[Paper Review] Statistical Analysis and Parameter Selection for Mapper

Mathieu Carrière, Bertrand Michel+1 more|arXiv (Cornell University)|Jun 1, 2017

Topological and Geometric Data Analysis27 references59 citations

TL;DR

The Mapper is shown to converge to the Reeb graph and is an optimal estimator, enabling automatic parameter tuning and confidence regions for topological features.

ABSTRACT

In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a method to automatically tune its parameters and compute confidence regions on its topological features, such as its loops and flares. This allows to circumvent the issue of testing a large grid of parameters and keeping the most stable ones in the brute-force setting, which is widely used in visualization, clustering and feature selection with the Mapper.

Motivation & Objective

• Motivate the use of Mapper as a topological data analysis tool for unsupervised learning and visualization.
• Establish a statistical convergence framework relating Mapper to the continuous Reeb graph.
• Derive parameter selection guidance and rates of convergence based on filter regularity and data standardness.
• Provide methods to construct confidence regions for Mapper topological features such as loops and flares.

Proposed method

• Define Mapper with a specific 1-skeleton (Rips) and a regular cover of the filter range, using intervals with fixed length r and fixed overlap g.
• Use extended persistence diagrams and the persistence metric dΔ to compare Mapper outputs with Reeb graphs.
• Prove an approximation inequality bounding dΔ(R_f(X), M_n) by r + 2ω(δ), under reach/convexity assumptions and modulus of continuity ω of the filter f.
• Derive minimax convergence rates for Mapper as n grows, depending on standardness parameters (a, b) and the modulus of continuity of f.
• Describe two settings: exact filter with known generative model and inferred filter (estimators) with corresponding risk bounds.
• Present corollaries and subsampling strategies to handle unknown generative parameters and provide stability results.

Experimental results

Research questions

• RQ1 Does Mapper consistently approximate the Reeb graph of a space under Morse-type filters?
• RQ2 What are the rates of convergence for Mapper to the Reeb graph as the sample size grows, given regularity of the filter and data distribution?
• RQ3 How should one select Mapper parameters (r, g, δ) to optimize estimation error and avoid artifacts?
• RQ4 How can one compute confidence regions for topological features in Mapper (loops, flares) via extended persistence?
• RQ5 How do exact-filter and inferred-filter settings compare in terms of estimator risk and practical parameter tuning?

Key findings

• Mapper with a specific parameter choice satisfies dΔ(R_f(X), M_n) ≤ r + 2ω(δ), providing a concrete approximation bound.
• The convergence rate of Mapper to the Reeb graph scales with the modulus of continuity ω of the filter and the data dimension parameter b, yielding minimax optimality up to log factors.
• Under standardness assumptions (a, b) and Lipschitz or concave modulus of continuity, Mapper achieves convergence rates comparable to known rates for related set estimation problems.
• Corollaries show robustness to filter estimation error, with bounds incorporating the estimator deviation ω(δ) and sample-induced discrepancies.
• Subsampling strategies enable parameter tuning when the true generative model parameters are unknown, preserving convergence guarantees.