Research Projects

  1. Spectral relaxations of persistent rank invariants

    TopologyPersistenceLinear Algebra

    We introduce a framework for constructing families of continuous relaxations of the persistent rank invariant for persistence modules indexed over the real line. Applications to multi-parameter persistence, parameter optimization, and shape classification are also presented.

  2. Move Schedules: Fast persistence computations in coarse dynamic settings


    Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are quadratically many such transpositions, this maintenance procedure exhibits limited scalability and often is too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations that requires only subquadratic time and linear space to construct.

  3. Efficient Multiscale Simplicial Complex Generation for Mapper

    ClusteringTopologyR Package

    The primary result of the Mapper framework is the geometric realization of a simplicial complex, depicting topological relationships and structures suitable for visualizing, analyzing, and comparing high dimensional data...

  4. Automating Point of Interest Discovery in Geospatial Contexts

    ClusteringGeospatial analysisNetwork modeling

    With the rapid development and widespread deployment of sensors dedicated to location-acquisition, new types of models have emerged to predict macroscopic patterns that manifest in large data sets representing "significant" group behavior. Partially due to the immense scale of geospatial data, current approaches to discover these macroscopic patterns are primarily driven by inherently heuristic detection methods. Although useful in practice, the inductive bias adopted by such mainstream detection schemes is often unstated or simply unknown. Inspired by recent theoretical advances in efficient non-parametric density level set estimation techniques, in this research effort we describe a semi-supervised framework for automating point of interest discovery in geospatial contexts. We outline the flexibility and utility of our approach through numerous examples, and give a systematic framework for incorporating semisupervised information while retaining finite-sample estimation guarantees.

  5. Bringing High Performance Density-based Clustering to R

    ClusteringR PackageHigh performance computing

    Density-based clustering techniques have become extremely popular in the past decade. It's often conjectured that the reason for the success of these methods is due to their ability of identify 'natural groups' in data. These groups are often non-convex (in terms of shape), deviating the typical premise of 'minimal variance' that underlies parametric, model-based approaches, and often appear in very large data sets. As the era of 'Big Data' continues to rise in popularity, it seems that typical notions having access to scalable, easy-to-use, and scalable implementations of these density-based methods is paramount. In this research effort, we provide fast, state-of-the-art density-based algorithms in the form of an open-source package in R. We also provide several related density-based clustering tools to help bring make state of the art density-based clustering accessible to people with large, computationally difficult problems.

  6. Towards Autonomous Aerial Refueling: Massive Parallel Iterative Closest Point

    GeometryPoint registrationHigh performance computing

    The Iterative Closest Point (ICP) problem is now a well-studied problem that seeks to align a given query point cloud to a fixed reference point cloud. The ICP problem computationally is dominated by the first phase, a pairwise distance minimization. The ''brute-force'' approach, an embarrassingly parallel problem amenable to GPU-acceleration..