Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions
In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes t...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
14.07.2021
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Subjects | |
Online Access | Get full text |
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Summary: | In this paper, we introduce the signed barcode, a new visual representation
of the global structure of the rank invariant of a multi-parameter persistence
module or, more generally, of a poset representation. Like its unsigned
counterpart in one-parameter persistence, the signed barcode decomposes the
rank invariant as a $\Z$-linear combination of rank invariants of indicator
modules supported on segments in the poset. We develop the theory behind these
decompositions, both for the usual rank invariant and for its generalizations,
showing under what conditions they exist and are unique. We also show that,
like its unsigned counterpart, the signed barcode reflects in part the
algebraic structure of the module: specifically, it derives from the terms in
the minimal rank-exact resolution of the module, i.e., its minimal projective
resolution relative to the class of short exact sequences on which the rank
invariant is additive. To complete the picture, we show some experimental
results that illustrate the contribution of the signed barcode in the
exploration of multi-parameter persistence modules. |
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DOI: | 10.48550/arxiv.2107.06800 |