Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent Homology
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Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K[x,y]-module M, where K is a field. The algorithm assumes that M is given implicitly: It takes as input a short chain complex of free bipersistence modules F^2 ∂^2 F^1 ∂^1 F^0 such that M∂^1/im∂^2. The algorithm runs in time O(∑_i |F^i|^3) and requires O(∑_i |F^i|^2) storage, where |F^i| denotes the size of a basis of F^i. Given the presentation, the bigraded Betti numbers of the module are readily computed. We also present a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In preliminary experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.
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