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Workshop: Geometrical and Topological Representation Learning

CubeRep: Learning Relations Between Different Views of Data

Rishi Sonthalia · Anna Gilbert · Matthew Durham

Keywords: [ Markov chain Monte Carlo ] [ graph learning ] [ Multiview learning ] [ hierarchically hyperbolic spaces ]

Abstract: Multi-view learning tasks typically seek an aggregate synthesis of multiple views or perspectives of a single data set. The current approach assumes that there is an ambient space $X$ in which the views are images of $X$ under certain functions and attempts to learn these functions via a neural network. Unfortunately, such an approach neglects to consider the geometry of the ambient space. Hierarchically hyperbolic spaces (HHSes) do, however, provide a natural multi-view arrangement of data; they provide geometric tools for the assembly of different views of a single data set into a coherent global space, a \emph{CAT(0) cube complex}. In this work, we provide the first step toward theoretically justifiable methods for learning embeddings of multi-view data sets into CAT(0) cube complexes. We present an algorithm which, given a finite set of finite metric spaces (views) on a finite set of points (the objects), produces the key components of an HHS structure. From this structure, we can produce a \emph{CAT(0) cube complex} that encodes the hyperbolic geometry in the data while simultaneously allowing for Euclidean features given by the detected relations among the views.

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