Deep Learning with Topological Signatures

By Christoph Hofer, Roland Kwitt, Marc Niethammer, Andreas Uhl. Neural Information Processing Systems, 2017

315 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/9c9ba700f9871a41cfc007658c9ff24f84dc4dcb

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In Computational Theory and Mathematics

Citations in the corpus

Wasserstein Stability for Persistence Diagrams (2020, unknown): “This includes persistent scale space, C T = 1 [36, 31], (4) Betti curves [37, 40], (5) learned/optimized representations [28, 29], (6) persistent homology rank function [38], C T = 1 (Corollary 6.6).”
Topology-Preserving Deep Image Segmentation (2019, unknown): “[21] proposed a CNN-based topological classifier.”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “To overcome this issue, Hofer et al. (2017) proposes a deep learning approach that allows to learn the relevant topological features for a given task.”