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Diffstat (limited to 'src/Rips_complex/doc/Intro_rips_complex.h')
| -rw-r--r-- | src/Rips_complex/doc/Intro_rips_complex.h | 29 |
1 files changed, 19 insertions, 10 deletions
diff --git a/src/Rips_complex/doc/Intro_rips_complex.h b/src/Rips_complex/doc/Intro_rips_complex.h index 712d3b6e..97d66fbd 100644 --- a/src/Rips_complex/doc/Intro_rips_complex.h +++ b/src/Rips_complex/doc/Intro_rips_complex.h @@ -39,7 +39,7 @@ namespace rips_complex { * <a target="_blank" href="https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex">(Wikipedia)</a> * is an abstract simplicial complex * defined on a finite metric space, where each simplex corresponds to a subset - * of point whose diameter is smaller that some given threshold. + * of points whose diameter is smaller that some given threshold. * Varying the threshold, we can also see the Rips complex as a filtration of * the \f$(n-1)-\f$dimensional simplex, where the filtration value of each * simplex is the diameter of the corresponding subset of points. @@ -53,7 +53,10 @@ namespace rips_complex { * The number of simplices in the full Rips complex is exponential in the * number of vertices, it is thus usually restricted, by excluding all the * simplices with filtration value larger than some threshold, and keeping only - * the dim_max-skeleton. + * the dim_max-skeleton. It may also be a good idea to start by making the + * point set sparser, for instance with + * `Gudhi::subsampling::sparsify_point_set()`, since small clusters of points + * have a disproportionate cost without affecting the persistence diagram much. * * In order to build this complex, the algorithm first builds the graph. * The filtration value of each edge is computed from a user-given distance @@ -89,21 +92,27 @@ namespace rips_complex { * The sparse Rips filtration was introduced by Don Sheehy * \cite sheehy13linear. We are using the version described in * \cite buchet16efficient (except that we multiply all filtration values - * by 2, to match the usual Rips complex), which proves a - * \f$\frac{1+\epsilon}{1-\epsilon}\f$-interleaving, although in practice the + * by 2, to match the usual Rips complex), for which \cite cavanna15geometric proves a + * \f$(1,\frac{1}{1-\epsilon})\f$-interleaving, although in practice the * error is usually smaller. * A more intuitive presentation of the idea is available in * \cite cavanna15geometric, and in a video \cite cavanna15visualizing. * * The interface of `Sparse_rips_complex` is similar to the one for the usual - * `Rips_complex`, except that one has to specify the approximation factor, and - * there is no option to limit the maximum filtration value (the way the - * approximation is done means that larger filtration values are much cheaper - * to handle than low filtration values, so the gain would be too small). + * `Rips_complex`, except that one has to specify the approximation factor. + * There is an option to limit the minimum and maximum filtration values, but + * they are not recommended: the way the approximation is done means that + * larger filtration values are much cheaper to handle than low filtration + * values, so the gain in ignoring the large ones is small, and + * `Gudhi::subsampling::sparsify_point_set()` is a more efficient way of + * ignoring small filtration values. * * Theoretical guarantees are only available for \f$\epsilon<1\f$. The * construction accepts larger values of ε, and the size of the complex * keeps decreasing, but there is no guarantee on the quality of the result. + * Note that while the number of edges decreases when ε increases, the + * number of higher-dimensional simplices may not be monotonous when + * \f$\frac12\leq\epsilon\leq 1\f$. * * \section ripspointsdistance Point cloud and distance function * @@ -116,8 +125,8 @@ namespace rips_complex { * * \include Rips_complex/example_one_skeleton_rips_from_points.cpp * - * When launching (Rips maximal distance between 2 points is 12.0, is expanded until dimension 1 - one skeleton graph - * in other words): + * When launching (Rips maximal distance between 2 points is 12.0, is expanded + * until dimension 1 - one skeleton graph in other words): * * \code $> ./Rips_complex_example_one_skeleton_from_points * \endcode |