This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the purpose of exposition we focus the discussion of applications on homological algebra and computation, with additional remarks in lattice theory, linear algebra, and abelian categories. For motivation, we recall that the methods of algebraic topology have driven remarkable progress in the qualitative study of large, noisy bodies of data over the past 15 years. A primary tool in Topological Data Analysis [TDA] is the homological persistence module, which leverages categorical structure to compare algebraic shape descriptors across multiple scales of measurement. Our principle application to computation is a novel algorithm to calculate persistent homology which, in certain cases, improves the state of the art by several orders of magnitude. Included are novel results in discrete, spectral, and algebraic Morse theory, and on the strong maps of matroid theory. The defining theme throughout is interplay between the combinatorial theory matroids and the algebraic theory of categories. The nature of these interactions is remarkably simple, but their consequences in homological algebra, quiver theory, and combinatorial optimization represent new and widely open fields for interaction between the disciplines.
The to compute persistent homology was introduced for coefficients in the two element field by Edelsbrunner, Letscher, and Zomorodian in @cite_72 . The adaptation of this algorithm for arbitrary field coefficients was presented by Carlsson and Zomorodian in @cite_28 . The standard algorithm is known to have worst case cubic complexity, a bound that was shown to be sharp by Morozov in @cite_80 . Under certain sparsity conditions, the complexity of this algorithm is less than cubic. An algorithm by Milosavljevi 'c, Morozov, and Skraba has been shown to perform the same computation in @math time, where @math is the matrix-multiplication coefficient @cite_3 .
@cite_28 @cite_80 @cite_72 @cite_3
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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. We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension. We approach the problem of the computation of persistent homology for large datasets by a divide-and-conquer strategy. Dividing the total space into separate but overlapping components, we are able to limit the total memory residency for any part of the computation, while not degrading the overall complexity much. Locally computed persistence information is then merged from the components and their intersections using a spectral sequence generalizing the Mayer-Vietoris long exact sequence. We describe the Mayer-Vietoris spectral sequence and give details on how to compute with it. This allows us to merge local homological data into the global persistent homology. Furthermore, we detail how the classical topology constructions inherent in the spectral sequence adapt to a persistence perspective, as well as describe the techniques from computational commutative algebra necessary for this extension. The resulting computational scheme suggests a parallelization scheme, and we discuss the communication steps involved in this scheme. Furthermore, the computational scheme can also serve as a guideline for which parts of the boundary matrix manipulation need to co-exist in primary memory at any given time allowing for stratified memory access in single-core computation. The spectral sequence viewpoint also provides easy proofs of a homology nerve lemma as well as a persistent homology nerve lemma. In addition, the algebraic tools we develop to approch persistent homology provide a purely algebraic formulation of kernel, image and cokernel persistence (D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and D. Morozov. Persistent homology for kernels, images, and cokernels. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1011-1020. Society for Industrial and Applied Mathematics, 2009.) We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by G ", which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we improve the performance through parallelized computation.