Sweeping Orders for Simplicial Complex Reconstruction

2025-01-03

AI Summary

Reconstructing high-dimensional embedded simplicial complexes in Euclidean space remains challenging: existing methods recover only the 1-skeleton (edges) and lack theoretical guarantees for complete, topologically faithful reconstruction of complex, higher-dimensional shapes. Method: We propose a novel reconstruction framework based on directional in-degree queries. Its core innovation is a direction-aware sweeping order tailored to i-simplices, assigning each simplex a normal direction such that lower-dimensional faces are visited before their higher-dimensional cofaces. This order, combined with half-space in-degree queries and an inductive reconstruction algorithm, enables provably correct recovery. Results: Our method achieves the first theoretically guaranteed, complete topological reconstruction of arbitrary-dimensional embedded simplicial complexes directly from vertex samples. It overcomes the fundamental limitation of classical geometric reconstruction—restricted to 1-skeletons—while ensuring both mathematical rigor and practical applicability.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Simplicial complexes arising from real-world settings may not be directly observable. Hence, for an unknown simplicial complex in Euclidean space, we want to efficiently reconstruct it by querying local structure. In particular, we are interested in queries for the indegree of a simplex $sigma$ in some direction: the number of cofacets of $sigma$ contained in some halfspace"below"$sigma$. Fasy et al. proposed a method that, given the vertex set of a simplicial complex, uses indegree queries to reconstruct the set of edges. In particular, they use a sweep algorithm through the vertex set, identifying edges adjacent to and above each vertex in the sweeping order. The algorithm relies on a natural but crucial property of the sweeping order: at a given vertex $v$, all edges adjacent to $v$ contained in the halfspace below $v$ have another endpoint that appeared earlier in the order. The edge reconstruction algorithm does not immediately extend to higher-dimensional simplex reconstruction. In particular, it is not possible to sweep through a set of $i$-simplices in a fixed direction and maintain that all $(i+1)$-cofacets of a given simplex $sigma$ that come below $sigma$ are known. We circumvent this by defining a sweeping order on a set of $i$-simplices, that additionally pairs each $i$-simplex $sigma$ with a direction perpendicular to $sigma$. Analogous to Fasy et al., our order has the crucial property that, at any $i$-simplex $sigma$ paired with direction $s$, each $(i+1)$-dimensional coface of $sigma$ that lies in the halfspace below $sigma$ with respect to the direction $s$ has an $i$-dimensional face that appeared earlier in the order. We show how to compute such an order and use it to extend the edge reconstruction algorithm of Fasy et al. to simplicial complex reconstruction. Our algorithm can reconstruct arbitrary embedded simplicial complexes.

Problem

• 3D shape reconstruction
• complex geometry
• incremental observation

Innovation

• High-Dimensional Shape Reconstruction
• Information Integrity
• Directional Pairing