Fast Quadratic Manifold Learning For Nonlinear Dimensionality Reduction in Large-scale Systems using Riemannian Optimization

Abstract

The effectiveness of dimensionality reduction with quadratic manifolds hinges on the choice of a reduced basis and the associated quadratic correction terms. Existing approaches typically rely on subspaces spanned by the leading principal components of the training data. Although optimal for linear approximation, such bases are inherently suboptimal for quadratic manifold learning. Greedy basis-selection methods can significantly improve the representational capacity of quadratic manifolds by searching over a larger pool of candidate principal components, but the combinatorial cost limits the basis sizes that can be used in practice. This work proposes FastQM, an approach that treats the identification of an optimal quadratic approximation as a continuous optimization problem on the Stiefel manifold. By rotating the reduced basis within a candidate span of singular vectors, FastQM learns an ideal coordinate alignment tailored to quadratic manifold approximation. A feature-space formulation ensures that the optimization cost scales independently of the full state-space dimension. The efficacy of the proposed method is demonstrated on a turbulent airfoil-wake large-eddy simulation.