🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.001053s)
  1. Topological Data Analysis: Concepts, Computation, and Applications in Chemical Engineering (2021)

    Alexander D. Smith, Paweł Dłotko, Victor M. Zavala
    Abstract A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional descriptors. The key properties of these descriptors (also known as topological features) are that they provide multiscale information and that they are stable under perturbations (e.g., noise, translation, and rotation). In this work, we review the key mathematical concepts and methods of TDA and present different applications in chemical engineering.
  2. Topologically Densified Distributions (2020)

    Christoph Hofer, Florian Graf, Marc Niethammer, Roland Kwitt
    Abstract 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.
  3. Statistical Topological Data Analysis - A Kernel Perspective (2015)

    Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich Bauer
    Abstract We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in two-sample hypothesis testing on synthetic as well as real-world data.