Dmitriy Morozov

Vines and vineyards by updating persistence in linear time

Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithms analysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to the study of protein folding trajectories.

Zigzag persistent homology and real-valued functions

We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.

Zigzag persistent homology in matrix multiplication time

We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n×n matrices in M(n) time, our algorithm runs in O(M(n) + n2 log2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n3) time in the worst case.

Dualities in persistent (co)homology

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.

Persistence-sensitive simplification functions on 2-manifolds

We continue the study of topological persistence [5] by investigating the problem of simplifying a function <i>f</i> in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function <i>g</i> an ε-<i>simplification</i> of another function <i>f</i> if ¦¦<i>f−g</i>¦¦_∞≤ε, and the persistence diagrams of <i>g</i> are the same as those of <i>f</i> except all points within <i>L</i>₁-distance at most ε from the diagonal have been removed. We prove that for functions <i>f</i> on a 2-manifold such ε-simplification exists, and we give an algorithm to construct them in the piecewise linear case.

Inferring Local Homology from Sampled Stratified Spaces

We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.

Distributed merge trees

Improved simulations and sensors are producing datasets whose increasing complexity exhausts our ability to visualize and comprehend them directly. To cope with this problem, we can detect and extract significant features in the data and use them as the basis for subsequent analysis. Topological methods are valuable in this context because they provide robust and general feature definitions. As the growth of serial computational power has stalled, data analysis is becoming increasingly dependent on massively parallel machines. To satisfy the computational demand created by complex datasets, algorithms need to effectively utilize these computer architectures. The main strength of topological methods, their emphasis on global information, turns into an obstacle during parallelization. We present two approaches to alleviate this problem. We develop a distributed representation of the merge tree that avoids computing the global tree on a single processor and lets us parallelize subsequent queries. To account for the increasing number of cores per processor, we develop a new data structure that lets us take advantage of multiple shared-memory cores to parallelize the work on a single node. Finally, we present experiments that illustrate the strengths of our approach as well as help identify future challenges.

Quantifying Transversality by Measuring the Robustness of Intersections

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

Metric graph reconstruction from noisy data

Many real-world data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs [16]. Building on the notions of correspondence and Gromov-Hausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric graph. We present a novel algorithm that takes as an input such a data set, and outputs the underlying metric graph with guarantees. We also implement the algorithm, and evaluate its performance on a variety of real world data sets.

The Stability of the Apparent Contour of an Orientable 2-Manifold

The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :𝕄→ℝ2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming 𝕄 is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.