In my previous blog post, I introduced the newly emerged topological data analysis (TDA). Unlike most of the other data analytic algorithms, TDA, concerning the topology as its name tells, cares for the connectivity of points, instead of the distance (according to a metric, whether it is Euclidean, Manhattan, Minkowski or any other). What is the best tools to describe topology?
Physicists use a lot homotopy. But for the sake of computation, it is better to use a scheme that are suited for discrete computation. It turns out that there are useful tools in algebraic topology: homology. But to understand homology, we need to understand what a simplicial complex is.
Gunnar Carlsson [Carlsson 2009] and Afra Zomorodian [Zomorodian 2011] wrote good reviews about them, although from a different path in introducing the concept. I first followed Zomorodian’s review [Zomorodian 2011], then his book [Zomorodian 2009] that filled in a lot of missing links in his review, to a certain point. I recently started reading Carlsson’s review.
One must first understand what a simplicial complex is. Without giving too much technical details, simplicial complex is basically a shape connecting points together. A line is a 1-simplex, connecting two points. A triangle is a 2-simplex. A tetrahedron is a 3-complex. There are other more complicated and unnamed complexes. Any subsets of a simplicial complex are faces. For example, the sides of the triangle are faces. The faces and the sides are the faces of the tetrahedron. (Refer to Wolfram MathWorld for more details. There are a lot of good tutorials online.)
Implementing Simplicial Complex
We can easily encoded this into a python code. I wrote a class SimplicialComplex in Python to implement this. We first import necessary libraries:
import numpy as np from itertools import combinations from scipy.sparse import dok_matrix from operator import add
The first line imports the numpy library, the second the iteration tools necessary for extracting the faces for simplicial complex, the third the sparse matrix implementation in the scipy library (applied on something that I will not go over in this blog entry), and the fourth for some reduce operation.
We want to describe the simplicial complexes in the order of some labels (which can be anything, such as integers or strings). If it is a point, then it can be represented as tuples, as below:
(1,)
Or if it is a line (a 1-simplex), then
(1, 2)
Or a 2-simplex as a triangle, then
(1, 2, 3)
I think you get the gist. The integers 1, 2, or 3 here are simply labels. We can easily store this in the class:
class SimplicialComplex:
def __init__(self, simplices=[]):
self.import_simplices(simplices=simplices)
def import_simplices(self, simplices=[]):
self.simplices = map(lambda simplex: tuple(sorted(simplex)), simplices)
self.face_set = self.faces()
You might observe the last line of the codes above. And it is for calculating all the faces of this complex, and it is implemented in this way:
def faces(self):
faceset = set()
for simplex in self.simplices:
numnodes = len(simplex)
for r in range(numnodes, 0, -1):
for face in combinations(simplex, r):
faceset.add(face)
return faceset
The faces are intuitively sides of a 2D shape (2-simplex), or planes of a 3D shape (3-simplex). But the faces of a 3-simplex includes the faces of all its faces. All the faces are saved in a field called faceset. If the user wants to retrieve the faces in a particular dimension, they can call this method:
def n_faces(self, n):
return filter(lambda face: len(face)==n+1, self.face_set)
There are other methods that I am not going over in this blog entry. Now let us demonstrate how to use the class by implementing a tetrahedron.
sc = SimplicialComplex([('a', 'b', 'c', 'd')])
If we want to extract the faces, then enter:
sc.faces()
which outputs:
{('a',),
('a', 'b'),
('a', 'b', 'c'),
('a', 'b', 'c', 'd'),
('a', 'b', 'd'),
('a', 'c'),
('a', 'c', 'd'),
('a', 'd'),
('b',),
('b', 'c'),
('b', 'c', 'd'),
('b', 'd'),
('c',),
('c', 'd'),
('d',)}
We have gone over the basis of simplicial complex, which is the foundation of TDA. We appreciate that the simplicial complex deals only with the connectivity of points instead of the distances between the points. And the homology groups will be calculated based on this. However, how do we obtain the simplicial complex from the discrete data we have? Zomorodian’s review [Zomorodian 2011] gave a number of examples, but I will only go through two of them only. And from this, you can see that to establish the connectivity between points, we still need to apply some sort of distance metrics.
