Overview
This research bridges Pure Mathematics (Differential Geometry) and Representation Learning (Manifold Embeddings).
The project constructs an automated experiment measuring how topological embeddings deform when subsets of data (e.g. digit classes in MNIST) are removed and incrementally restored.
Methodology & Mathematical Formulation
1. Manifold Embedding & Class Perturbations
- Fit reference UMAP embeddings on controlled high-dimensional image subsets.
- Incrementally drop and re-introduce individual classes across controlled fractional stages, refitting embeddings deterministically.
2. Concave Boundary & Periodic Spline Curvature
- Fit enclosing boundary curves via alpha-shape concave hulls around target embedding clusters.
- Parameterize boundaries as smooth, periodic 2D splines and sample uniformly by arc length.
- Compute local signed curvature (\kappa(s)) along the boundary curve from first and second parametric derivatives.
3. Scale-Invariance & Wasserstein Distance
- Enforce scale-invariance by normalizing curvature distributions with the boundary’s root-mean-square (RMS) radius.
- Quantify topological deformation by computing the Wasserstein distance (Earth Mover’s Distance) between perturbed curvature distributions and the unperturbed reference embedding.