GeomTop Seminar: Applying Persistent Homology in Machine Learning

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We present some recent applications of persistent homology
in machine learning. First, we introduce a metric shape space
based on a topological representation of 2D/3D objects.
The metric allows to use classic metric-based machine learning
algorithms, e.g., k-nearest neighbors or k-means clustering.

Next, we establish a relation between end-to-end learnable
deep neural networks and persistence barcodes.
The key contribution here is the construction of parametrized
vectorization schemes which respect the stability properties of
persistent homology computation. These vectorization schemes
can be implemented as a learnable input layer for neural
networks, yielding an approach for supervised end-to-end learning
in the regime of persistence barcodes.

Finally, we leverage that Vietoris-Rips persistent homology is locally differentiable and apply this insight to impose topological constraints on the latent representations learned by an autoencoder.
These representations show beneficial properties
for kernel density based one-class learning.

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