Distributional Reduction: Unifying Dimensionality Reduction and
  Clustering with Gromov-Wasserstein Projection
Hugues Van Assel Cédric Vincent-Cuaz Nicolas Courty Rémi Flamary Pascal Frossard Titouan Vayer
Unsupervised learning aims to capture the underlying structure of potentially
large and high-dimensional datasets. Traditionally, this involves using
dimensionality reduction methods to project data onto interpretable spaces or
organizing points into meaningful clusters. In practice, these methods are used
sequentially, without guaranteeing that the clustering aligns well with the
conducted dimensionality reduction. In this work, we offer a fresh perspective:
that of distributions. Leveraging tools from optimal transport, particularly
the Gromov-Wasserstein distance, we unify clustering and dimensionality
reduction into a single framework called distributional reduction. This allows
us to jointly address clustering and dimensionality reduction with a single
optimization problem. Through comprehensive experiments, we highlight the
versatility and interpretability of our method and show that it outperforms
existing approaches across a variety of image and genomics datasets.