Parallel computation of interval bases for persistence module decomposition
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866918187059118080 |
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| author | De Gregorio, Alessandro Guerra, Marco Scaramuccia, Sara Vaccarino, Francesco |
| author_facet | De Gregorio, Alessandro Guerra, Marco Scaramuccia, Sara Vaccarino, Francesco |
| contents | A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_11884 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Parallel computation of interval bases for persistence module decomposition De Gregorio, Alessandro Guerra, Marco Scaramuccia, Sara Vaccarino, Francesco Algebraic Topology Computational Geometry 13P20, 55N31, 62R40 A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition. |
| title | Parallel computation of interval bases for persistence module decomposition |
| topic | Algebraic Topology Computational Geometry 13P20, 55N31, 62R40 |
| url | https://arxiv.org/abs/2106.11884 |