Discrete Morse Theory and Classifying Spaces
The aim of this paper is to develop a refinement of Forman’s discrete Morse theory. To an acyclic partial matching µ on a finite regular CW complex X, Forman introduced a discrete analogue of gradient flows. Although Forman’s gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of X. Forman also proved the existence of a CW complex which is homotopy equivalent to X and whose cells are in one-to-one correspondence with the critical cells of µ, but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman’s gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of X, while retaining a combinatorial description. The critical difference from Forman’s gradient flows is the existence of a partial order on the set of flow paths, from which a 2-category C(µ) is constructed. It is shown that the classifying space of C(µ) is homotopy equivalent to X by using homotopy theory of 2-categories. This result may be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory from the early 90’s.
Status: Submitted (2016), the arxiv preprint is here.
Discrete Morse Theory and Localization
Incidence relations among the cells of a regular CW complex produce a 2-category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching on (the cells of) such a complex corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex. This flow category forms a combinatorial and computable counterpart to the one described here by Cohen, Jones and Segal in the context of smooth Morse theory.
Status: Submitted (2016), also available on the arxiv here.
Discrete Morse Theory for Computing Cellular Sheaf Cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for computation of (ordinary) cohomology of a cell complex.
Justin Curry, Robert Ghrist and Vidit Nanda. Discrete Morse Theory for Computing Cellular Sheaf Cohomology. Foundations of Computational Mathematics Volume 16, Issue 4, pp 875 - 897, August 2016.
Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
We provide explicit and efficient algorithms based on discrete Morse theory to compute homology of a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that underlying function is subject to measurement errors. We introduce a new Morse theoretic algorithm for deriving chain maps from these set-valued maps, and hence an effective scheme for computing the map induced on homology by the approximated continuous function.
Shaun Harker, Konstantin Mischaikow, Marian Mrozek and Vidit Nanda. Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps. Foundations of Computational Mathematics Volume 14, Issue 1, pp 151-184, February 2014.
Morse Theory for Filtrations and Efficient Computation of Persistent Homology
With Konstantin Mischaikow.
We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations of cell complexes.
This paper provides the theoretical basis for the Perseus software project designed to compute persistent homology of various types of filtrations.
Konstantin Mischaikow and Vidit Nanda. Morse Theory for Filtrations and Efficient Computation of Persistent Homology. Discrete & Computational Geometry, Volume 50, Issue 2, pp 330-353, September 2013.
Higher Interpolation and Extension for Persistence Modules
The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. As a consequence of such “higher interpolation”, it becomes possible to compare Vietoris-Rips and Cech complexes built within the space of persistence modules.
Status: Submitted (2016), see also the Arxiv version.
Topological Signatures of Singularities in Simplicial Ricci Flow
We apply the methods of persistent homology to a selection of two and three-dimensional geometries evolved by simplicial Ricci flow. We construct a triangular mesh for a sample of points so that the scalar curvature along the edges of the triangulation serves as a filtration parameter at each time step. We present and analyze the persistent homology of a two–dimensional rotational solid that collapses and of three-dimensional dumbbells which manifest neck-pinch singularities. We conclude that persistent homology identifies geometric criticality. Finally, we discuss the interpretation and implication of these results and present future applications
Status: Submitted (2016), also available here on the Arxiv.
Geometry in the Space of Persistence Modules
With Vin de Silva.
We study the geometry of the space of persistence modules and diagrams, with special attention to Cech and Rips complexes. The metric structures are determined in terms of interleaving maps (of modules) and matchings (between diagrams). We show that the relationship between the Cech and Rips complexes is governed by the relationship between the corresponding interleavings and matchings.
Vin de Silva and Vidit Nanda. Geometry in the Space of Persistence Modules, Proceedings of the 29th Annual Symposuim on Computational Geometry, ACM, pp 397-404, June 2013.
Reconstructing Functions from Random Samples
By a fantastic result of Partha Niyogi, Steve Smale and Shmuel Weinberger from this paper, it is possible to construct a simplicial complex homotopy-equivalent to a compact Riemannian submanifold of Euclidean space using only a finite point sample lying on that manifold. We describe a corresponding result for Lipschitz-continuous functions between such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy groups with high confidence using only finite sampled data from the domain, the codomain, and the image of every sampled point from the domain. We provide bounds on sizes of uniform samples required to guarantee success of such a reconstruction up to any desired probability smaller than 1. We show that this reconstruction is robust to bounded sampling and evaluation noise.
Steve Ferry, Konstantin Mischaikow and Vidit Nanda. Reconstructing Functions from Random Samples. Journal of Computational Dynamics Volume 1, Issue 2, pp 233-248, December 2014.
A Topological Measurement of Protein Compressibility
Given X-ray crystallography data of a protein molecule from the PDB, we build a van der Waal weighted alpha shape representation of that protein molecule by constructing cells around each atom center. Thus, to each protein we assoicate a set of persistence diagrams (one for each dimension). Using elementary physical principles, we identify certain structural features of molecules that are conjectured to impact compressibility. A simple parameter search through the persistence diagrams isolates these features and provides a robust measure which exhibits remarkable linear correlation with experimentally computed protein compressibility.
Status: Appeared in the Japan Journal of Industrial and Applied Mathematics. Springer maintains an official version of the accepted article here.
Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar, Konstantin Mischaikow, Vidit Nanda, A Topological Measurement of Protein Compressibility, Japan Journal of Industrial and Applied Mathematics, Volume 32, Issue 1, pp 1-17, March 2015.
Simplicial Models and Topological Inference in Biological Systems
With Radmila Sazdanovic.
This article is a user's guide to algebraic topological methods for data analysis with a particular focus on applications to datasets arising in experimental biology. We begin with the combinatorics and geometry of simplicial complexes and outline the standard techniques for imposing filtered simplicial structures on a general class of datasets. From these structures, one computes topological statistics of the original data via the algebraic theory of (persistent) homology. These statistics are shown to be computable and robust invariants of the shape underlying a dataset. Finally, we showcase some appealing instances of topology-driven inference in biological settings -- from the detection of a new type of breast cancer to the analysis of various neural structures.
Status: Apeared as Chapter 6 of the 2014 Springer book Discrete and Topological Models in Molecular Biology, editors N. Jonoska and M. Saito. Springer also offers e-prints of the article here.
Vidit Nanda and Radmila Sazdanovic, Simplicial Models and Topological Inference in Biological Systems, Discrete and Topological Models in Molecular Biology, Natural Computing Series (Springer), pp 109-141, January 2014.
Discrete Morse Theory for Filtrations
My Ph.D. dissertation from October 2012. The main results presented here are from the paper titled "Morse theory for filtrations and efficient computation of persistent homology". The final chapter outlines a bonus application: the filtered Morse theory may be used towards simplifying the construction of long exact sequences via the Zigzag lemma. Here is Rutgers' official copy of the document, which has been formatted with awful (but mandatory) double-spacing, to say nothing of the gigantic margins into which many marvelous proofs would fit.