🍩 Database of Original & Non-Theoretical Uses of Topology
(found 5 matches in 0.00135s)
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Graph Filtration Learning (2020)
Christoph Hofer, Florian Graf, Bastian Rieck, Marc Niethammer, Roland KwittAbstract
We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we leverage persistent homology computed via a real-valued, learnable, filter function. We establish the theoretical foundation for differentiating through the persistent homology computation. Empirically, we show that this type of readout operation compares favorably to previous techniques, especially when the graph connectivity structure is informative for the learning problem. -
Topologically Densified Distributions (2020)
Christoph Hofer, Florian Graf, Marc Niethammer, Roland KwittAbstract
We study regularization in the context of small sample-size learning with over-parametrized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal representations before a linear classifier. Specifically, we impose a topological constraint on samples drawn from the probability measure induced in that space. This provably leads to mass concentration effects around the representations of training instances, i.e., a property beneficial for generalization. By leveraging previous work to impose topological constrains in a neural network setting, we provide empirical evidence (across various vision benchmarks) to support our claim for better generalization. -
Deep Learning With Topological Signatures (2017)
Christoph Hofer, Roland Kwitt, Marc Niethammer, Andreas Uhl -
Constructing Shape Spaces From a Topological Perspective (2017)
Christoph Hofer, Roland Kwitt, Marc Niethammer, Yvonne Höller, Eugen Trinka, Andreas UhlAbstract
We consider the task of constructing (metric) shape space(s) from a topological perspective. In particular, we present a generic construction scheme and demonstrate how to apply this scheme when shape is interpreted as the differences that remain after factoring out translation, scaling and rotation. This is achieved by leveraging a recently proposed injective functional transform of 2D/3D (binary) objects, based on persistent homology. The resulting shape space is then equipped with a similarity measure that is (1) by design robust to noise and (2) fulfills all metric axioms. From a practical point of view, analyses of object shape can then be carried out directly on segmented objects obtained from some imaging modality without any preprocessing, such as alignment, smoothing, or landmark selection. We demonstrate the utility of the approach on the problem of distinguishing segmented hippocampi from normal controls vs. patients with Alzheimer’s disease in a challenging setup where volume changes are no longer discriminative.