Current projects

We are currently working on the following projects:

  1. Analysis of density-based algorithms for clustering and anomaly detection used in CS and data science (and lacking statistics guarantees)
    • DBSCAN, OPTICS, local outlier factor, isolation forests.
    • We can use these algorithm to get e.g. cluster trees.
  2. Streaming/online density estimation with applications to TDA
    • Maybe sliding windows can work.
  3. Use of concave-hull and alpha-concave hull for support recovery, shape estimation, clustering, etc
    • This project would be more geometrical.
  4. Confidence sets in the persistent homology framework described in this paper:
    • Omer Bobrowski, Sayan Mukherjee, Jonathan Taylor, “Topological Consistency via Kernel Estimation” [ arXiv ].
    • Filtration is constructed differently.

  5. Investigate the statistical use/advantages of alpha-witness complexes

  6. Come up and study continuous geometric signatures (as opposed to the discrete ones used in TDA)
    • Such as interpoint distance distribution studied in the paper “Shape classification based on interpoint distance distributions” [ link ].
    • Continuous features would be easier to analyze.
  7. Bifiltrations for persistent homology.
    • In particular, the bifiltration arising form letting the level and the bandwidth change in density estimation.
  8. Intensity functions for handling persistence diagrams and their usage.
    • Darren and Yen-Chi wrote a nice conference paper as a first step, but this can be further developed.
    • It would be useful when dataset consists of many replications.
  9. Mode trees.
    • Jussi Klemela, Mode Trees for Multivariate Data [ link ].
  10. Mapper.
    • Now Mapper is on R CRAN as package TDAmapper [ CRAN ].

Papers to be considered

Several other things to be considered