Topics in Persistent Homology: From Morse Theory for Minimal Surfaces to Efficient Computation of Image Persistence

We study some problems and develop some theory related to persistent homology, separated into two lines of investigation. In the first part, we introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes by de Silva, Morozov, and Vejdemo-Johansson. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for an efficient algorithm to compute barcodes of images and induced matchings of such morphisms, which performs computations in terms of relative cohomology and then translates to absolute homology via the aforementioned dualities. Our method is based on a previous algorithm by Cohen-Steiner, Edelsbrunner, Harer, and Morozov that did not make use of relative cohomology. Using it is crucial, however, because our algorithm applies the clearing optimization introduced by Chen and Kerber, which works particularly well in the context of relative cohomology. We provide an implementation of our algorithm for inclusions of filtrations of Vietoris–Rips complexes in the framework of the software Ripser by Ulrich Bauer. In the second part, we introduce local connectedness conditions on a broad class of functionals that ensure that the persistent homology of their associated sublevel set filtration is q-tame, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the unstable minimal surface theorem given by Morse and Tompkins in terms of persistent Čech homology in a modern and rigorous framework. Moreover, we show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a filtration consisting of paracompact Hausdorff spaces is 0 if it satisfies a similar local connectedness condition to the one used to ensure q-tameness, generalizing a result by Mardešić for locally connected spaces to the setting of filtrations. In contrast to singular homology, the persistent Čech homology of a compact filtration is always upper semi-continuous, which has structural implications in the q-tame case: using a result by Chazal, Crawley-Boevey, and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper semi-continuous q-tame persistence module can be decomposed as a product of interval modules

Slope And Geometry In Variational Mathematics

Structure permeates both theory and practice in modern optimization. To make progress, optimizers often presuppose a particular algebraic description of the problem at hand, namely whether the functional components are affine, polynomial, smooth, sparse, etc., and a qualification (transversality) condition guaranteeing the components do not interact wildly. This thesis deals with structure as well, but in an intrinsic and geometric sense, independent of functional representation. On one hand, we emphasize the slope - the fastest instantaneous rate of decrease of a function - as an elegant and powerful tool to study nonsmooth phenomenon. The slope yields a verifiable condition for existence of exact error bounds - a Lipschitz-like dependence of a function's sublevel sets on its values. This relationship, in particular, will be key for the convergence analysis of the method of alternating projections and for the existence theory of steepest descent curves (appropriately defined in absence of differentiability). On the other hand, the slope and the derived concept of subdifferential may be of limited use in general due to various pathologies that may occur. For example, the subdifferential graph may be large (full-dimensional in the ambient space) or the critical value set may be dense in the image space. Such pathologies, however, rarely appear in practice. Semi-algebraic functions - those functions whose graphs are composed of finitely many sets, each defined by finitely many polynomial inequalities - nicely represent concrete functions arising in optimization and are void of such pathologies. To illustrate, we will see that semi-algebraic subdifferential graphs are, in a precise mathematical sense, small. Moreover, using the slope in tandem with semi-algebraic techniques, we significantly strengthen the convergence theory of the method of alternating projections and prove new regularity properties of steepest descent curves in the semi-algebraic setting. To illustrate, under reasonable conditions, bounded steepest descent curves of semi-algebraic functions have finite length and converge to local minimizers - properties that decisively fail in absence of semi-algebraicity. We conclude the thesis with a fresh new look at active sets in optimization from the perspective of representation independence. The underlying idea is extremely simple: around a solution of an optimization problem, an "identifiable" subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are "minimal". In the context of standard nonlinear programming, this concept reduces to the active-set philosophy. On the other hand, identifiability is much broader, being independent of functional representation of the problem. This new notion lays a broad and intuitive variational-analytic foundation for optimality conditions, sensitivity, and active-set methods. In the last chapter of the thesis, we illustrate the robustness of the concept in the context of eigenvalue optimization