Persistent magnitude

Govc, Dejan; Hepworth, Richard

doi:10.1016/j.jpaa.2020.106517

“…Therefore c k,n takes the form prescribed in the theorem by Proposition 2.8. We finish the proof by showing that β rk(γ)−(n+1)/2,n+1 does not contribute in Equation ( 13) and that there is no logarithmic term when expanding the ξ-derivatives in Equation (13)…”

Section: ≥3mentioning

confidence: 73%

“…We presume no prerequisites from the reader on magnitude in this paper, but for the convenience of the reader we summarize the implications to magnitude here and expand on this relation in the follow up paper [12]. From its category-theoretic origin, magnitude has found unexpected applications from algebraic topology [13,27,40] and applied category theory [8,35] to data science [6,7] and mathematical biology [24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Semiclassical analysis of a nonlocal boundary value problem related to magnitude

Gimperlein¹,

Goffeng²,

Louca³

2022

Preprint

0

View full text Add to dashboard Cite

We study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in short-time expansions for heat kernels.

show abstract

“…A related line of work studies what can be proven about the topology of random point clouds, typically as the number of points in the point cloud goes to infinity (Kahle, 2011;Adler et al, 2014;Bobrowski and Kahle, 2014;Bobrowski et al, 2017). The magnitude (Leinster, 2013) and magnitude homology (Hepworth and Willerton, 2017;Leinster and Shulman, 2017) of a metric space measure both local and global properties; recent and ongoing work is being done to connect magnitude with persistent homology (Otter, 2018;Govc and Hepworth, 2021). See also Weinberger (2019) for connections between sublevel set persistent homology and the geometry of spaces of functions, including Lipschitz constants of functions.…”

Section: Theory Of How Persistent Homology Measures Local Geometrymentioning

confidence: 99%

Topology Applied to Machine Learning: From Global to Local

Adams

¹

,

Moy

²

2021

Front. Artif. Intell.

View full text Add to dashboard Cite

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.

show abstract

“…A related line of work studies what can be proven about the topology of random point clouds, typically as the number of points in the point cloud goes to infinity [Adler et al, 2014, Bobrowski and Kahle, 2014, Bobrowski et al, 2015, Kahle, 2011. The magnitude [Leinster, 2013] and magnitude homology Willerton, 2017, Leinster andShulman, 2017] of a metric space measure both local and global properties; recent and ongoing work is being done to connect magnitude with persistent homology [Otter, 2018, Govc andHepworth, 2021]. See also Weinberger [2019] for connections between sublevel set persistent homology and the geometry of spaces of functions, including Lipschitz constants of functions.…”

Section: Theory Of How Persistent Homology Measures Local Geometrymentioning

confidence: 99%

Topology Applied to Machine Learning: From Global to Local

Adams¹,

Moy²

2021

Preprint

0

View full text Add to dashboard Cite

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.

show abstract

Persistent magnitude

Cited by 16 publications

References 26 publications

Semiclassical analysis of a nonlocal boundary value problem related to magnitude

Semiclassical analysis of a nonlocal boundary value problem related to magnitude

Topology Applied to Machine Learning: From Global to Local

Topology Applied to Machine Learning: From Global to Local

Contact Info

Product

Resources

About