topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
In topological machine learning one tries to combine methods from both topological data analysis and machine learning for application to data analysis.
A popular approach (originating around RHBK15, KHNL15) is to use persistent homology as a pre-processing stage and train neural networks not on full data sets, but on their persistence diagram/barcode (which may still be a large amount of data), by making these amenable to kernel methods.
On feeding persistence diagrams into machine learning-algorithms by equipping them with kernels:
Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt, A Stable Multi-Scale Kernel for Topological Machine Learning, NIPS’15: Proceedings of the 28th International Conference on Neural Information Processing System, 2 (2015 3070–3078 $[$arXiv:1412.6821$]$
Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich Bauer, Statistical Topological Data Analysis – A Kernel Perspective, in: Advances in Neural Information Processing Systems (NIPS 2015) $[$ISBN:9781510825024, doi:10.5555/2969442.2969582$]$
Bastian Rieck, Filip Sadlo, Heike Leitte, Topological Machine Learning with Persistence Indicator Functions, In: Topological Methods in Data Analysis and Visualization V TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$arXiv:1907.13496, doi:10.1007/978-3-030-43036-8_6$]$
Raphael Reinauer, Matteo Caorsi, Nicolas Berkouk, Persformer: A Transformer Architecture for Topological Machine Learning $[$arXiv:2112.15210$]$
Review:
See also:
Created on June 13, 2022 at 11:32:07. See the history of this page for a list of all contributions to it.