Embeddings of persistence diagrams into Hilbert spaces

Bubenik, Peter; Wagner, Ana Paula

doi:10.1007/s41468-020-00056-w

Cited by 23 publications

(10 citation statements)

References 16 publications

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“…The space of persistence barcodes is not immediately appropriate for machine learning. Indeed, averages of barcodes need not be unique (Mileyko et al, 2011), and the space of persistence barcodes does not coarsely embed into any Hilbert space (Bubenik and Wagner, 2020). These limitations have initiated a large amount of research on transforming persistence barcodes into more natural formats for machine learning.…”

Section: Machine Learningmentioning

confidence: 99%

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Topology Applied to Machine Learning: From Global to Local

Adams

Moy

2021

Front. Artif. Intell.

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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.

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Section: Machine Learningmentioning

confidence: 99%

“…In practice, a different metric is sometimes chosen to map landscapes into a Hilbert space, though the restrictions ofBubenik and Wagner (2020) apply.…”

mentioning

confidence: 99%

Topology Applied to Machine Learning: From Global to Local

Adams

Moy

2021

Front. Artif. Intell.

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“…The theory of vectorization of persistence diagrams is an active area of research, with recent results showing the impossibility of full embeddability. Indeed, even though the space of persistence diagrams with exactly n points can be coarsely embedded in a Hilbert space ( Mitra and Virk, 2021 ), this ceases to be true if the number of points is allowed do vary ( Wagner, 2019 ; Bubenik and Wagner, 2020 ). That said, partial featurization is still useful as we will demonstrate here.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study of Machine Learning Methods for Persistence Diagrams

Barnes

Polanco

Perea

2021

Front. Artif. Intell.

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Many and varied methods currently exist for featurization, which is the process of mapping persistence diagrams to Euclidean space, with the goal of maximally preserving structure. However, and to our knowledge, there are presently no methodical comparisons of existing approaches, nor a standardized collection of test data sets. This paper provides a comparative study of several such methods. In particular, we review, evaluate, and compare the stable multi-scale kernel, persistence landscapes, persistence images, the ring of algebraic functions, template functions, and adaptive template systems. Using these approaches for feature extraction, we apply and compare popular machine learning methods on five data sets: MNIST, Shape retrieval of non-rigid 3D Human Models (SHREC14), extracts from the Protein Classification Benchmark Collection (Protein), MPEG7 shape matching, and HAM10000 skin lesion data set. These data sets are commonly used in the above methods for featurization, and we use them to evaluate predictive utility in real-world applications.

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“…Turner et al showed that the space of Euclidean persistence diagrams with the 2-Wasserstein metric is an Alexandrov space with curvature bounded below [40]. More recently, Bubenik and Wagner [9] studied the roundness and asymptotic dimension of the space of Euclidean persistence diagrams with the bottleneck distance. Our understanding of such spaces is, however, still fragmentary.…”

Section: Introductionmentioning

confidence: 99%

Metric Geometry of Spaces of Persistence Diagrams

Che¹,

Galaz-García²,

Guijarro³

et al. 2021

Preprint

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Persistence diagrams are geometric objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors Dp, 1 ≤ p ≤ ∞, that assign, to each metric pair (X, A), a pointed metric space Dp(X, A). Moreover, we show that D∞ is continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that Dp preserves several useful metric properties, such as completeness and separability, for p ∈ [1, ∞], and geodesicity and non-negative curvature in the sense of Alexandrov, for p = 2. As an application of our framework, we prove that the space of Euclidean persistence diagrams, D2(R 2n , ∆n), has infinite covering, Hausdorff, and asymptotic dimensions.

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Embeddings of persistence diagrams into Hilbert spaces

Cited by 23 publications

References 16 publications

Topology Applied to Machine Learning: From Global to Local

Topology Applied to Machine Learning: From Global to Local

A Comparative Study of Machine Learning Methods for Persistence Diagrams

Metric Geometry of Spaces of Persistence Diagrams

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