def rips_chain_complex(simplices):
"""Construct the boundary operators for the Rips complex
Constructs the boundary operators for the Rips complex for
a given a list of simplex arrays.
Parameters
----------
simplices : list of arrays
List of length D, where simplices[i] is the simplex
array representing the i-dimensional simplices.
Returns
-------
Bs : list of sparse matrices
List of length D + 1 where Bs[i] is the boundary operator
for the i-dimensional simplices.
Examples
--------
TODO
"""
Bs = []
dtype = 'float32'
simplices = [asarray(x) for x in simplices]
B0 = csc_matrix( (1, len(simplices[0]) ), dtype=dtype )
Bs.append( B0 )
if len(simplices) == 1:
B1 = csc_matrix( (len(simplices[0]), 1), dtype=dtype )
Bs.append(B1)
return Bs
B1 = empty( 2*len(simplices[1]), dtype=dtype )
B1[0::2] = -1
B1[1::2] = 1
B1 = csc_matrix( ( B1, simplices[1].ravel(), arange(0, 2*(len(simplices[1]) + 1), 2)), \
shape=(len(simplices[0]), len(simplices[1])) )
Bs.append( B1 )
for f,s in zip(simplices[1:-1],simplices[2:]):
k = f.shape[1]
faces = empty(( len(s), (k+1), k),dtype=s.dtype)
for i in range(k+1):
faces[:,i,:i] = s[:, : i ]
faces[:,i,i:] = s[:,i+1: ]
indptr = arange(0, (k+1)*(len(s) + 1), k+1)
indices = simplex_array_searchsorted(f, faces.reshape(-1,k) )
data = empty( faces.shape[:-1], dtype=dtype )
for i in range(k+1):
data[:,i] = (-1)**i
data = data.ravel()
Bs.append( csc_matrix( (data,indices,indptr), shape=(len(f),len(s)) ) )
return Bs
def __init__(self, simplices):
"""Construct an abstract simplicial complex
Parameters
----------
simplices : list of arrays
Maximal simplices of each dimension
TODO
Examples
--------
>>> from pydec.dec import abstract_simplicial_complex
>>> from numpy import array
>>> simplices = [array([[4]]), array([[0,3]])]
>>> asc = abstract_simplicial_complex(simplices)
TODO
>>> print rc.simplices[0]
>>> print rc.simplices[1]
Notes
-----
TODO explain input handling
"""
# convert array-like objects to arrays
simplices = [atleast_2d(s) for s in simplices]
# find top simplex dimension
D = max([s.shape[1] for s in simplices]) - 1
# convert simplices list to canonical simplex_array list representation
old_simplices = simplices
simplices = [None] * (D + 1)
for s in old_simplices:
simplices[s.shape[1] - 1] = s
# top most simplex array
s = simplices[-1].copy()
parity = simplex_array_parity(s)
s.sort()
chain_complex = [None] * (D + 1)
for d in range(D - 1, -1, -1):
s,B = simplex_array_boundary(s,parity)
if simplices[d] is not None:
old_s = s
# sort columns to ensure user-defined faces are in canonical format
simplices[d].sort()
# merge user-defined faces with boundary faces
s = vstack((s,simplices[d]))
# sort rows to bring equivalent elements together
s = s[lexsort(s.T[::-1])]
# find unique simplices
mask = -hstack((array([False]),alltrue(s[1:] == s[:-1],axis=1)))
s = s[mask]
# indices of the boundary faces in the full face array
remap = simplex_array_searchsorted(s, old_s)
# remap indices of boundary operator
B = B.tocoo(copy=False)
B = sparse.coo_matrix((B.data, (remap[B.row], B.col)), (s.shape[0], B.shape[1]))
B = B.tocsr()
# faces are already in canonical format, so parity is even
parity = zeros(s.shape[0],dtype=s.dtype)
simplices[d] = s
chain_complex[d+1] = B
# compute 0-simplices and boundary operator
simplices[0] = arange(simplices[0].max() + 1).reshape(-1,1)
chain_complex[0] = sparse.csr_matrix( (1,len(s)), dtype='uint8')
# store the cochain complex
Bn = chain_complex[-1]
cochain_complex = [ B.T for B in chain_complex[1:] ]
cochain_complex += [ sparse.csc_matrix( (1, Bn.shape[1]), dtype=Bn.dtype) ]
# store the data members
self.simplices = simplices
self._chain_complex = chain_complex
self._cochain_complex = cochain_complex
