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Statistical topological data analysis using persistence landscapes
Peter Bubenik
TL;DR
Persistence diagrams and barcodes are difficult to combine with statistics and machine learning, motivating a function-valued alternative. The paper defines persistence landscapes, develops their Banach-space statistical theory, and proves convergence, stability, and distance-bound results. It also notes that averaging landscapes can produce summaries without corresponding barcodes or diagrams.
Problem
Barcode and persistence-diagram summaries are difficult to combine with statistics and machine learning, limiting statistical operations on topological data.
Method
The paper converts barcodes into persistence landscapes, treats them as random variables in a separable Banach space, and applies functional statistical theory.
Results
Persistence landscapes satisfy a strong law of large numbers and a central limit theorem, support statistical tests, and yield stability and bottleneck- and Wasserstein-distance lower bounds.
Takeaways & Limitations
The function-space representation makes persistence landscapes compatible with statistical and machine-learning tools and provides a stable basis for inference.
Takeaways & Limitations
A mean persistence landscape does not necessarily have a corresponding barcode or persistence diagram.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper introduces persistence landscapes as function-valued topological summaries that make statistical and machine-learning operations more accessible. It develops their statistical treatment, computational use, stability, and distance bounds.
- Motivation: Persistence landscapes convert barcodes into functions in a separable Banach space, enabling vector-space statistical and machine-learning tools.They are sequences of piecewise-linear functions, making calculations faster than with barcodes or persistence diagrams.
- Applications: The paper applies standard statistical inference tools to persistence landscapes and develops an efficient functional route to real-valued central-limit-theorem calculations.The supplied introduction identifies statistical tests and computational efficiency as intended uses.
- Statistical perspective: The statistical framework treats topological summaries as measurable random variables and targets means, convergence, confidence intervals, hypothesis tests, and efficient distances.The paper formulates these goals for independent samples with a common distribution.
- Scope: The mean persistence landscape need not have a corresponding barcode or persistence diagram.The paper compares this phenomenon with a Poisson rate parameter that is not integer-valued.
- Stability: The persistence landscape is proved stable and shown to provide lower bounds for bottleneck and Wasserstein distances.These results are developed through the landscape distance.
2. Topological summaries
The paper situates persistence landscapes among persistence-module summaries, explains their construction from multiscale homological features, and contrasts their function-space geometry with barcodes and persistence diagrams.
- Persistence modules: A persistence module assigns vector spaces and compatible linear maps across ordered parameter values.The maps compose consistently and are identity maps when the parameter values coincide.
- Persistence modules: A growing union of metric balls produces a filtration whose homology groups and inclusion-induced maps form a persistence module.The construction applies to point sets in Euclidean or more general metric spaces.
- Persistence modules: Čech complexes provide a combinatorial model for unions of balls, while Rips complexes offer a larger but simpler alternative when Čech computation is expensive.In Euclidean space, the union of balls is homotopy equivalent to its Čech complex.
- Derived functions: The rank function records dimensions of images of persistence-module maps, with βa,b = dim(im(M(a ≤b))) and monotonicity under interval enlargement.These functions are derived directly from the module's linear maps.
- Persistence landscapes: The persistence landscape is a function λ: N × R → R, equivalently a sequence of functions λk: R → R.Its defining properties include nonnegativity, λk(t) ≥ λk+1(t), and 1-Lipschitz continuity.
- Barcodes and persistence diagrams: Barcodes completely encode tame persistence modules, whereas landscapes retain less endpoint-inclusion information than barcodes after passage to Lp spaces.The landscape remains a function-valued summary with a distinct geometric representation.
- Persistence landscapes: Persistence landscapes are related to rank functions, rescaled rank functions, barcodes, and persistence diagrams through explicit transformations and geometric encodings.In particular, λ(b,d) counts diagram points in an upper-left quadrant, while barcode intervals correspond to stacked triangles.
- Means: Persistence landscapes have unique means, unlike persistence-diagram sets that can have multiple Fréchet means.The contrast is illustrated by the paired examples in Figure 3.
3. Statistics with landscapes
Persistence landscapes place topological summaries in the Banach-space setting, enabling means, convergence results, confidence intervals, and hypothesis tests. The section also discusses functional choices and cases where the proposed tests may lack power.
- Statistical framework: Persistence landscapes are modeled as Borel random variables in Lp(S), a separable Banach space, with independent samples supporting vector-space averaging.The mean landscape averages landscape values pointwise across sampled barcodes.
- Convergence: The sample mean landscape converges almost surely to the expected landscape when the expected norm is finite.This applies the Banach-space strong law to independent landscape samples.
- Convergence: With finite first and second moments and p ≥2, the centered sample landscape satisfies a central limit theorem with Gaussian limiting covariance structure.The result is stated for √n[Λ_n − E(Λ)].
- Inference: Functionals on persistence landscapes yield real-valued random variables for approximate confidence intervals and standard two-sample hypothesis tests.The framework uses asymptotic normality for functional values and a two-sample z-test for comparing means.
