913 research outputs found
Combinatorial presentation of multidimensional persistent homology
A multifiltration is a functor indexed by that maps any
morphism to a monomorphism. The goal of this paper is to describe in an
explicit and combinatorial way the natural -graded -module structure on the homology of a multifiltration of simplicial
complexes. To do that we study multifiltrations of sets and vector spaces. We
prove in particular that the -graded -modules
that can occur as -spans of multifiltrations of sets are the direct sums of
monomial ideals.Comment: 21 pages, 3 figure
Stratifying multiparameter persistent homology
A fundamental tool in topological data analysis is persistent homology, which
allows extraction of information from complex datasets in a robust way.
Persistent homology assigns a module over a principal ideal domain to a
one-parameter family of spaces obtained from the data. In applications data
often depend on several parameters, and in this case one is interested in
studying the persistent homology of a multiparameter family of spaces
associated to the data. While the theory of persistent homology for
one-parameter families is well-understood, the situation for multiparameter
families is more delicate. Following Carlsson and Zomorodian we recast the
problem in the setting of multigraded algebra, and we propose multigraded
Hilbert series, multigraded associated primes and local cohomology as
invariants for studying multiparameter persistent homology. Multigraded
associated primes provide a stratification of the region where a multigraded
module does not vanish, while multigraded Hilbert series and local cohomology
give a measure of the size of components of the module supported on different
strata. These invariants generalize in a suitable sense the invariant for the
one-parameter case.Comment: Minor improvements throughout. In particular: we extended the
introduction, added Table 1, which gives a dictionary between terms used in
PH and commutative algebra; we streamlined Section 3; we added Proposition
4.49 about the information captured by the cp-rank; we moved the code from
the appendix to github. Final version, to appear in SIAG
Chunk Reduction for Multi-Parameter Persistent Homology
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation
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