Statistical topological data analysis using persistence landscapes

By Peter Bubenik. Journal of machine learning research, 2012

1,037 citations recorded by the source corpus.

We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/14d0faa1e9e3e33c400176a33c657992ab332a88

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In Computational Theory and Mathematics

Citations in the corpus

Visualizing High-Dimensional Data: Advances in the Past Decade (2017, neutral): “Bubenik [139] introduces a visualization called the persistence landscape as an alternative to the persistence diagrams and barcodes used by both Carlsson and Ghrist.”
What are higher-order networks? (2021, background): “landscapes, or can be based on kernel methods, which can then be interpreted with tools from statistics and machine learning [3, 45, 58, 164].”
Topological Data Analysis (2016, methodology): “Next, they convert the persistence diagrams into a set of functions called landscapes as defined in Bubenik (2015). Turning the diagram into a set of one-dimensional functions makes it easier to use standard statistical tools.”
Persistence Images: A Stable Vector Representation of Persistent Homology (2015, methodology): “To resolve these issues, considerable eﬀort has been made to map PDs into spaces which are suitable for other ML tools (Bubenik, 2015; Reininghaus et al., 2015; Bendich et al., 2014; Adcock et al., 2013; Donatini et al., 1998; Ferri et al., 1997; Chung et al., 2009; Pachauri et al., 2011; Bendich…”
Beyond Deep Residual Learning for Image Restoration: Persistent Homology-Guided Manifold Simplification (2016, background): “However, for the intermediate size of the training data, there still exist gaps between the ERM and the risk minimization.”
Statistical Analysis of Persistence Intensity Functions (2015, methodology): “The second method is based on converting the diagram into a set of one dimensional functions called landscapes [1].”
Deep Learning with Topological Signatures (2017, neutral): “Along another line of research, Bubenik [6] proposes a mapping of barcodes into a Banach space.”
Topological Consistency via Kernel Estimation (2014, background): “ details and formal deﬁnitions. Another perspective of persistence homology is as a summary statistic of point cloud data that is robust to certain invariances, this perspective has been developed in [10, 14, 49, 69]. A useful way to describe persistent homology is via the notion of barcodes. A barc”
A roadmap for the computation of persistent homology (2015, methodology): “Such methods include persistence landscapes [16] (see also the tutorial [18]), the space of algebraic functions [3], and persistence images [28].”
A stable multi-scale kernel for topological machine learning (2014, methodology): “In [3], Bubenik introduced persistence landscapes, a representation of persistence diagrams as functions in the Banach space Lp(JR2).”
Wasserstein Stability for Persistence Diagrams (2020, neutral): “It is worth noting that [6] does have a limited version of Wasserstein stability using [17] .”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, methodology): “[Bubenik 2015] Let X and ˜ X be two compact sets.”
Persistent Convolution: A Topological Framework for AI Alignment Testing and Semantic Space Characterization (2026, unknown): “Definition 2.3 (Chazal et al., 2016) Given any 𝑓 ∶ 𝑋 → ℝ and 𝑎 ∈ ℝ , where 𝕍 (𝑓) (𝑎 ≤ 𝑏) is induced by inclusion (Bubenik et al., 2015).”