General Information

Student:	Andrew Xie
Office:	CoRE 444 96 Frelinghuysen Rd, Piscataway, NJ 08854
School:	Rutgers University
E-mail:	andrew.x@rutgers.edu
Project:	Invariants for Dynamic Data via Curvature Sets

Project Description

A dynamic metric space is a finite set of points equipped with a pseudometric that varies through time. The goal of this project is to extend curvature-set persistent homology to dynamic metric spaces, providing a computationally efficient method to track how the topology of data changes over time. We aim to develop a construction that is provably stable and representable, offering a principled technique to analyze dynamic data, such as motions of particles or neural activity.

Weekly Log

Week 0:

Much of this week was spent learning background material and defining possible research directions.

In particular, we read about poset-indexed persistence modules (most importantly zigzag persistence), which allows for non-monotone filtrations used to analyze dynamic data. We also reviewed Tom Needham's book on applied algebraic topology for general persistence theory.

In addition, Prof. Memoli has given us a few talks on Gromov-Hausdorff distance and dynamic metric spaces, and posed a few interesting questions about G-H distance between spheres and intervals/trees.

Week 1:

This week we found an interesting topic to work on, which is extending the notion of curvature set persistent homology to a dynamic (time-evolving) setting. At the moment, we have a rough roadmap of what to focus on next few weeks:

• Stability: given dynamic metric spaces \( \gamma_X \) and \( \gamma_Y \), the \( k \)th persistent homology of size \( n \) curvature sets is stable, i.e. "if \( \gamma_X \) and \( \gamma_Y \) are Gromov-Hausdorff close, then the persistence diagrams of the curvature sets must be similar". Formally:
  \[ d^{\mathcal{D}}_H(D_{n,k}(\gamma_X), D_{n,k}(\gamma_Y)) \leq C d^{\text{dyn}}_{\text{GH}}(\gamma_X, \gamma_Y) \]
  for some universal constant \( C > 0 \).
• Representation: find an efficient way to summarize "thin" persistent vector spaces indexed by \( \mathbf{Int} \times \mathbb{R}_{+} \), where \( \mathbf{Int} \) is the poset category of intervals in \( \mathbb{R} \).

In addition, we gave a brief talk about our project to the rest of the DIMACS cohort.

Week 2:

This week I proved a partial result towards the classification task: for a persistence module \( f: \mathbf{Int} \times \mathbb{R}_{+} \rightarrow \textbf{Vec} \) arising from our particular construction, the structure maps \( f(I_1, \delta_1) \rightarrow f(I_2, \delta_2) \) are uniquely determined by \( \dim f(I_1, \delta_1) \) and \( \dim f(I_2, \delta_2) \). This allows us to summarize \( f \) as just a mapping, instead of a functor.

We also thought a bit more about Nadya's question of an explicit example of two spaces that are not homotopic but have the same Vietoris-Rips barcode. Prof. Memoli suggested adapting the classic \( S^2 \vee S^1 \vee S^1 \) and \( S^1 \times S^1 \) example with a modified metric, which seems to work.

Week 3:

This week we focused on characterizing the interleaving distance \( d_I(D_{n, k}(\gamma_X), D_{n, k}(\gamma_Y)) \).

By an earlier result, our persistence modules are uniquely determined by their "support", i.e. all the \( (I, \delta) \) where \( f(I, \delta) \cong \mathbb{F} \). I came up with a metric \( d \) that depends only on the geometry of the supports that lower bounds \( d_I \). This metric should be very easy to compute, if we allow ourselves to discretize the support.

I was hoping that \( d = d_I \) (at least for the restricted class of modules we study), but this isn't true: if we ignore our construction and only work at a module level there is a very obvious counterexample (but indexed over \(\mathbb{C}\)). Sriram and Dr. M\'emoli made a good point about this not being realizable, but I later found a more subtle counterexample that can be constructed with our \((2k+2)\) approach. I believe that given our current analysis, if we work in characteristic \( 2 \) (e.g. fix \( \mathbb{F} = \mathbb{Z} / 2 \mathbb{Z} \)) the obstruction causing \( d \neq d_I \) is quite simple. Perhaps a slightly modified definition of \( d \) will suffice.

