IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations

This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We present a systematic and detailed construction of a metric representation for locally distorted metric spaces that captures complex data structures more accurately than the previous schemes. Our approach addresses limitations in existing methods by accommodating non-uniform data distributions and intricate local geometries. We validate its performance through extensive experiments on examples of various geometric objects and benchmark real-world datasets, demonstrating significant improvements in representation quality.

💡 Research Summary

The paper introduces IsUMap, a novel manifold learning and visualization method that merges the strengths of UMAP and Isomap while leveraging Vietoris‑Rips filtrations to construct a globally consistent intrinsic metric. The authors start from the observation that existing low‑dimensional embedding techniques struggle with non‑uniform sampling and complex local geometry. UMAP builds a probabilistic fuzzy simplicial set from a single global distance function, which can be distorted in regions of high curvature or density variation. Isomap computes geodesic distances on a k‑nearest‑neighbor graph, but its reliance on a simple graph structure ignores higher‑order relationships among points.

IsUMap addresses these issues through a four‑step pipeline:

1. Local distance distortion – For each data point, a locally adapted distance function is derived, allowing the method to respect heterogeneous density and curvature.
2. Vietoris‑Rips filtration construction – Using the locally distorted distances, a family of Vietoris‑Rips complexes (V_R(X,r)) is built across a range of scales (r). Unlike ordinary graph‑based approaches, these complexes contain higher‑dimensional simplices (triangles, tetrahedra, etc.), capturing multi‑way relationships.
3. Weighted simplicial complex definition – Each simplex (\sigma) receives a weight (W(\sigma)=\min{r \mid \sigma\in V_R(X,r)}), i.e., the smallest scale at which the simplex appears. This weight is propagated to edges so that the weight of a simplex equals the maximum weight of its 1‑dimensional faces, ensuring consistency.
4. Metric realization – Following Spivak’s metric realization theory (2009), the weighted complex is transformed into an extended metric space ((Y,d)) whose vertex set coincides with the original data points. The resulting distance satisfies symmetry, the triangle inequality, and reflexivity, thereby providing an intrinsic metric that reflects the global geometry implied by all local distortions.

With this intrinsic metric in hand, the authors embed the data into a low‑dimensional Euclidean space using Multidimensional Scaling (MDS). They justify the choice of MDS by citing Lim & Memoli (2022), which shows that when an intrinsic metric is available, MDS minimizes stress and preserves global relationships optimally.

The theoretical contributions are twofold. First, the paper revisits the Hausmann–Latschev theorem, establishing that for sufficiently small scales the Vietoris‑Rips complex of a compact Riemannian manifold is homotopy equivalent to the manifold itself, guaranteeing that the filtration captures the true topology. Second, the authors formulate the construction in categorical language, describing an equivalence between the category of “uber metric spaces” (UM) and the category of weighted Vietoris‑Rips filtrations. This abstraction clarifies the conditions under which a decreasing function (\Phi) can convert a fuzzy simplicial set into a weighted simplicial complex corresponding to a genuine metric.

Algorithmic details are provided in a step‑by‑step pseudocode (Algorithm 1). Input includes the raw point cloud (X), an initial distance matrix (d), a scale range (