Can you do geometry with persistent homology?

Question

Setup

In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.

However most filtrations (Vietoris, Čech, ...) depend on the metric on $M$ and the resulting Persistent Modules also depend on that metric, and not just on the homology of $M$.

Questions

Thus I wonder, what properties of the Riemannian metric on $M$ can be reconstructed from persistent homology? Can we use persistent homology to construct some invariant of the metric?

Do you know of any research in that direction?

There is the paper Persistent homology detects curvature which seems to be a step in that direction.

'Persistent homology of $M$'

Of course persistent homology starts with data $X$ sampled from $M$ and it is not immediate how to get an object only depending on $M$ and not on a specific sampled dataset. For that, one could use the average of a vectorization of the persistence modules. For instance the average persistence landscape would only depend on $M$ and might hold some information of the metric.

Here is the same question on math.exchange

Answer 1

One thing you can get from a suitable metric on the manifold is a Hodge Laplacian on the differential forms. You can try to approximate it by using discrete versions of the Laplacian on the persistent cohomology cochains obtained from a point cloud sampled from the manifold. Then you could do some kind of spectral geometry, approximating eigenvalues of the Hodge Laplacian of the manifold. There are some significant and nontrivial mathematical questions: for example, persistent homology most commonly is studied and calculated with coefficients in a finite field, but the most familiar facts of Hodge theory (every cohomology class has a unique harmonic representative, etc.) require a characteristic zero coefficient field, and figuring out how to do Hodge theory with positive-characteristic coefficients on your cochains takes a bit of thought.

Here is a paper by Catanzaro and Vose about these questions, motivated by applications to persistent homology: https://arxiv.org/abs/2110.10885 Their "Related Work" section also has references to other papers which import some Hodge theory into persistent homology in various ways.

Answer 2

Perhaps my paper From Geometry to Topology: Inverse Theorems for Distributed Persistence with Wagner and Bendich gives a positive answer to your question on reconstructing geometry from topology.

The paper takes a metric space and computes the persistent homology of many "small" subsets. The paper has three kinds of results: (1) similar metric spaces give similar sets of persistence diagrams, (2) the persistent homology of all of these subsets determines the isometry type of the metric space, and (3) if two metric spaces have similar persistent homologies for their collections of aligned subsets, they must be similar (in the Gromov-Hausdorff sense).

(1) is a pretty standard stability result, not much new there. (2) is an inverse result, and is an answer to your question. It allows you to vary the cardinalities of the subsets, so they actually be fairly big, but you need to use multiple cardinalities at once, as depends on the skeleton being used for persistence calculations, and (3) shows that the map from geometry to "persistence of subsets" has a Lipschitz inverse, where the Lipschitz constant depends on the cardinality size you choose to work with.

In practice, using subsets that are large enough to capture structure about the space but small enough to compute quickly interpolates well between geometry and topology and can be used in topological optimization, as in our other paper: Improving Metric Dimensionality Reduction with Distributed Topology.