Research map / Learning theory and algorithms / Optimal transport
Wasserstein convergence of Cech persistence diagrams for samplings of submanifolds
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Shows that Cech persistence diagrams of samples on an m-dimensional submanifold converge in OT_p exactly when p>m, improves the bottleneck stability bound to quadratic order, and proves laws of large numbers for alpha-total persistence.
Problem
Stability and convergence of Cech persistence diagrams with respect to Wasserstein distances OT_p, and boundedness of alpha-total persistence, for samples drawn from submanifolds of R^d.
Approach
Uses the manifold hypothesis: the intrinsic dimension m replaces the ambient dimension d in total persistence bounds; combines the reach-based distance comparison (quadratic improvement) with genericity (Morse-type structure of the distance function) and random-sampling results to get OT_p convergence iff p>m.
Key results
- Improved bottleneck stability: points of dgm_i(A) in region (2)/(3) are matched within O(eps^2), versus O(eps) in the classical theorem.
- For i.i.d. samples with density bounded away from 0 and infinity, E[OT_p^p(dgm_i(A_n), dgm_i(M))] -> 0 for p>m, with alpha-total persistence expansion Pers_alpha = Pers_alpha(dgm_i(M)) + C_i n^{1-alpha/m} + o(n^{1-alpha/m}) + O((log n / n)^{1/m}).
- For generic submanifolds and delta-dense samples, OT_p^p(dgm_i(A), dgm_i(M)) <= C_2 eps^{p-m} and Pers_alpha stays bounded as eps -> 0 for alpha >= m.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an m-dimensional manifold hypothesis for the sampled data | Cech persistence diagrams of manifold samplings · Submanifold samplings | OT_p convergence: p>m | Helped |
| Sample n i.i.d. points with density bounded away from 0 and infinity | Cech persistence diagrams of random submanifold samples · Random submanifold samplings | Expected OT_p^p: E[OT_p^p(μ_{n,i}, μ_{f,i})] → 0 | Helped |
| Require p>m for random samples when i<m | Cech persistence diagrams of random submanifold samples · Random submanifold samplings | Expected OT_p^p: n^{1-p/m} | Hurt |
| Use a generic submanifold and a (δ, ε)-dense point cloud with ε/δ bounded | Cech persistence diagrams of point-cloud approximations · Generic submanifold samplings | OT_p^p: C_2 ε^{p-m} | Helped |
| Use a generic submanifold and a (δ, ε)-dense point cloud with ε/δ bounded | Cech persistence diagrams of point-cloud approximations · Generic submanifold samplings | p-total persistence: Pers_p(dgm_i(A))≤C_3(C_4^p+ε^{p-m}) | Helped |
| Assume the underlying compact submanifold is generic | Cech persistence diagrams of dense point-cloud approximations · Generic submanifold samplings | Number of points in region (3): C_1 | Helped |
| Exploit positive reach of the underlying submanifold | Bottleneck matching between Cech persistence diagrams of A and M · Hausdorff approximation of a submanifold | Bottleneck matching coordinate error: Cε^2 | Helped |
Surprising findings
- There exist C^2 curves whose Cech PD has infinite alpha-total persistence for small alpha, so total persistence cannot be controlled without extra assumptions.
- The number of small-persistence points in a dense sample grows as O(n) (or O(1/delta^m)), so the first-region contribution to OT_p diverges in the random case unless p>m.
Limitations
- Results for the large-persistence regions require genericity assumptions on the submanifold.
- Experiments are used only to illustrate the theory; the paper is primarily theoretical.
Experimental setup
Datasets: synthetic point clouds sampled on submanifolds
Metrics: OT_p distance alpha-total persistence bottleneck distance
Baselines: standard bottleneck stability theorem Hiraoka-Shirai-Trinh limit laws
Builds on: bottleneck stability theorem Goel-Trinh-Tsunoda limit laws Divol-Polonik total persistence generic distance function genericity theorem (Division-Thomas style Morse theory
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