Learning Hyperbolic Representations of Topological Features
This addresses a limitation in topological data analysis for machine learning applications by enabling more expressive representations that handle essential features better.
The paper tackles the problem of learning representations for persistence diagrams in topological data analysis by proposing hyperbolic representations on the Poincare ball to preserve the importance of infinite persistence features, achieving performance on par with or exceeding state-of-the-art methods on graph and image classification tasks.
Learning task-specific representations of persistence diagrams is an important problem in topological data analysis and machine learning. However, current state of the art methods are restricted in terms of their expressivity as they are focused on Euclidean representations. Persistence diagrams often contain features of infinite persistence (i.e., essential features) and Euclidean spaces shrink their importance relative to non-essential features because they cannot assign infinite distance to finite points. To deal with this issue, we propose a method to learn representations of persistence diagrams on hyperbolic spaces, more specifically on the Poincare ball. By representing features of infinite persistence infinitesimally close to the boundary of the ball, their distance to non-essential features approaches infinity, thereby their relative importance is preserved. This is achieved without utilizing extremely high values for the learnable parameters, thus the representation can be fed into downstream optimization methods and trained efficiently in an end-to-end fashion. We present experimental results on graph and image classification tasks and show that the performance of our method is on par with or exceeds the performance of other state of the art methods.