🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.001115s)

  1. Bayesian Computation Meets Topology (2024)

    Julius von Rohrscheidt, Bastian Rieck, Sebastian M. Schmon
    Abstract Computational topology recently started to emerge as a novel paradigm for characterising the ‘shape’ of high-dimensional data, leading to powerful algorithms in (un)supervised representation learning. While capable of capturing prominent features at multiple scales, topological methods cannot readily be used for Bayesian inference. We develop a novel approach that bridges this gap, making it possible to perform parameter estimation in a Bayesian framework, using topology-based loss functions. Our method affords easy integration into topological machine learning algorithms. We demonstrate its efficacy for parameter estimation in different simulation settings.

  2. Topological Singularity Detection at Multiple Scales (2023)

    Julius von Rohrscheidt, Bastian Rieck
    Abstract The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.