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Algorithms and Software

Applications of Topological Data Analysis

  • Violeta Kovacev-Nikolic, Peter Bubenik, Dragan Nikolić, and Giseon Heo. Using cycles in high dimensional data to analyze protein binding. Statistical Applications in Genetics and Molecular Biology, 15 (2016) no. 1, 19–38. Also available at arXiv:1412.1394 [stat.ME].We show that persistent homology is able to differentiate between open and closed forms of the maltose-binding protein, a large biomolecule consisting of 370 amino acid residues. We also observe that the majority of active site residues and allosteric pathway residues are located in the vicinity of the most persistent loop in the corresponding filtered Vietoris-Rips complex.
  • Moo K. Chung, Peter Bubenik and Peter T. Kim. Persistence Diagrams of Cortical Surface Data. Information Processing in Medical Imaging 2009. Lecture Notes in Computer Science, 5636 (2009), 386-397. We apply the techniques of the paper “Statistical topology via Morse theory, persistence and nonparametric estimation” to brain imaging data. In this case, the manifold is the brain cortex, and the unknown function is the cortical thickness. We use the resulting persistence diagrams to differentiate between the autistic and control subjects.

Statistics and Topology

  • Peter Bubenik. Statistical topological data analysis using persistence landscapes, Journal of Machine Learning Research, 16 (2015), 77–102. Also available at arXiv:1207.6437 [math.AT].
    I define a new descriptor for persistent homology, which I call the persistence landscape, for the purpose of facilitating statistical inference. This descriptor may be thought of as an embedding of the usual descriptors, barcodes and persistence diagrams, into a space of functions. The persistence landscape is a piecewise linear function. The linear structure of the function space allows simple and fast calculations. In fact the function space is a separable Banach space and so has a nice probability theory. For examples, I calculate mean landscapes for random geometric complexes, random clique complexes and Gaussian random fields.
  • Peter Bubenik, Gunnar Carlsson, Peter T. Kim and Zhi-Ming Luo. Statistical topology via Morse theory, persistence, and nonparametric estimation. Algebraic Methods in Statistics and Probability II. Contemporary Mathematics, 516 (2010), 75-92. Also available at arXiv:0908.3688[math.ST].
    Given data of the form (x_1, y_1), …, (x_N, y_N) where the x_i are points on a manifold, we assume that the data is generated by y_i = f(x_i) + epsilon_i, where f is some unknown function (from some fixed class of functions, e.g. Lipschitz) and epsilon_i is Gaussian noise with mean 0 and some fixed variance. We would like to use the data to recover the persistent homology of the lower excursion sets of f. Using a stability theorem for persistent homology, we can do this by estimating f with respect to the supremum norm. We do this by triangulating the manifold and filtering the triangulation using an estimator obtained by smoothing the data using kernels. The persistent homology of this filtered simplicial complex is the desired estimate of the persistent homology of the sublevel sets of f. This construction is asymptotically optimal, with specified rate and constant, in the minimax sense.
  • Peter Bubenik and Peter T. Kim. A statistical approach to persistent homology. Homology, Homotopy, and Applications, 9 (2007), No. 2, pp.337-362. Also available at arXiv:math/0607634 [math.AT].
    Consider a finite set of points randomly sampled from a Riemannian manifold according to some unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using these two filtrations we calculate the persistent homology of several directional densities. Using statistical estimators we can recover the persistent homology of the underlying distribution from the sample, for certain examples.We also show that the classical theory of spacings can be used to calculate the exact expectations of the persistent homology of samples from the uniform distribution on the circle, together with their asymptotic behavior.Some nice topological aspects in this paper include the following. The Morse filtrations of the von Mises-Fisher, Watson, and Bingham distributions correspond to different CW structures of the n-sphere. Similarly, the matrix von Mises distribution corresponds to a relative CW structure on SO(3), where this last decomposition is obtained using the Hopf fibration S^0 -> S^3 -> RP^3.

