doc/python/rips.rst
 author Dmitriy Morozov Thu, 14 May 2009 17:43:19 -0700 branch dev changeset 140 9851fee5a33b parent 134 c270826fd4a8 child 147 d39a20acb253 permissions -rw-r--r--
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:class:Rips class
======================

.. class:: Rips

.. method:: __init__(distances)

Initializes :class:Rips with the given distances whose main purpose
is to return the distance of two points given their indices. See
Distances_ below.

.. method:: generate(k, max, functor[, seq])

Calls functor with every simplex in the k-skeleton of the Rips
complex :math:VR (max). If seq is provided, then the complex is
restricted to the vertex indices in the sequence.

.. method:: vertex_coface(v, k, max, functor[, seq])

Calls functor with every coface of the vertex v in the k-skeleton
of the Rips complex :math:VR (max). If seq is provided, then the
complex is restricted to the vertex indices in the sequence.

.. method:: edge_cofaces(u, v, k, max, functor[, seq])

Calls functor with every coface of the edge (u, v) in the
k-skeleton of the Rips complex :math:VR (max). If seq is
provided, then the complex is restricted to the vertex indices in the
sequence.

.. method:: cmp(s1, s2)

Compares simplices s1 and s2 with respect to their ordering in the
Rips complex.  Note that like Python's built in cmp this is a three
possible outsome comparison (-1,0,1) for (:math:\leq, =, \geq,
respectively).

.. method:: eval(s)

Returns the size of simplex s, i.e. the length of its longest edge.

.. _distances:

Distances
---------

An instance of distances passed to the constructor of :class:Rips should
know its length and the distances between the points. The length should be
retrievable via len(distance) and it determines how many points the complex
is built on. The distances between the points are inferred by the class
:class:Rips by calling distances with a pair of vertices as arguments.

For example, the following class represents 10 points on an integer lattice::

class Distances:
def __len__(self):
return 10

def __call__(self, x, y):
return math.fabs(y-x)

The bindings provide a pure Python class :class:PairwiseDistances to deal with
explicit points in a Euclidean space. It is defined in
:sfile:bindings/python/dionysus/distances.py::

class PairwiseDistances:
def __init__(self, points, norm = l2):
self.points = points
self.norm = norm

def __len__(self):
return len(self.points)

def __call__(self, x, y):
return self.norm([x - y for (x,y) in zip(self.points[p1], self.points[p2])])

Another distances class is available that speeds up the computation of the Rips
complex at the expense of the memory usage: :class:ExplicitDistances. It is
initialized with an instance of any class that behaves like a distances class,
and it stores all of its distances explicitly to not have to recompute them in
the future::

distances = PairwiseDistances(points)
distances = ExplicitDistances(distances)

Example
-------

The following example reads in points from a file, and fills the list
simplices with the simplices of the 2-skeleton of the Rips complex built on
those vertices with distance cutoff parameter 50. Subsequently it computes the
persistence of the resulting filtration (defined by rips.cmp)::

points = [for p in points_file('...')]
distances = PairwiseDistances(points)
rips = Rips(distances)
simplices = []
rips.generate(2, 50, simplices.append)

f = Filtration(simplices, rips.cmp)
p = StaticPersistence(f)
p.pair_simplices()

Essentially the same example is implemented in
:sfile:examples/rips/rips-pairwise.py, although it reads the k and max
parameters for the Rips complex on the command line, and uses a trick to speed
up the computation.