(Upcoming) Constraint-Driven Diffusion Maps for Automatic Reduced-Space Learning
Date:
The Diffusion Maps (DMaps) algorithm is a manifold discovery technique that provides latent space coordinates for reduced-order models (ROMs) via spectral decomposition of a local kernel graph Laplacian which approximates the continuous Laplace-Beltrami operator. However, this method suffers from challenges inherent to kernel methods and harmonic analysis. First, success depends heavily on tuning the kernel width parameter. Second, the algorithm frequently discovers duplicate higher-frequency modes before uncovering coordinates that span new dimensions, meaning Laplacian spectral properties do not reliably indicate the data's intrinsic dimensionality. These issues are further exacerbated in real-world datasets that are highly nonlinear, noisy, sparse, or have varying point densities.
Consequently, using DMaps for ROMs requires human-in-the-loop inspection of the computed latent space coordinates. This bottleneck limits its practical application to 3D latent spaces, where dimensionality can be visually verified. To overcome this, we introduce a nonlinear optimization scheme to automatically discover both optimal parameters and diffusion coordinates. By formulating an objective function based on the geometric properties of a Laplacian reconstructed in the latent space, our approach enables the automated discovery of complex, n-D manifold structure, eliminating the need for manual inspection.