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RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Tasks

Eduard Tulchinskii, Daria Voronkova, Ilya Trofimov, Evgeny Burnaev, Serguei Barannikov

TL;DR

RTD-Lite tackles the challenge of scalable topological comparison between two weighted graphs with a one-to-one vertex correspondence by avoiding full persistent homology calculations. It introduces a minimal spanning tree–based approach that constructs an auxiliary graph $C$ with edge weights $c_e = \min(a_e, b_e)$ and outputs a barcode-like RTD-Lite representation whose complexity is $O(n^2)$. The paper provides both a full RTD-Lite barcode algorithm and a faster summed-intervals variant, analyzes their computational properties, and demonstrates their utility in comparing neural representations and as a differentiable loss in learning tasks. Empirical results on synthetic and real data show substantial speedups over RTD while preserving topological fidelity, enabling applications in dimensionality reduction and neural-network training; the authors also release publicly available code at https://github.com/ArGintum/RTD-Lite.

Abstract

Topological methods for comparing weighted graphs are valuable in various learning tasks but often suffer from computational inefficiency on large datasets. We introduce RTD-Lite, a scalable algorithm that efficiently compares topological features, specifically connectivity or cluster structures at arbitrary scales, of two weighted graphs with one-to-one correspondence between vertices. Using minimal spanning trees in auxiliary graphs, RTD-Lite captures topological discrepancies with $O(n^2)$ time and memory complexity. This efficiency enables its application in tasks like dimensionality reduction and neural network training. Experiments on synthetic and real-world datasets demonstrate that RTD-Lite effectively identifies topological differences while significantly reducing computation time compared to existing methods. Moreover, integrating RTD-Lite into neural network training as a loss function component enhances the preservation of topological structures in learned representations. Our code is publicly available at https://github.com/ArGintum/RTD-Lite

RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Tasks

TL;DR

RTD-Lite tackles the challenge of scalable topological comparison between two weighted graphs with a one-to-one vertex correspondence by avoiding full persistent homology calculations. It introduces a minimal spanning tree–based approach that constructs an auxiliary graph with edge weights and outputs a barcode-like RTD-Lite representation whose complexity is . The paper provides both a full RTD-Lite barcode algorithm and a faster summed-intervals variant, analyzes their computational properties, and demonstrates their utility in comparing neural representations and as a differentiable loss in learning tasks. Empirical results on synthetic and real data show substantial speedups over RTD while preserving topological fidelity, enabling applications in dimensionality reduction and neural-network training; the authors also release publicly available code at https://github.com/ArGintum/RTD-Lite.

Abstract

Topological methods for comparing weighted graphs are valuable in various learning tasks but often suffer from computational inefficiency on large datasets. We introduce RTD-Lite, a scalable algorithm that efficiently compares topological features, specifically connectivity or cluster structures at arbitrary scales, of two weighted graphs with one-to-one correspondence between vertices. Using minimal spanning trees in auxiliary graphs, RTD-Lite captures topological discrepancies with time and memory complexity. This efficiency enables its application in tasks like dimensionality reduction and neural network training. Experiments on synthetic and real-world datasets demonstrate that RTD-Lite effectively identifies topological differences while significantly reducing computation time compared to existing methods. Moreover, integrating RTD-Lite into neural network training as a loss function component enhances the preservation of topological structures in learned representations. Our code is publicly available at https://github.com/ArGintum/RTD-Lite

Paper Structure

This paper contains 37 sections, 1 theorem, 5 equations, 13 figures, 3 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. DEFINITION AND TWO ALGORITHMS
  4. Direct algorithm
  5. Computational Complexity.
  6. Simplified computation
  7. Computational Complexity.
  8. Application to comparison of neural representations
  9. Gradient Optimization
  10. RTD-LITE COMPARES MULTI-SCALE CLUSTERS OF TWO GRAPHS
  11. EXPERIMENTS
  12. Comparing point clouds and neural representations
  13. Experiments with synthetic data
  14. Visualization of RTD-Lite-Barcodes
  15. Comparing representations from UMAP
  16. ...and 22 more sections

Key Result

Lemma H.1

Dimensionality of $\ker{(r_{0, \alpha})}$ equals to the number of intervals in RTD-Lite barcode that contains ⁠ $\alpha$.

Figures (13)

  • Figure 1: RTD-Lite along with RTD perfectly detect cluster structures, while rival measures fail. Five connected components (rings) are compared with 1-5 rings.
  • Figure 2: Cross-Barcodes for comparison of 1 cluster (A) vs. 3 clusters (B).
  • Figure 3: Comparing representations of MNIST by UMAP with varying n_neighbors. Left: representations with n_neighbors = 10, 50, 200. Right: comparison of representations by metrics.
  • Figure 4: Comparing layers from All-CNN networks trained on CIFAR10 with different seeds.
  • Figure 5: RTD-Lite scores between layers from GNNs trained with different seeds.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Lemma H.1
  • proof