We study the algorithmic complexity of computing persistent homology of a randomly chosen filtration. Specifically, we prove upper bounds for the average fill-up (number of nonzero entries) of the boundary matrix on Erdös-Renyi filtrations and Vietoris-Rips filtrations after matrix reduction. Our bounds show that, in both cases, the reduced matrix is expected to be significantly sparser than what the general worst-case predicts. Our bounds are based on previous results on the expected first Betti numbers of corresponding complexes. We establish a link between these results to the fill-up of the boundary matrix. Our bound for Vietoris-Rips complexes is asymptotically tight up to logarithmic factors.
We study the algorithmic complexity of computing persistent homology of a randomly chosen filtration. Specifically, we prove upper bounds for the average fill-up (number of non-zero entries) of the boundary matrix on Erdős-Rényi and Vietoris-Rips filtrations after matrix reduction. Our bounds show that, in both cases, the reduced matrix is expected to be significantly sparser than what the general worst-case predicts. Our method is based on a link between the fillup of the boundary matrix and expected Betti numbers of random filtrations. Our bound for Vietoris-Rips complexes is asymptotically tight up to logarithmic factors. We also provide an Erdős-Rényi filtration realising the worst-case.
CCS Concepts• Mathematics of computing → Computations on matrices; Algebraic topology; • Theory of computation → Computational complexity and cryptography.
We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at the chain complex level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes several cases that usually TDA handles separately, such as persistence modules, zigzag modules, and commutative ladders. We extract new invariants in this category using a model structure and various minimal cofibrant approximations. Such approximations and their invariants retain some of the topological, and not just homological, aspects of the objects they approximate.
The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the situation for multiparameter persistence modules is more challenging, since there is no generalisation of the barcode.Here we introduce a general framework to study stability questions in multiparameter persistence. We introduce amplitudes -invariants that arise from assigning a non-negative real number to each persistence module, and which are monotone and subadditive in an appropriate sense -and then study different ways to associate distances to such invariants. Our framework is very comprehensive, as many different invariants that have been introduced in the Topological Data Analysis literature are examples of amplitudes, and furthermore many known distances for multiparameter persistence can be shown to be distances from amplitudes. Finally, we show how our framework can be used to prove new stability results.
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