Matrix method for persistence modules on commutative ladders of finite type
The theory of persistence modules on the commutative ladders \(CL_n(\tau )\) provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module M on \(CL_n(\tau )\) as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case (\(n\le 4\)), we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of M, and thus its persistence diagram, is obtained.
KeywordsPersistence modules Commutative ladders Computational topology Algorithms
Mathematics Subject Classification68W30 16G20 55N99
On behalf of all authors, the corresponding author states that there is no conflict of interest. This work was partially supported by JST CREST Mathematics 15656429. H.T. is supported by JSPS KAKENHI Grant number JP16J03138.
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