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- W3217506786 abstract "In the field of topological data analysis, persistence modules are used to express geometrical features of data sets. The matching distance $d_mathcal{M}$ measures the difference between $2$-parameter persistence modules by taking the maximum bottleneck distance between $1$-parameter slices of the modules. The previous fastest algorithm to compute $d_mathcal{M}$ exactly runs in $O(n^{8+omega})$, where $omega$ is the matrix multiplication constant. We improve significantly on this by describing an algorithm with expected running time $O(n^5 log^3 n)$. We first solve the decision problem $d_mathcal{M}leq lambda$ for a constant $lambda$ in $O(n^5log n)$ by traversing a line arrangement in the dual plane, where each point represents a slice. Then we lift the line arrangement to a plane arrangement in $mathbb{R}^3$ whose vertices represent possible values for $d_mathcal{M}$, and use a randomized incremental method to search through the vertices and find $d_mathcal{M}$. The expected running time of this algorithm is $O((n^4+T(n))log^2 n)$, where $T(n)$ is an upper bound for the complexity of deciding if $d_mathcal{M}leq lambda$." @default.
- W3217506786 created "2021-12-06" @default.
- W3217506786 creator A5047215403 @default.
- W3217506786 creator A5086819896 @default.
- W3217506786 date "2021-11-19" @default.
- W3217506786 modified "2023-09-27" @default.
- W3217506786 title "Asymptotic Improvements on the Exact Matching Distance for 2-parameter Persistence." @default.
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