Source-linked AI summary
Stochastic Convergence of Persistence Landscapes and Silhouettes
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman
TL;DR
The paper addresses the difficulty of statistically analyzing distributions of persistence diagrams by studying functional summaries of those diagrams. It establishes statistical results for persistence landscapes, develops bootstrap confidence bands, and introduces the silhouette with analogous statistical theory.
Problem
Statistical analysis of distributions of persistence diagrams is difficult because persistence diagrams form a general metric space.
Method
The paper studies random landscapes from persistence diagrams, analyzes their average and bootstrap behavior, and introduces the silhouette as an alternate functional summary.
Results
The average persistence landscape converges weakly to a Gaussian process, bootstrap confidence bands are valid, and analogous statistical theory is derived for silhouettes.
Takeaways & Limitations
Persistence landscapes and silhouettes provide functional summaries for statistical inference on samples of persistence diagrams.
Takeaways & Limitations
For subsampling, the paper provides inference for the mean function but does not yet estimate the difference between the subsampling mean landscape and the original large-dataset landscape.
Abstract
from arXiv · showhide
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory.
1 Introduction
The paper addresses the difficulty of statistically analyzing distributions of persistence diagrams by encoding them as function-valued persistence landscapes. It studies landscape averages, bootstrap inference, and introduces the silhouette as an alternate functional summary.
- Statistical analysis is difficult because persistence diagrams form a general metric space.
- Persistence landscapes encode diagrams as real-valued one-Lipschitz functions that can be analyzed with nonparametric statistical techniques.
- The paper studies mean landscapes from random samples of compact sets or Morse functions.The target is the mean landscape µ = E(λi).
- For subsamples of a large dataset, the analysis estimates the mean landscape and focuses on variation among subsample landscapes.The paper does not address the difference between the subsample mean µ and the original landscape λ.
- The average landscape converges weakly to a Gaussian process, bootstrap confidence bands are valid, and the paper introduces the silhouette.
2 Persistence Diagrams and Landscapes
A persistence landscape converts a persistence diagram into ordered functions obtained from tent-shaped functions associated with its points. The resulting summary is one-Lipschitz and, in this paper, is primarily studied at k = 1.
- A finite persistence diagram is a finite set of intervals representing features by their birth and death values.
- Each persistence point is converted into a tent-shaped function Λp before the functions are overlaid.
- The landscape λD(k,t) is the kth largest value among the overlaid functions at t.If fewer than k values exist, λD(k,t) is set to 0.
- Each landscape component λD(k,·) is one-Lipschitz because the functions Λp are one-Lipschitz.
- The paper focuses on λ(t) = λD(1,t), although the results also hold for k > 1.
3 Weak Convergence of Landscapes
This section treats the sample average landscape as an estimator of the unknown mean landscape and analyzes its empirical-process behavior. The normalized process converges weakly to a Gaussian process, with a stated convergence rate under a positive-variance condition.
- Mean-landscape estimation: The sample average landscape estimates the unknown mean landscape µ and is pointwise unbiased because E(λ_n(t)) = µ(t).The paper extends earlier pointwise convergence and Central Limit Theorem results to uniform convergence of the average landscape.
- Empirical-process formulation: For each t ∈ [0,T], the evaluation function f_t maps a landscape λ to λ(t), so the average landscape can be represented as an empirical process indexed by these functions.The empirical measure P_n assigns mass 1/n to each observed landscape, while P denotes the underlying distribution.
- Weak convergence: The normalized empirical process G_n converges weakly to a Brownian bridge, a mean-zero Gaussian process characterized by its covariance function.Weak convergence is defined through convergence of expectations of bounded continuous functions, with the paper using outer expectation for technical reasons.
- Inference: The convergence analysis focuses on the maximum of G_n because this quantity is relevant for statistical inference.The limiting comparison concerns the maximum of the normalized empirical process and the maximum of its limiting distribution.
- Uniform central limit theorem: Under σ(t) > c > 0 on an interval [t*,t*] ⊂ [0,T], a uniform central limit theorem provides a convergence result for the process.The positivity assumption may be weakened to positivity over a finite collection of sub-intervals without changing the result.
4 The Bootstrap for Landscapes
The paper constructs multiplier-bootstrap confidence bands for the unknown mean persistence landscape, including both fixed-width and adaptive-width bands. The adaptive construction allows band width to vary with the estimated standard deviation while retaining asymptotic coverage under a positivity condition.
- Uniform confidence bands: The multiplier bootstrap constructs asymptotic (1−α) confidence bands for the unknown mean landscape µ(t).The procedure uses observed landscapes and simulated Gaussian multipliers to approximate the relevant critical value.
- Interpretation: Confidence bands quantify and visualize uncertainty about µ and can help screen out topological noise.The paper illustrates their construction through an explicit algorithm.
- Uniform confidence bands: The uniform band has constant width across t, even though estimator accuracy may vary over the domain.This motivates a more refined adaptive band.
- Adaptive confidence bands: The adaptive confidence band varies its width with the standard deviation function σ and its estimate.The standardized empirical process and multiplier bootstrap process determine the adaptive critical value.
