Statistical Computational Topology

Persistent homology is a method for probing topological properties of point clouds and function. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be “topological noise.” I am interested in bringing statistical ideas to persistent homology in order to distinguish topological signal from topological noise and to derive meaningful, yet computable, summaries of large datasets. For more information, please see the CMU TopStat website.

publications and preprints

presentations and poster

  • SoCG 2014: Stochastic Convergence of Persistence Landscapes and Silhouettes [Slides]
  • JMM 2014: The Intersection of Statistics and Topology [Slides]
  • Statistical Inference for Persistent Homology [Poster]
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Map Construction and Comparison

In today's society, there is an over-abundance of data. In particular, the movement of every cell phone carried by a person walking or every vehicle driven through a city can be traced by GPS and then recorded. Given the GPS trajectories of, for example, all taxis driving through a major city, I investigate how to synthesize the overabundance of information that these trajectories provide. One way of synthesizing the set of trajectories is to use them to estimate the underlying road network. This is called map (re-)construction. I focus on evaluating map construction algorithms. Given two road networks, A and B, represented as graphs embedded in a compact subspace of R^2, the task is to compute a meaningful distance between the two road networks that takes into consideration both the local and the global differences. For example, A may be the ground truth road network and B may be the road network reconstructed from GPS trajectory data. Moreover, this research applies to more networks than just to road networks. For example, one may also consider biological networks, filaments of galaxies throughout the universe, and hiking paths.

publications and preprints

presentations

  • Local Homology Based Distance [Slides]
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Other

Modes of Gaussian Mixtures. To many researcher's surprise, n+1 isotropic Gaussian kernels in R^n can have n+2 modes. I call these additional modes "ghost modes." More generally, there exist finite configurations of isotropic Gaussian kernels with superlinearly many modes. Moreover, even the most simple configuration exhibits an exponential number of critical points.

Curves in Space. Researchers across many fields are interested in the topological and geometric properties of shapes in Euclidean space. In a 2011 Journal paper, I bound the difference of lengths of curves in Euclidean space by a function of the total curvature and the Fréchet distance between the curves.

Heat Equation Homotopy. Persistence homology is a tool used to classify topological features that are present in data sets and functions. I am interested in using persistence to explore the deep structure, or scale space, of images. The scale space of an image is a family of related images obtained through convolution with the Gaussian kernel. Viewing each image as a real-valued function, I stack the corresponding persistence diagrams. This creates a vineyard of curves that connect the points in the diagrams. I am interested in using the vineyard arising from the heat equation homotopy to define a distance between two images.

Computer Science Education. I created virtual worlds using a 3D interactive animation environment, Alice. Alice had previously been used as a program visualization tool for introductory computer science classes. The worlds I created enable use of this tool at the intermediate programming level by introducing the concepts of lists and arrays. This research has provoked interest in expanding the use of Alice to higher level CS courses.

Homotopy Classification Problems. A basic problem in mathematics is the classification problem. In group theory we have the Classification Problem for Groups: Given a collection G of groups, classify the groups G in G up to isomorphism. And in homotopy theory, we have the Homotopy Classification Problem for Spaces: Given a collection T of topological spaces, classify the spaces X in T up to homotopy equivalence. In some cases, these classification problems are actually equivalent. I am interested in understanding these cases. In particular, I look at examples classifying groups by the centralizers and classifying the path components of function spaces.

publications and preprints

presentations

  • SoCG 2012: Ghost Modes in Gaussian Mixtures [Slides]
  • Prelim 2010: Discovering Metrics and Scale Space [Slides]
  • Expanding Alice [Slides]
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