Rapid and Precise Topological Comparison with Merge Tree Neural Networks

TL;DR

The Merge Tree Neural Network (MTNN) is introduced, a learned neural network model designed for merge tree comparison that enables rapid and high-quality similarity computation and demonstrates the approach's superiority in accuracy and efficiency.

Abstract

Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than $100\times$ on the benchmark datasets while maintaining an error rate below $0.1\%$.

Figures (13)

• Figure 1: An illustration of merge tree. On the left, a scalar function is represented by an orange line, highlighting the critical points, and showcasing how these points merge and connect topologically. On the right, the corresponding merge tree of sub-level set filtration.
• Figure 2: Edit distance on merge trees example. The transformation from merge tree 1 to merge tree 2 needs two nodes to relabel operations, where the relabeled node is highlighted in the red circle.
• Figure 3: Example of the GNNs process for a single node. Left: a graph with node features. Right: message function and update function for the target node 3. This node receives information from its directly connected neighbors, nodes 1, 2, and 4. Notably, node 4 further communicates a message from its adjacent node 5. In the end, the feature of node 3 is updated by aggregating all incoming messages.
• Figure 4: An illustration of two merge trees and their graph structure. From left to right: critical points (dots) in two scalar functions, showcasing how these points merge and connect topologically. In the middle are corresponding merge trees, highlighting structural differences. Finally, the graph structure of the two merge trees is identical but differs in function value labels.
• Figure 5: The architecture of our merge tree neural network (MTNN) operating on a pair of merge trees ($MT_{1,2}$). First, adjacency matrices for each merge tree with node feature are weighted by function value and are fed into a GIN. This produces node embeddings for both. Next, our topological attention produces tree embeddings from the node embeddings. A persistence-weighted adjacency matrix is an additional input for this step. The node embeddings are then compared, and tree embeddings are fed into a neural tensor network. These outputs are concatenated to form a joint embedding. This joint embedding is fed into a multi-layered perceptron (MLP) to produce a similarity score. This score is compared to the ground truth distance (normalized) in the training loss function.