[Paper Review] Statistical topological data analysis using persistence landscapes

Peter Bubenik|arXiv (Cornell University)|Jul 27, 2012

Topological and Geometric Data AnalysisComputer Science37 references524 citations

TL;DR

This paper introduces the persistence landscape, a vector-space-valued topological summary that enables statistical analysis of persistent homology by transforming barcodes into functions. It establishes strong statistical laws (LLN, CLT), enables hypothesis testing, and proves stability with lower bounds for bottleneck and Wasserstein distances, overcoming key limitations in applying topological data analysis to statistics and machine learning.

ABSTRACT

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.

Motivation & Objective

• Address the challenge of integrating topological data analysis (TDA) with statistics and machine learning by transforming non-vectorial persistence diagrams into a vector space.
• Overcome the statistical incompatibility of standard TDA summaries like barcodes and persistence diagrams, which lack vector space structure and hinder statistical inference.
• Enable statistical inference via convergence laws (LLN, CLT), hypothesis testing, and confidence intervals by embedding topological summaries in a separable Banach space.
• Provide theoretical guarantees for stability and lower bounds on Wasserstein and bottleneck distances using the persistence landscape.
• Facilitate efficient computation by representing topological summaries as piecewise-linear functions, enabling faster calculations than traditional barcode methods.

Proposed method

• Define the persistence landscape as a transformation of a barcode into a sequence of piecewise-linear functions, embedding it in a separable Banach space.
• Represent each persistence interval (b,d) as a triangular function λ₁(t) = (h - |t - m|)+, where h = (d-b)/2 and m = (b+d)/2.
• Construct the persistence landscape Λ(D) as a sequence of such functions, with ℓ_p-norms used to measure distances between landscapes.
• Apply the theory of random variables with values in Banach spaces to derive strong laws of large numbers and central limit theorems for the landscape.
• Prove stability by bounding the ℓ_p-norm of the landscape difference in terms of the p-Wasserstein distance between persistence diagrams.
• Derive lower bounds for bottleneck and Wasserstein distances using the landscape’s ℓ_p-norm, showing that small landscape differences imply small distances in the original space.

Experimental results

Research questions

• RQ1 Can a topological summary be constructed that lies in a vector space to enable standard statistical tools like hypothesis testing and confidence intervals?
• RQ2 Does the persistence landscape satisfy strong laws of large numbers and central limit theorems when treated as a random variable in a Banach space?
• RQ3 Can the persistence landscape provide stable, computable lower bounds for the bottleneck and Wasserstein distances between persistence diagrams?
• RQ4 How does the computational efficiency of the persistence landscape compare to standard barcode and persistence diagram representations?
• RQ5 To what extent can the persistence landscape be used to infer global topological features of data under sampling uncertainty?

Key findings

• The persistence landscape satisfies a strong law of large numbers and a central limit theorem when treated as a random variable in a separable Banach space.
• The landscape enables statistical inference: sample means converge to population means, and approximate confidence intervals can be computed.
• The ℓ_p-norm of the difference between two persistence landscapes provides a lower bound for the p-Wasserstein distance between their corresponding persistence diagrams.
• The persistence landscape is stable: the ℓ_p-norm of the landscape difference is bounded by a function of the persistence lengths and the p-Wasserstein distance between diagrams.
• For diagrams with bounded persistence, the landscape provides a lower bound on the p-Wasserstein distance proportional to the ℓ_p-norm of the landscape difference.
• The method allows efficient computation via piecewise-linear functions, significantly outperforming direct barcode or diagram calculations in terms of speed and compatibility with statistical algorithms.