5 The Weighted Silhouette
The paper introduces weighted silhouettes as an alternative function-valued summary of persistence diagrams. Power weighting controls whether the summary emphasizes low-persistence pairs or the most persistent pair, while preserving one-Lipschitz regularity and the landscape theory.
- Definition and construction: Weighted silhouettes are introduced as a new family of function-valued summaries for persistence diagrams.They are formed from weighted averages of the diagram’s triangle functions.
- Definition and construction: Power-weighted silhouettes assign weights wj=|dj−bj|^p, favoring pairs with greater persistence.The parameter p controls how strongly persistence differences affect the weighted average.
- Role of p: Small p emphasizes low-persistence pairs, whereas large p makes the silhouette dominated by the most persistent pair.The paper presents examples for different p values in Figure 3.
- Statistical theory: The power-weighted silhouette is one-Lipschitz for every non-negative weighting scheme.This regularity permits the results for landscapes to transfer to weighted silhouettes.
- Statistical theory: Applying the landscape theorems yields weak convergence and bootstrap confidence-band results for weighted silhouettes, subject to positive variance on an interval.The stated corollary identifies a Brownian-bridge limit for the empirical process.
6 Examples
The examples apply the landscape and silhouette confidence-band procedures to earthquake epicenters and a synthetic torus-and-circle dataset. They show adaptive bands alongside fixed-width bands and illustrate how alternative summaries expose persistent topological features.
- Earthquake data: For 8000 earthquake epicenters, the authors repeatedly sample m=400 points, compute Betti-1 persistence diagrams and landscapes, and repeat this n=30 times.The Vietoris–Rips filtration uses Euclidean distance.
- Earthquake data: Both uniform and adaptive 95% confidence bands have coverage around 95% for the mean earthquake-data landscape.The same 30 diagrams support corresponding bands for the mean weighted silhouette with p=0.01.
- Earthquake data: For most t∈[0,T], the adaptive confidence band is tighter than the fixed-width confidence band.This comparison is shown for both the mean landscape and mean weighted silhouette bands in the earthquake example.
- Toy example: rings: The synthetic example embeds a torus in R^3, links it to a radius-5 circle, samples N=11,800 points, and generates 30 subsample diagrams.Each subsample contains m=600 points and produces first and third landscapes plus silhouettes for p=0.1 and p=4.
- Toy example: rings: The third landscape and the silhouette with p=0.1 detect features corresponding to the torus’s two cycles and the circle’s cycle.These features are less visible in the first landscape because many persistence-diagram points are hidden by it.
7 Discussion
The paper establishes bootstrap confidence bands for persistence landscapes and silhouettes, while identifying extensions and an unresolved comparison involving subsampling.
- Bootstrap methods provide confidence bands for Bubenik’s persistence landscape and the paper’s persistence silhouette.
- Planned extensions include countably infinite persistence diagrams, unbounded T, and additional functional summaries.
- For subsampling, the paper provides accurate inference for the mean function μ but leaves estimating the difference between μ and the original-data landscape λ for future work.
A Results from Chernozhukov et al. (2013)
This appendix collects empirical-process and Gaussian-process results used in the paper’s statistical analysis, including covering-number conditions, concentration, anti-concentration, and multiplier-process constructions.
- The appendix defines covering numbers, VC-type function classes, envelopes, variance bounds, and empirical-process suprema.
- The appendix defines Gaussian multiplier variables independent of the sample and the supremum of the resulting multiplier process.
- Gaussian anti-concentration inequalities control suprema of separable Gaussian processes and tight Gaussian random elements under variance conditions.
- The stated results provide concentration bounds for empirical-process suprema under uniform boundedness and covering-number assumptions.
B Technical Tools
The technical development constructs Gaussian and bootstrap processes for persistence landscapes, establishes function-class complexity bounds, and derives uniform convergence ingredients under positive variance assumptions.
- The analysis uses Brownian bridges and standardized Gaussian processes to represent landscape fluctuations over an interval.
- Under σ(t) > c > 0 on [t*, t*], Proposition 13 establishes supremum convergence for the relevant Gaussian process.
- The technical lemmas control bootstrap-process approximation, estimated standard deviations, quantiles, and associated error probabilities.
- The standardized class Gc is shown to be VC type with constants a = (T^2 + 2c^2)/c^2 and v = 1 and envelope T/(2c).
- Lipschitzness of landscape evaluations in t supports grid constructions and covering-number bounds for the function classes.
C Main Proofs
The main proofs combine Lipschitz-based entropy bounds with empirical-process concentration and Gaussian anti-concentration to establish uniform and adaptive confidence-band results.
- The landscape class admits grid-based ε-nets because |ft(λ) − fu(λ)| ≤ |t − u|, yielding covering-number bounds.
- Empirical-process inequalities are applied with envelope and variance parameters to control the relevant suprema for large n.
- Gaussian anti-concentration converts process approximation bounds into probability bounds for supremum events.
- Theorem 3’s uniform-band proof follows from Theorem 2 and Proposition 13, while the adaptive-band proof additionally uses quantile-control results.