428 research outputs found
Persistence Modules on Commutative Ladders of Finite Type
We study persistence modules defined on commutative ladders. This class of
persistence modules frequently appears in topological data analysis, and the
theory and algorithm proposed in this paper can be applied to these practical
problems. A new algebraic framework deals with persistence modules as
representations on associative algebras and the Auslander-Reiten theory is
applied to develop the theoretical and algorithmic foundations. In particular,
we prove that the commutative ladders of length less than 5 are
representation-finite and explicitly show their Auslander-Reiten quivers.
Furthermore, a generalization of persistence diagrams is introduced by using
Auslander-Reiten quivers. We provide an algorithm for computing persistence
diagrams for the commutative ladders of length 3 by using the structure of
Auslander-Reiten quivers.Comment: 48 page
Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension
While persistent homology has taken strides towards becoming a widespread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies
Summand-injectivity of interval covers and monotonicity of interval resolution global dimensions
Recently, there is growing interest in the use of relative homology algebra
to develop invariants using interval covers and interval resolutions (i.e.,
right minimal approximations and resolutions relative to interval-decomposable
modules) for multi-parameter persistence modules. In this paper, the set of all
interval modules over a given poset plays a central role. Firstly, we show that
the restriction of interval covers of modules to each indecomposable direct
summand is injective. This result suggests a way to simplify the computation of
interval covers. Secondly, we show the monotonicity of the interval resolution
global dimension, i.e., if is a full subposet of , then the interval
resolution global dimension of is not larger than that of . Finally, we
provide a complete classification of posets whose interval resolution global
dimension is zero.Comment: 23 page
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