55 research outputs found
Induced Matchings and the Algebraic Stability of Persistence Barcodes
We define a simple, explicit map sending a morphism of
pointwise finite dimensional persistence modules to a matching between the
barcodes of and . Our main result is that, in a precise sense, the
quality of this matching is tightly controlled by the lengths of the longest
intervals in the barcodes of and . As an
immediate corollary, we obtain a new proof of the algebraic stability of
persistence, a fundamental result in the theory of persistent homology. In
contrast to previous proofs, ours shows explicitly how a -interleaving
morphism between two persistence modules induces a -matching between
the barcodes of the two modules. Our main result also specializes to a
structure theorem for submodules and quotients of persistence modules, and
yields a novel "single-morphism" characterization of the interleaving relation
on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational
Geometry. Includes a proof that no definition of induced matching can be
fully functorial (Proposition 5.10), and an extension of our single-morphism
characterization of the interleaving relation to multidimensional persistence
modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde
Multidimensional Interleavings and Applications to Topological Inference
This work concerns the theoretical foundations of persistence-based
topological data analysis. We develop theory of topological inference in the
multidimensional persistence setting, and directly at the (topological) level
of filtrations rather than only at the (algebraic) level of persistent homology
modules.
Our main mathematical objects of study are interleavings. These are tools for
quantifying the similarity between two multidimensional filtrations or
persistence modules. They were introduced for 1-D filtrations and persistence
modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce
generalizations of the definitions of interleavings given by Chazal et al. and
use these to define pseudometrics, called interleaving distances, on
multidimensional filtrations and multidimensional persistence modules.
We present an in-depth study of interleavings and interleaving distances. We
then use them to formulate and prove several multidimensional analogues of a
topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These
results hold directly at the level of filtrations; they yield as corollaries
corresponding results at the module level.Comment: Late stage draft of Ph.D. thesis. 176 pages. Expands upon content in
arXiv:1106.530
Exact Computation of the Matching Distance on 2-Parameter Persistence Modules
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational
Computing the Multicover Bifiltration
Given a finite set , let Cov denote the set of
all points within distance to at least points of . Allowing and
to vary, we obtain a 2-parameter family of spaces that grow larger when
increases or decreases, called the \emph{multicover bifiltration}.
Motivated by the problem of computing the homology of this bifiltration, we
introduce two closely related combinatorial bifiltrations, one polyhedral and
the other simplicial, which are both topologically equivalent to the multicover
bifiltration and far smaller than a \v Cech-based model considered in prior
work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid
tiling of Edelsbrunner and Osang, and can be efficiently computed using a
variant of an algorithm given by these authors. Using an implementation for
dimension 2 and 3, we provide experimental results. Our simplicial construction
is useful for understanding the polyhedral construction and proving its
correctness.Comment: 25 pages, 8 figures, 4 tables. Extended version of a paper accepted
to the 2021 Symposium on Computational Geometr
Efficient two-parameter persistence computation via cohomology
Clearing is a simple but effective optimization for the standard algorithm of
persistent homology (PH), which dramatically improves the speed and scalability
of PH computations for Vietoris--Rips filtrations. Due to the quick growth of
the boundary matrices of a Vietoris--Rips filtration with increasing dimension,
clearing is only effective when used in conjunction with a dual (cohomological)
variant of the standard algorithm. This approach has not previously been
applied successfully to the computation of two-parameter PH.
We introduce a cohomological algorithm for computing minimal free resolutions
of two-parameter PH that allows for clearing. To derive our algorithm, we
extend the duality principles which underlie the one-parameter approach to the
two-parameter setting. We provide an implementation and report experimental run
times for function-Rips filtrations. Our method is faster than the current
state-of-the-art by a factor of up to 20.Comment: This is an extended version of a conference paper that appeared at
SoCG 2023, see https://drops.dagstuhl.de/opus/volltexte/2023/1786
LIPIcs
Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness
- …