Foundations of Applied Topology

  • Peter Bubenik, Vidit de Silva and Vidit Nanda. Higher interpolation and extension of persistence modules, 12pp. Available at arXiv:1603.07406 [math.AT].
    A fundamental result in the theory of persistent homology, is that for any interleaving of two persistence modules, there exists a 1-parameter family that interpolates between these two persistence modules. That is, the metric space of persistence modules has geodesics between any two points. (Warning: these are not unique.) Casting this in the language of category theory, we see that this greatly generalizes: any family of interleaved persistence modules can be interpolated and extended, as long as the collections of interleavings is compatible (i.e. functorial). The solution is given by Kan extensions.
  • Peter Bubenik, Vin de Silva and Jonathan Scott. Metrics for generalized persistence modules, Foundations of Computational Mathematics, 15 (2015), no. 6, 1501–1531. Also available at arXiv:1312.3829 [math.AT].
    We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between `soft’ and `hard’ stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.
  • Peter Bubenik and Jonathan A. Scott. Categorification of persistent homology, Discrete and Computational Geometry, 51 (2014), no. 3, pp. 600-627. Also available at arXiv:1205.3669 [math.AT].
    We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.

Topology and Physics

  • Yuliy Baryshnikov, Peter Bubenik, and Matthew Kahle. Min-type Morse theory for configuration spaces of hard spheres. International Mathematics Research Notices, 2014 (2014), no. 9, pp. 2577-2592. Also available at arXiv:1108.3061 [math.AT].
    Hard spheres are important models for matter in statistical physics. Surprisingly, the answers to a number of basic topological questions of the configuration space of hard spheres are unknown. We develop a Morse theory for studying the homotopy theory of this configuration space. The critical points and critical submanifolds in this theory correspond to mechanically balanced configurations of spheres.

Topology and Concurrent Computing

  • Peter Bubenik. Simplicial models for concurrency. Electronic Notes in Theoretical Computer Science, 283 (2012) pp. 3-12. Also available at arXiv:1011.6599 [cs.DC].
    I give a simplicial model for concurrent programs, and a simplicial model for the possible executions from one state to another. The latter use necklaces of simplices, an idea used by Jacob Lurie in his study of higher dimensional category theory. It should be possible to give a more economical cubical version of these models.
  • Peter Bubenik. Models and van Kampen theorems for directed homotopy theory. Homology, Homotopy and Applications, 11 (2009), 185-202. Also available at arXiv:0810.4164 [math.AT].In order to model concurrent parallel computing, I study topological spaces with a distinguished set of paths, called directed paths. The topological space models the state space of the system, and the directed paths model the execution paths. To reduce to the essentially different executions, we reduce to homotopy classes of directed paths. To reduce the size of the state space we apply future retracts and past retracts. We also prove some theorems for doing these constructions in a piece-by-piece fashion.More technically, since the directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. I define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, I prove van Kampen theorems for subcategories, retracts, and models of the fundamental category.
  • Peter Bubenik and Krzysztof Worytkiewicz. A model category structure for local po-spaces. Homology, Homotopy and Applications, 8 (2006), pp. 263-292. Also available at arXiv:math/0506352 [math.AT].Local po-spaces are topological spaces together with a local partial-order. Maps between these spaces are continuous maps which respect the order. Some computer scientists use local po-spaces to model concurrent systems and would like a good theory of equivalences for these spaces.In this paper we construct a homotopy theory for these spaces. Spaces which are trivial or nearly trivial in the classical undirected case can be highly non-trivial when homotopies are forced to respect directions. Our aim is to construct a (Quillen) model category for local po-spaces as a framework for a directed homotopy theory. This is technically difficult because local po-spaces are not closed under colimits, which is a necessary condition in a model category. Our method is inspired by the construction of Voevodsky’s A1-homotopy theory and Dugger’s universal homotopy theories. We pass to the simplicial presheaf category and apply Jardine’s model structure. We analyze the weak equivalences in this model category. Finally one uses the right context (see “Context for models of concurrency”) and localizes with respect to a class of equivalences – for example, the dihomotopy equivalences. Further work will analyze the weak equivalences in this model category.
  • Peter Bubenik. Context for models of concurrency. Electronic Notes in Theoretical Computer Science, 230 (2009), 3-21. Preliminary version in Proceedings of the Workshop on Geometry and Topology in Concurrency and Distributed Computing, BRICS Notes NS-04-2, pp.33-49). Also available at arXiv:math/0608733 [math.AT].In this paper I examine many simple examples of partially-ordered spaces which model concurrent systems. It would be useful to be able to replace a given model with a simpler one. What one wants is a good notion of a directed homology equivalence. However, I show that the usual definition of directed homotopy equivalence is too coarse. To solve this problem I introduce the notion of context. More precisely, I show that dihomotopy equivalences are best defined in the category of po-spaces under a po-space A where the choice of A depends on the ‘pastings’ that one would like to consider.

