Curvature-Weighted Embeddings for Non-Linear Data Fusion

CAMEL introduces curvature-augmented metrics for dimensionality reduction, while Zhang’s hierarchical simplicial learning preserves global topology via nested complexes. Neither fully addresses multi-scale curvature constraints in data fusion (e.g., combining medical imaging and genomics). This research proposes Curvature-Weighted Simplicial Embeddings (CWSE): First, use CAMEL’s Riemannian curvature to weight simplicial complex construction, ensuring high-curvature regions (e.g., tumor boundaries) are densely sampled. Second, apply Zhang’s hierarchical reduction to guarantee topological fidelity. By embedding data into a stratified manifold where curvature dictates local dimension, we outperform methods like UMAP on heterogeneous datasets. Novelty: Integrates differential geometry (curvature) with algebraic topology (simplicial hierarchies) for structured multi-modal fusion.

References:

1. Hierarchical simplicial manifold learning. Wei Zhang, Yi-Hsuan Shih, Jr-Shin Li (2024). PNAS Nexus.
2. UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction. Leland McInnes, John Healy (2018). arXiv.org.
3. CAMEL: Curvature-Augmented Manifold Embedding and Learning. Nan Xu, Yongming Liu (2023). arXiv.org.