Alpha Complex
An alpha complex is the nerve of the cover of the restricted Voronoi regions. (Refer the details to Zomorodian’s review [Zomorodian 2011], this Wolfram MathWorld entry, or this Wolfram Demonstration.) We can extend the class SimplicialComplex to get a class AlphaComplex:
from scipy.spatial import Delaunay, distance
from operator import or_
from functools import partial
def facesiter(simplex):
for i in range(len(simplex)):
yield simplex[:i]+simplex[(i+1):]
def flattening_simplex(simplices):
for simplex in simplices:
for point in simplex:
yield point
def get_allpoints(simplices):
return set(flattening_simplex(simplices))
def contain_detachededges(simplex, distdict, epsilon):
if len(simplex)==2:
return (distdict[simplex[0], simplex[1]] > 2*epsilon)
else:
return reduce(or_, map(partial(contain_detachededges, distdict=distdict, epsilon=epsilon), facesiter(simplex)))
class AlphaComplex(SimplicialComplex):
def __init__(self, points, epsilon, labels=None, distfcn=distance.euclidean):
self.pts = points
self.labels = range(len(self.pts)) if labels==None or len(labels)!=len(self.pts) else labels
self.epsilon = epsilon
self.distfcn = distfcn
self.import_simplices(self.construct_simplices(self.pts, self.labels, self.epsilon, self.distfcn))
def calculate_distmatrix(self, points, labels, distfcn):
distdict = {}
for i in range(len(labels)):
for j in range(len(labels)):
distdict[(labels[i], labels[j])] = distfcn(points[i], points[j])
return distdict
def construct_simplices(self, points, labels, epsilon, distfcn):
delaunay = Delaunay(points)
delaunay_simplices = map(tuple, delaunay.simplices)
distdict = self.calculate_distmatrix(points, labels, distfcn)
simplices = []
for simplex in delaunay_simplices:
faces = list(facesiter(simplex))
detached = map(partial(contain_detachededges, distdict=distdict, epsilon=epsilon), faces)
if reduce(or_, detached):
if len(simplex)>2:
for face, notkeep in zip(faces, detached):
if not notkeep:
simplices.append(face)
else:
simplices.append(simplex)
simplices = map(lambda simplex: tuple(sorted(simplex)), simplices)
simplices = list(set(simplices))
allpts = get_allpoints(simplices)
for point in (set(labels)-allpts):
simplices += [(point,)]
return simplices
The scipy package already has a package to calculate Delaunay region. The function contain_detachededges is for constructing the restricted Voronoi region from the calculated Delaunay region.
This class demonstrates how an Alpha Complex is constructed, but this runs slowly once the number of points gets big!
Vietoris-Rips (VR) Complex
Another commonly used complex is called the Vietoris-Rips (VR) Complex, which connects points as the edge of a graph if they are close enough. (Refer to Zomorodian’s review [Zomorodian 2011] or this Wikipedia page for details.) To implement this, import the famous networkx originally designed for network analysis.
import networkx as nx
from scipy.spatial import distance
from itertools import product
class VietorisRipsComplex(SimplicialComplex):
def __init__(self, points, epsilon, labels=None, distfcn=distance.euclidean):
self.pts = points
self.labels = range(len(self.pts)) if labels==None or len(labels)!=len(self.pts) else labels
self.epsilon = epsilon
self.distfcn = distfcn
self.network = self.construct_network(self.pts, self.labels, self.epsilon, self.distfcn)
self.import_simplices(map(tuple, list(nx.find_cliques(self.network))))
def construct_network(self, points, labels, epsilon, distfcn):
g = nx.Graph()
g.add_nodes_from(labels)
zips = zip(points, labels)
for pair in product(zips, zips):
if pair[0][1]!=pair[1][1]:
dist = distfcn(pair[0][0], pair[1][0])
if dist<epsilon:
g.add_edge(pair[0][1], pair[1][1])
return g
The intuitiveness and efficiencies are the reasons that VR complexes are widely used.
For more details about the Alpha Complexes, VR Complexes and the related Čech Complexes, refer to this page.
More…
There are other commonly used complexes used, including Witness Complex, Cubical Complex etc., which I leave no introductions. Upon building the complexes, we can analyze the topology by calculating their homology groups, Betti numbers, the persistent homology etc. I wish to write more about it soon.
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