Topology and Algebra

  • Peter Bubenik and Leah Gold. Graph products of spheres, associative graded algebras and Hilbert series. Mathematische Zeitschrift, 268 (2011), pp. 821–836. Also available at arXiv:0901.4493 [math.AT].
    Given a finite, simple, vertex–weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Noncommutative Gr¨obner bases play an important role in our proof.There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment–angle complex. We apply our result to the loop–space homology of this space.
  • Peter Bubenik. Separated Lie models the homotopy Lie algebra. Journal of Pure and Applied Algebra 212 (2008), no.2, 401-410. Also available atarXiv:math/0406405 [math.AT].For a simply connected topological space X there is a differential graded Lie algebra, called the Quillen model, which determines the rational homotopy-type of X. Its homology is isomorphic to the rational homotopy groups of the loop space on X, called the homotopy Lie algebra. In this paper I show that for spaces with finite (rational) LS category, these can be assumed to satisfy a condition I call separated, which is useful for calculations of the homotopy Lie algebra. This separated condition implies the free condition introduced in “Free and semi-inert cell attachments”.The separated Lie models give a nice characterization of the rational homotopy Lie algebra in the case where the “top” differential creates at least two new homology classes: the radical is contained in previous homology and the rational homotopy Lie algebra contains a free Lie algebra on two generators. So it satisfies the Avramov-Felix conjecture.Any rational space constructed using a sequence of cell attachments of length N is equivalent to a space constructed using a sequence of free cell attachments of length N+1. This is shown by proving a similar result for differential graded Lie algebras (dgLs). As a results, one obtains a method for calculating the homotopy Lie algebra, and the homology of certain dgLs.
  • Peter Bubenik. Free and semi-inert cell attachments. Transactions of the American Mathematical Society 357 (2005) pp. 4533-4553. Also available at arXiv:math/0312387 [math.AT].In this paper I give some new results on the cell attachment problem, which was perhaps first studied by J.H.C. Whitehead around 1940: If one attaches one or more cells to a topological space, what is the effect on the homology of the loop space, and on the homotopy-type?I introduce the free and semi-inert conditions under which I determine the loop space homology as a module and as an algebra respectively. Under a further condition I determine the homotopy-type of the space.
  • Peter Bubenik. Cell attachments and the homology of loop spaces and differential graded algebras. Thesis, University of Toronto, 2003. Available at arXiv:math/0601421 [math.AT].

Statistics and Geometry

  • Peter Bubenik and John Holbrook. Densities for Random Balanced Sampling, Journal of Multivariate Analysis 98 (2007) pp. 350-369. Also available at arXiv:math/0608737 [math.ST].
    A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors X lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (2,3,4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n goes to infinity). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, isolation of nonlinearities, and goodness of fit testing. We also show that the previously known densities (for n less than or equal to 4) are in fact the only solutions in a natural and very large class of potential RBS densities. It follows that the new methods lead in another direction entirely.