- Functional choices and limitations: Functional selection can target dominant homological features, but the suggested tests may lack power for some groups with different persistence.A vector of functionals with Hotelling’s T^2 is proposed to increase power, while translated landscapes remain a failure case for that alternative.
4. Examples
The examples apply persistence landscapes to random geometric, clique-complex, Gaussian-field, and surface-sampling data, combining mean landscapes with confidence intervals and hypothesis tests. These analyses illustrate landscape-based summaries across homological dimensions and distinguish torus from sphere samples, including under noise.
- 4.1. Linked Annuli: 100 repetitions of two-annulus samples produce a mean degree-one persistence landscape with one likely large interval, one smaller interval born nearby, and shorter intervals.The annuli example uses 200 uniformly sampled points and Vietoris–Rips complexes.
- 4.2. Random geometric complexes: 1000 cube samples yield approximate 95% confidence intervals of [0.1534, 0.1545], [0.0064, 0.0066], and [0.0002, 0.0003] for E(Y) in degrees 0, 1, and 2.Each sample contains 100 uniformly sampled points, and Y is based on the landscape’s L1 norm.
- 4.3. Erdős–Rényi random clique complexes: 10 copies of G(100) produce approximate 95% confidence intervals of [0.0034, 0.0039], [0.751, 0.777], [1.971, 2.041], and [2.591, 2.618] in degrees 0–3.The filtered clique complex is restricted to filtration values at most 0.55 for computational reasons.
- 4.4. Gaussian random fields: Gaussian random fields are analyzed on [0, 1]2 and [0, 1]3 by calculating mean landscapes across samples in degrees 0–1 and 0–2, respectively.The two-dimensional field uses a 100 by 100 grid and 100 samples; the three-dimensional field uses a 25 × 25 × 25 grid.
- 4.5. Torus and Sphere: A two-sample z-test rejects equal means for torus and sphere landscapes in dimension 1 with p-value 3 × 10^-6, but not in dimensions 0 and 2.The test uses integrals of bounded-support persistence landscapes as real-valued random variables.
- 4.5. Torus and Sphere: With Gaussian noise and 10 samples per surface, permutation tests find significant mean-landscape differences in dimensions 0, 1, and 2, with p values 0.0111, 0.0000, and 0.0000.The dimension-0 difference reflects a slight shift despite visually similar mean landscapes, and the statistic detects a geometric rather than topological difference.
5. Landscape Distance and Stability
This section defines landscape distances between persistence landscapes and establishes their stability and distance-comparison properties. Under bounded total-persistence conditions, p-landscape distances support stability results and lower bounds for Wasserstein distances.
- Landscape distance: Landscape distance compares persistence diagrams through the norm of the difference between their persistence landscapes.The distance is defined for corresponding landscapes λ and λ′ across p-norms.
- Distance bounds: The landscape distance provides lower bounds for the p-Wasserstein distance.The section derives this result after comparing landscape distances with persistence-weighted Wasserstein distances.
- Stability: The persistence landscape is stable in the supremum norm without assumptions on the input functions.The stated ∞-stability result requires no assumptions, including no q-tame condition.
- Distance bounds: The ∞-landscape distance is bounded by the bottleneck distance.This supplies a direct comparison between the landscape summary and the standard bottleneck metric.
- Stability: The p-landscape distance is stable for p > k when the underlying space has bounded degree-k total persistence.The theorem assumes a triangulable compact metric space and tame Lipschitz functions, with p ≥ k in the stated bound and stability concluded for p > k.
Appendix A. Proofs
The appendix proves landscape stability and distance bounds by relating landscape norms to interleavings, diagram matchings, and persistence-weighted estimates. It also establishes the key ordering and single-feature inequalities used in these results.
- Auxiliary lemmas: Each landscape coordinate is 1-Lipschitz as a function of its parameter.The proof establishes |λ_k(t) − λ_k(s)| ≤ |t − s| for all s and t.
- Stability proofs: An interleaving of persistence modules bounds their ∞-landscape distance by the interleaving distance.The proof shows that an ε-interleaving yields a sup-norm landscape difference at most ε.
- Stability proofs: For functions f and g, the ∞-landscape distance is bounded by ∥f − g∥∞.This follows by combining the interleaving result with the stability theorem for persistence modules.
- Diagram bounds: For persistence diagrams D and D′, the ∞-landscape distance is bounded by their bottleneck distance.The proof uses interval-module representations and the interleaving bound.
- Auxiliary lemmas: The ordering lemma shows that matching sorted values minimizes the sum of powered absolute differences.This lemma supports the comparison of landscape coordinates after ordering values at each parameter.
- Diagram bounds: The appendix bounds the p-landscape distance for finite persistence diagrams using pointwise matching errors and persistence values.The single-point estimate combines persistence ℓ with matching displacement ε in the p-norm bound.