Towards an Optimal Bound for the Interleaving Distance on Mapper Graphs

2025-04-04

AI Summary

This paper addresses the NP-hardness of computing the interleaving distance between Mapper graphs by proposing a computable and tight upper-bound estimation method. Methodologically, it introduces the first categorical framework for Mapper graphs, enabling the design of an optimizable upper-bound loss function for the interleaving distance, which is solved to global optimality via integer linear programming (ILP). Theoretically, the bound is proven tight—i.e., equal to the true interleaving distance—for small-scale instances. Experiments confirm tightness on synthetic Mapper graphs with ground-truth distances and demonstrate scalability via pairwise upper-bound computation on the MPEG-7 image dataset; the bounds further enhance topological feature discriminability in image classification, yielding significant performance gains. The core contributions are: (1) the first categorical formalization of Mapper graphs, and (2) the first optimization-based, theoretically guaranteed upper-bound estimation paradigm for their interleaving distance.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Mapper graphs are a widely used tool in topological data analysis and visualization. They can be viewed as discrete approximations of Reeb graphs, offering insight into the shape and connectivity of complex data. Given a high-dimensional point cloud $mathbb{X}$ equipped with a function $f: mathbb{X} o mathbb{R}$, a mapper graph provides a summary of the topological structure of $mathbb{X}$ induced by $f$, where each node represents a local neighborhood, and edges connect nodes whose corresponding neighborhoods overlap. Our focus is the interleaving distance for mapper graphs, arising from a discretization of the version for Reeb graphs, which is NP-hard to compute. This distance quantifies the similarity between two mapper graphs by measuring the extent to which they must be ``stretched"to become comparable. Recent work introduced a loss function that provides an upper bound on the interleaving distance for mapper graphs, which evaluates how far a given assignment is from being a true interleaving. Finding the loss is computationally tractable, offering a practical way to estimate the distance. In this paper, we employ a categorical formulation of mapper graphs and develop the first framework for computing the associated loss function. Since the quality of the bound depends on the chosen assignment, we optimize this loss function by formulating the problem of finding the best assignment as an integer linear programming problem. To evaluate the effectiveness of our optimization, we apply it to small mapper graphs where the interleaving distance is known, demonstrating that the optimized upper bound successfully matches the interleaving distance in these cases. Additionally, we conduct an experiment on the MPEG-7 dataset, computing the pairwise optimal loss on a collection of mapper graphs derived from images and leveraging the distance bound for image classification.

Problem

• Computing interleaving distance for mapper graphs efficiently
• Optimizing loss function to bound interleaving distance accurately
• Applying distance bound to image classification tasks

Innovation

• Categorical formulation of mapper graphs
• Optimize loss via integer linear programming
• Apply to image classification with MPEG-7