a theoretical standpoint, this work may catalyze further research into streaming versions of other nonlinear dimensionality reduction techniques and hybrid approaches combining multiple feature maps and manifold types.
• Future Research Directions: Future developments could explore deeper integration with GPU-accelerated computing and increased generalization to broader classes of manifolds and feature maps. The ongoing challenge will be achieving seamless scalability as data sizes continue to grow exponentially, potentially drawing more from advances in high-performance computing and machine learning frameworks.

Overall, the paper advances the state of data-driven modeling in scientific computing and offers a robust methodology for practitioners grappling with large datasets, solidifying the practical relevance of manifold learning in this domain.