68 lines
2.0 KiB
ReStructuredText
68 lines
2.0 KiB
ReStructuredText
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Topological Data Analysis
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=========================
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The module :code:`topology` implements Vietoris--Rips persistent
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homology and the bottleneck distance between persistence diagrams.
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Vietoris--Rips Filtration
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-------------------------
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For a finite point cloud :math:`X = \{x_1, \dots, x_n\} \subset \mathbb{R}^d`
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and scale :math:`\varepsilon \ge 0`, the Vietoris--Rips complex is
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.. math::
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\mathrm{VR}_\varepsilon(X)
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\;=\;
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\big\{ \sigma \subseteq X : \mathrm{diam}(\sigma) \le \varepsilon \big\}.
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Increasing :math:`\varepsilon` yields a filtration; the persistent
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homology of this filtration produces, for each homological degree
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:math:`k`, a multiset of birth/death pairs
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.. math::
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D_k(X) = \big\{ (b_i, d_i) : 0 \le b_i < d_i \le \infty \big\}.
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Persistence Algorithm
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---------------------
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The boundary matrix :math:`\partial` is built over :math:`\mathbb{Z}/2`
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and reduced left-to-right: for each column :math:`j` we cancel its
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lowest entry by adding any earlier column with the same low. Pairs
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:math:`(\mathrm{low}(j), j)` give birth/death pairs.
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Bottleneck Distance
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-------------------
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For two diagrams :math:`D` and :math:`D'`,
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.. math::
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d_B(D, D')
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\;=\;
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\inf_{\eta : D \to D'}\;
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\sup_{x \in D}\, \|x - \eta(x)\|_\infty,
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where matchings may pair points with the diagonal
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:math:`\Delta = \{(t, t) : t \ge 0\}` at cost :math:`(d - b)/2`.
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The implementation binary-searches the threshold :math:`\varepsilon`
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and certifies a perfect matching by Hopcroft--Karp on the bipartite
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graph of admissible edges.
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API
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---
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.. code-block:: rust
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pub struct PersistencePair { pub dim: usize, pub birth: f64, pub death: f64 }
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pub struct PersistenceDiagram { pub pairs: Vec<PersistencePair> }
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pub fn vietoris_rips_filtration(points: &[Vec<f64>], max_dim: usize, max_eps: f64)
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-> Result<Vec<Simplex>>;
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pub fn persistent_homology(points: &[Vec<f64>], max_dim: usize, max_eps: f64)
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-> Result<PersistenceDiagram>;
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pub fn bottleneck_distance(d1: &[PersistencePair], d2: &[PersistencePair])
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-> Result<f64>;
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