Reproduce “Tangent Blow-Ups for Processing Non-Manifold Geometry” on OpenResearch

About this paper

Alice Petrov, Mohammad Sina Nabizadeh, Ana Dodik, Justin Solomon

The authors introduce a lifted point-cloud representation that pairs each sample’s position with a projector for its tangent plane, so neighborhoods account for both spatial proximity and tangent disagreement. This lets kernels and differential operators separate intersecting sheets without singularity detection. On a Klein bottle, curvature error remains low near crossings; the result assumes tangent information and does not establish noisy-tangent performance.

Key insights

• Non-manifold singularities can be inherently resolved by transforming geometric data into a higher-dimensional space that differentiates points not only by position but also by local geometric properties like tangent direction and curvature.
• The proposed product metric enables standard Euclidean point cloud algorithms (e.g., k-NN search, Laplacian construction) to be directly applied to the lifted data, effectively extending their applicability to complex non-manifold geometries without modification.
• Theoretical guarantees ensure that distinct geometric components become metrically separated in the lifted space and that the lifted manifold retains sufficient regularity to support differential calculus.

Summary by alphaXiv.