Merge trees are a valuable tool in 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 networks (MTNN), a
learned neural network model designed for merge tree comparison. The MTNN
enables rapid and high-quality similarity computation. We first demonstrate how
graph neural networks (GNNs), which emerged as an effective encoder for graphs,
can be trained to produce embeddings of merge trees in vector spaces that
enable 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 100x on the
benchmark datasets while maintaining an error rate below 0.1%.

通过使用图神经网络和拓扑注意机制，合并树神经网络 (MTNN) 可以快速高质量地计算合并树的相似性，该方法优于现有的合并树比较方法，并在不同领域的真实数据上展示了其强大性能。

快速而精确的拓扑比较与合并树神经网络

Rapid and Precise Topological Comparison with Merge Tree Neural Networks

This paper presents a computational framework for the Wasserstein
auto-encoding of merge trees (MT-WAE), a novel extension of the classical
auto-encoder neural network architecture to the Wasserstein metric space of
merge trees. In contrast to traditional auto-encoders which operate on
vectorized data, our formulation explicitly manipulates merge trees on their
associated metric space at each layer of the network, resulting in superior
accuracy and interpretability. Our novel neural network approach can be
interpreted as a non-linear generalization of previous linear attempts [65] at
merge tree encoding. It also trivially extends to persistence diagrams.
Extensive experiments on public ensembles demonstrate the efficiency of our
algorithms, with MT-WAE computations in the orders of minutes on average. We
show the utility of our contributions in two applications adapted from previous
work on merge tree encoding [65]. First, we apply MT-WAE to data reduction and
reliably compress merge trees by concisely representing them with their
coordinates in the final layer of our auto-encoder. Second, we document an
application to dimensionality reduction, by exploiting the latent space of our
auto-encoder, for the visual analysis of ensemble data. We illustrate the
versatility of our framework by introducing two penalty terms, to help preserve
in the latent space both the Wasserstein distances between merge trees, as well
as their clusters. In both applications, quantitative experiments assess the
relevance of our framework. Finally, we provide a C++ implementation that can
be used for reproducibility.

本文提出了一种基于合并树（MT-WAE）的 Wasserstein 自编码计算框架，该框架是传统自编码神经网络架构到 Wasserstein 度量空间的一种新扩展，在每个网络层上显式地处理相关度量空间上的合并树，从而提高了准确性和可解释性，并可用于数据压缩和降维的应用。

融合树（和持久图）的 Wasserstein 自编码器

Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)

Merge trees are a type of graph-based topological summary that tracks the
evolution of connected components in the sublevel sets of scalar functions.
They enjoy widespread applications in data analysis and scientific
visualization. In this paper, we consider the problem of comparing two merge
trees via the notion of interleaving distance in the metric space setting. We
investigate various theoretical properties of such a metric. In particular, we
show that the interleaving distance is intrinsic on the space of labeled merge
trees and provide an algorithm to construct metric 1-centers for collections of
labeled merge trees. We further prove that the intrinsic property of the
interleaving distance also holds for the space of unlabeled merge trees. Our
results are a first step toward performing statistics on graph-based
topological summaries.

本文主要研究了比较两种基于图形的拓扑摘要的问题，通过使用度量空间中的交错距离概念来解决该问题。我们进一步证明了交错距离在有标签的合并树空间中是内在的，也适用于无标签的合并树空间，这是开展图形拓扑摘要统计工作的第一步。