22 research outputs found
Computing Topological Persistence for Simplicial Maps
Algorithms for persistent homology and zigzag persistent homology are
well-studied for persistence modules where homomorphisms are induced by
inclusion maps. In this paper, we propose a practical algorithm for computing
persistence under coefficients for a sequence of general
simplicial maps and show how these maps arise naturally in some applications of
topological data analysis.
First, we observe that it is not hard to simulate simplicial maps by
inclusion maps but not necessarily in a monotone direction. This, combined with
the known algorithms for zigzag persistence, provides an algorithm for
computing the persistence induced by simplicial maps.
Our main result is that the above simple minded approach can be improved for
a sequence of simplicial maps given in a monotone direction. A simplicial map
can be decomposed into a set of elementary inclusions and vertex collapses--two
atomic operations that can be supported efficiently with the notion of simplex
annotations for computing persistent homology. A consistent annotation through
these atomic operations implies the maintenance of a consistent cohomology
basis, hence a homology basis by duality. While the idea of maintaining a
cohomology basis through an inclusion is not new, maintaining them through a
vertex collapse is new, which constitutes an important atomic operation for
simulating simplicial maps. Annotations support the vertex collapse in addition
to the usual inclusion quite naturally.
Finally, we exhibit an application of this new tool in which we approximate
the persistence diagram of a filtration of Rips complexes where vertex
collapses are used to tame the blow-up in size.Comment: This is the revised and full version of the paper that is going to
appear in the Proceedings of 30th Annual Symposium on Computational Geometr
Dist2Cycle: A Simplicial Neural Network for Homology Localization
Simplicial complexes can be viewed as high dimensional generalizations of
graphs that explicitly encode multi-way ordered relations between vertices at
different resolutions, all at once. This concept is central towards detection
of higher dimensional topological features of data, features to which graphs,
encoding only pairwise relationships, remain oblivious. While attempts have
been made to extend Graph Neural Networks (GNNs) to a simplicial complex
setting, the methods do not inherently exploit, or reason about, the underlying
topological structure of the network. We propose a graph convolutional model
for learning functions parametrized by the -homological features of
simplicial complexes. By spectrally manipulating their combinatorial
-dimensional Hodge Laplacians, the proposed model enables learning
topological features of the underlying simplicial complexes, specifically, the
distance of each -simplex from the nearest "optimal" -th homology
generator, effectively providing an alternative to homology localization.Comment: 9 pages, 5 figure
Homological scaffold via minimal homology bases
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks
Computing multiparameter persistent homology through a discrete Morse-based approach
Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes persistent homology to allow for the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent multiparameter persistent homology to be a feasible tool for analyzing large size data sets. We consider discrete Morse theory as a strategy to reduce the computation of multiparameter persistent homology by working on a reduced dataset. We propose a new preprocessing algorithm, well suited for parallel and distributed implementations, and we provide the first evaluation of the impact of multiparameter persistent homology on computations
Persistent Laplacians: properties, algorithms and implications
We present a thorough study of the theoretical properties and devise
efficient algorithms for the \emph{persistent Laplacian}, an extension of the
standard combinatorial Laplacian to the setting of pairs (or, in more
generality, sequences) of simplicial complexes , which was
recently introduced by Wang, Nguyen, and Wei. In particular, in analogy with
the non-persistent case, we first prove that the nullity of the -th
persistent Laplacian equals the -th persistent Betti number
of the inclusion . We then present an initial algorithm
for finding a matrix representation of , which itself helps
interpret the persistent Laplacian. We exhibit a novel relationship between the
persistent Laplacian and the notion of Schur complement of a matrix which has
several important implications. In the graph case, it both uncovers a link with
the notion of effective resistance and leads to a persistent version of the
Cheeger inequality. This relationship also yields an additional, very simple
algorithm for finding (a matrix representation of) the -th persistent
Laplacian which in turn leads to a novel and fundamentally different algorithm
for computing the -th persistent Betti number for a pair which can
be significantly more efficient than standard algorithms. Finally, we study
persistent Laplacians for simplicial filtrations and present novel stability
results for their eigenvalues. Our work brings methods from spectral graph
theory, circuit theory, and persistent homology together with a topological
view of the combinatorial Laplacian on simplicial complexes