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diff --git a/src/cython/doc/witness_complex_user.rst b/src/cython/doc/witness_complex_user.rst index 604c7357..071d8ef5 100644 --- a/src/cython/doc/witness_complex_user.rst +++ b/src/cython/doc/witness_complex_user.rst @@ -6,26 +6,80 @@ Definition .. include:: witness_complex_sum.rst + +Definitions +----------- + +Witness complex is a simplicial complex defined on two sets of points in :math:`\mathbb{R}^D`: + +- :math:`W` set of **witnesses** and +- :math:`L` set of **landmarks**. + +Even though often the set of landmarks :math:`L` is a subset of the set of witnesses :math:`W`, it is not a requirement +for the current implementation. + +Landmarks are the vertices of the simplicial complex and witnesses help to decide on which simplices are inserted via a +predicate "is witnessed". + +De Silva and Carlsson in their paper :cite:`de2004topological` differentiate **weak witnessing** and +**strong witnessing**: + +- *weak*: :math:`\sigma \subset L` is witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l) \leq d(w,l')` +- *strong*: :math:`\sigma \subset L` is witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l) \leq d(w,l')` + +where :math:`d(.,.)` is a distance function. + +Both definitions can be relaxed by a real value :math:`\alpha`: + +- *weak*: :math:`\sigma \subset L` is :math:`\alpha`-witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2` +- *strong*: :math:`\sigma \subset L` is :math:`\alpha`-witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2` + +which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and +**strong relaxed witness complex** respectively. + +.. figure:: img/swit.svg + :alt: Strongly witnessed simplex + :figclass: align-center + + Strongly witnessed simplex + + +In particular case of 0-relaxation, weak complex corresponds to **witness complex** introduced in +:cite:`de2004topological`, whereas 0-relaxed strong witness complex consists of just vertices and is not very +interesting. Hence for small relaxation weak version is preferable. +However, to capture the homotopy type (for example using Gudhi::persistent_cohomology::Persistent_cohomology) it is +often necessary to work with higher filtration values. In this case strong relaxed witness complex is faster to compute +and offers similar results. + Implementation -------------- -The principal class of this module is Gudhi::Witness_complex. +The two complexes described above are implemented in the corresponding classes + +- :doc:`witness_complex_ref` +- :doc:`strong_witness_complex_ref` +- :doc:`euclidean_witness_complex_ref` +- :doc:`euclidean_strong_witness_complex_ref` + +The construction of the Euclidean versions of complexes follow the same scheme: + +1. Construct a search tree on landmarks. +2. Construct lists of nearest landmarks for each witness. +3. Construct the witness complex for nearest landmark lists. + +In the non-Euclidean classes, the lists of nearest landmarks are supposed to be given as input. + +The constructors take on the steps 1 and 2, while the function 'create_complex' executes the step 3. + +Constructing weak relaxed witness complex from an off file +---------------------------------------------------------- + +Let's start with a simple example, which reads an off point file and computes a weak witness complex. -In both cases, the constructor for this class takes a {witness}x{closest_landmarks} table, where each row represents a -witness and consists of landmarks sorted by distance to this witness. +Example2: Computing persistence using strong relaxed witness complex +-------------------------------------------------------------------- -.. todo:: - This table can be constructed by two additional classes Landmark_choice_by_furthest_point and - Landmark_choice_by_random_point also included in the module. +Here is an example of constructing a strong witness complex filtration and computing persistence on it: -.. figure:: - img/bench_Cy8.png - :align: center - - Running time as function on number of landmarks. +.. include:: biblio.rst -.. figure:: - img/bench_sphere.png - :align: center - - Running time as function on number of witnesses for |L|=300. |