Week 4:

This week I worked on computing the interleaving distance \( d_I \) precisely: I came up with a metric \( d^* \) on the support sets such that \( d^*(S_\mathbb{V}, S_\mathbb{W}) = d_I(\mathbb{V}, \mathbb{W}) \). However, after speaking to my mentor we agreed that exactly computing the interleaving distance may not be necessary, and the less computationally intensive \( d \) may actually be preferable. Still, it was a somewhat interesting exercise.

We also thought about characterizing the support sets a bit more: we were able to rule out some very degenerate cases (e.g. the support set being a countable dense subset of a plane \( P \), where any two points in \( P \) are incomparable).

Finally, we discussed some computational (coding) details. A master's student (Max Goldberg) will be joining our team to work on experimental validation. To keep consistent with prior literature we will use Boids as synthetic data, which Max will help set up and integrate into our pipeline.

Week 5:

Since last week, we made some progress in understanding the structure of our modules, and produced a minimum viable implementation of our methods.

In particular, I proved that the modules \( \mathbb{V} \) produced by our construction are interval-decomposible, with the intervals being given by the order-connected components of the support: i.e. \( \mathbb{V} \cong \bigoplus_{\alpha \in A} \mathbf{1}_{S_\alpha} \) where \( \{S_\alpha\}_{\alpha \in A} \) are the connected components of \( S_\mathbb{V} \).

Moreover, we have the following "incomparability" result that may be of use later: \[ p \in S_\alpha, q \in S_\beta \implies p \not \leq q \text{ and } q \not \leq p \qquad \forall \alpha \neq \beta. \]

In addition, I was able to implement some python code that discretizes a DMS into \( S \) spatial steps and \( T \) time steps. This code can compute the support in \( O(T^2) \) time, and can compute \( d(S_{\mathbb{V}}, S_{\mathbb{W}}) \) in \( O(T^2 \log{T}) \) time.

This isn't a great time complexity, but for a research implementation it will probably suffice. Nadya was also able to produce an implementation of the erosion distance \( d_E \) in \( O(S^4 T^2) \) time, which although slow has the advantage of being very well-characterized in existing TDA literature (as opposed to \( d \)).

Week 6:

Much of this week was focused on (1) preparing for the final presentation, and (2) porting our python code to C++

Also, as the final presentation is coming up, I decided to prove the stability of our construction to ensure that we had some theoretical justification for the "validity" of our method. The proof ended up being more straightforward than expected, as there was already an existing stability result for a construction without curvature sets - making it work with curvature sets just required a technical lemma about Gromov-Hausdorff distance between subsets.

In addition, Nadya was able to find a very nice DP to compute \( d_E \) in \( O(S^2 T) \) time, which is now fast enough for us to run numerical experiments. Nadya and I worked on generalizing this algorithm to "arbitrary" multidimensional posets (well, not really arbitrary, just \( [n]^d \) with the product order, where \( [n] = \{1, \ldots, n\} \)). Hopefully this can broaden applicability, although the thinness constraint we impose is still quite a strong requirement.

Weeks 7-8:

Entering the final weeks of the REU, we focused on preparing our final presentation and project report. Thank you to Dr. M\'emoli and Sriram Raghunath for their valuable mentorship, and my collaborator Nadya for her helpful questions and ideas. I'd also like to thank the 2025 DIMACS cohort, Dr. Gallos and Larry Frolov for making this all possible!

UPDATE (March 2026)

As of week \( n \), we produced a preprint that we plan on submitting to some journals / conferences. Some additional updates were made: our generalized \( d_E \) algorithm on \( [n]^d \)-indexed modules now runs in \( O(n^{d-1} \cdot d \log{n}) \) time, and exposition was significantly improved. I came up with an approach to relax the thinness requirement, which I may write up.

As of week \( (n+1) \) a preprint of this report is now available on arXiv.

Additional Information

• My Mentors
  • Dr. Facundo Memoli
  • Sriram Raghunath
• My Personal Website