Deep Learning with Topological Signatures
This work addresses the problem of making topological data analysis more accessible and effective for machine learning practitioners, particularly in domains like shape classification and social network analysis, though it is incremental in improving existing representation strategies.
The paper tackles the challenge of integrating topological signatures, which have an impractical structure for machine learning, into deep neural networks by proposing a novel input layer that learns task-optimal representations, resulting in outperforming state-of-the-art methods on social network graphs by a large margin.
Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information, typically in the form of summary representations of topological features. However, such topological signatures often come with an unusual structure (e.g., multisets of intervals) that is highly impractical for most machine learning techniques. While many strategies have been proposed to map these topological signatures into machine learning compatible representations, they suffer from being agnostic to the target learning task. In contrast, we propose a technique that enables us to input topological signatures to deep neural networks and learn a task-optimal representation during training. Our approach is realized as a novel input layer with favorable theoretical properties. Classification experiments on 2D object shapes and social network graphs demonstrate the versatility of the approach and, in case of the latter, we even outperform the state-of-the-art by a large margin.