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 [79] 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 [79]. First, we apply MT-WAE to merge tree
compression, 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.Comment: arXiv admin note: text overlap with arXiv:2207.1096