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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