Persistence module
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features of an object across a range of scale parameters.
A persistence module often consists of a collection of homology groups (or vector spaces if using field coefficients) corresponding to a filtration of topological spaces, and a collection of linear maps induced by the inclusions of the filtration.
The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology.
[1] Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology.
[8] A single-parameter persistence module indexed by a discrete poset such as the integers can be represented intuitively as a diagram of spaces:
[9] Common choices of indexing sets include
One can alternatively use a set-theoretic definition of a persistence module that is equivalent to the categorical viewpoint: A persistence module is a pair
, we can define a multiparameter persistence module indexed by
[10] Multidimensional persistence modules were first introduced in 2009 by Carlsson and Zomorodian.
[11] Since then, there has been a significant amount of research into the theory and practice of working with multidimensional modules, since they provide more structure for studying the shape of data.
[12][13][14] Namely, multiparameter modules can have greater density sensitivity and robustness to outliers than single-parameter modules, making them a potentially useful tool for data analysis.
[15][16][17] One downside of multiparameter persistence is its inherent complexity.
This makes performing computations related to multiparameter persistence modules difficult.
In the worst case, the computational complexity of multidimensional persistent homology is exponential.
, by applying the homology functor at each index we obtain a persistence module
The vector spaces of the homology module can be defined index-wise as
[1] Homology modules are the most ubiquitous examples of persistence modules, as they encode information about the number and scale of topological features of an object (usually derived from building a filtration on a point cloud) in a purely algebraic structure, thus making understanding the shape of the data amenable to algebraic techniques, imported from well-developed areas of mathematics such as commutative algebra and representation theory.
[24][25][26] The notion of pointwise finite-dimensionality immediately extends to arbitrary indexing sets.
The definition of finite type can also be adapted to continuous indexing sets.
contains a finite number of unique vector spaces.
[27] Formally speaking, this requires that for all but a finite number of points
[4] A module satisfying only the former property is sometimes labeled essentially discrete, whereas a module satisfying both properties is known as essentially finite.
Note that this condition is redundant if the other finite type conditions above are satisfied, so it is not typically included in the definition, but is relevant in certain circumstances.
A persistence module that admits a decomposition as a direct sum of interval modules is often simply called "interval decomposable."
persistence module indexed over a totally ordered set is interval decomposable.
This is sometimes referred to as the "structure theorem for persistence modules.
is finite is a straightforward application of the structure theorem for finitely generated modules over a principal ideal domain.
, the first known proof of the structure theorem is due to Webb.
(or any totally ordered set containing a countable subset that is dense in
modules indexed over arbitrary totally ordered sets, was established by Botnan and Crawley-Boevey in 2019.
