Curve-Dimensional Relational Kernels for Low-Dimensional Geometric Embedding: A Preliminary Framework

This preprint investigates a preliminary modification of diffusion-map-based dimensionality reduction aimed at preserving ambient Euclidean distance characteristics in low-dimensional geometric embeddings. The central hypothesis is that incorporating curve-indexed geometric relationships into the affinity construction may provide additional information about the geometry of sampled objects. The proposed framework introduces the concept of curve-dimensions, in which each unique geometric curve associated with an object is treated as a dimension of its representation. Each point may be described through its membership in multiple curves and its position along each curve. Rather than changing the underlying spectral embedding procedure of diffusion maps, the framework modifies the pairwise relational structure used to construct the affinity kernel. Two pairwise relations are considered: a distance-based relation and a curve-dimensional relation. Their corresponding kernels can be combined multiplicatively before applying diffusion-map spectral embedding. The resulting low-dimensional coordinates are evaluated by comparing pairwise distances in the embedding with the original ambient Euclidean distances, using Pearson correlation, Spearman rank correlation, and scale-aligned stress. A preliminary synthetic experiment was conducted on an irregular non-planar triangulated surface containing 1,922 sampled vertices and 94 parameter curves. The experiment compares a geodesic-distance baseline with two curve-dimensional constructions: a coarse shared-curve scalar and a curve-signature relation incorporating along-curve positional information. The initial results show that the curve-signature construction captures more useful distance-related information than the coarse shared-curve scalar in this experiment. However, neither the curve-signature construction alone nor its multiplicative combination with the geodesic kernel outperforms the geodesic-only baseline. These findings therefore do not establish improved Euclidean distance preservation; instead, they provide an initial evaluation of the proposed representation and expose limitations in the current curve-dimensional distance definitions. The work establishes a preliminary mathematical and experimental framework for investigating whether curve membership, curve identity, and along-curve position can contribute useful relational information to low-dimensional geometric embedding. Future research will focus on refining the curve-dimensional relation, investigating alternative kernel-combination strategies, and evaluating Euclidean distance preservation across additional geometric datasets and established embedding methods. This is an ongoing research effort and does not claim a validated dimensionality-reduction algorithm or a theorem guaranteeing Euclidean distance preservation.