Fast (< n^2) clustering algorithm

Question

I have 1 million 5-dimensional points that I need to group into k clusters with k << 1 million. In each cluster, no two points should be too far apart (e.g. they could be

Answer 1

Below is a little test bench to see how fast scipy.spatial.cKDTree is on your data, and to get a rough idea of how the distances between nearby points scatter.

A nice way to run K-cluster for various K is to build an MST of nearest pairs, and remove the K-1 longest; see Wayne, Greedy Algorithms .

Visualizing the clusters would be fun -- project to 2d with PCA ?

(Just curious, is your K 10, 100, 1000 ?)

Added 17 Dec: real runtimes: 100000 x 5 10 sec, 500000 x 5 60sec

#!/usr/bin/env python
# time scipy.spatial.cKDTree build, query

from __future__ import division
import random
import sys
import time
import numpy as np
from scipy.spatial import cKDTree as KDTree
    # http://docs.scipy.org/doc/scipy/reference/spatial.html
    # $scipy/spatial/kdtree.py is slow but clean, 0.9 has cython
__date__ = "2010-12-17 dec denis"

def clumpiness( X, nbin=10 ):
    """ how clumpy is X ? histogramdd av, max """
        # effect on kdtree time ? not much
    N, dim = X.shape
    histo = np.histogramdd( X, nbin )[0] .astype(int)  # 10^dim
    n0 = histo.size - histo.astype(bool).sum()  # uniform: 1/e^lambda
    print "clumpiness: %d of %d^%d data bins are empty  av %.2g  max %d" % (
        n0, nbin, dim, histo.mean(), histo.max())

#...............................................................................
N = 100000
nask = 0  # 0: ask all N
dim = 5
rnormal = .9
    # KDtree params --
nnear = 2  # k=nnear+1, self
leafsize = 10
eps = 1  # approximate nearest, dist <= (1 + eps) * true nearest
seed = 1

exec "\n".join( sys.argv[1:] )  # run this.py N= ...
np.random.seed(seed)
np.set_printoptions( 2, threshold=200, suppress=True )  # .2f
nask = nask or N
print "\nkdtree:  dim=%d  N=%d  nask=%d  nnear=%d  rnormal=%.2g  leafsize=%d  eps=%.2g" % (
    dim, N, nask, nnear, rnormal, leafsize, eps)

if rnormal > 0:  # normal point cloud, .9 => many near 1 1 1 axis
    cov = rnormal * np.ones((dim,dim)) + (1 - rnormal) * np.eye(dim)
    data = np.abs( np.random.multivariate_normal( np.zeros(dim), cov, N )) % 1
        # % 1: wrap to unit cube
else:
    data = np.random.uniform( size=(N,dim) )
clumpiness(data)
ask = data if nask == N  else random.sample( data, sample )
t = time.time()

#...............................................................................
datatree = KDTree( data, leafsize=leafsize )  # build the tree
print "%.1f sec to build KDtree of %d points" % (time.time() - t, N)

t = time.time()
distances, ix = datatree.query( ask, k=nnear+1, eps=eps )
print "%.1f sec to query %d points" % (time.time() - t, nask)

distances = distances[:,1:]  # [:,0] is all 0, point to itself
avdist = distances.mean( axis=0 )
maxdist = distances.max( axis=0 )
print "distances to %d nearest: av" % nnear, avdist, "max", maxdist

# kdtree:  dim=5  N=100000  nask=100000  nnear=2  rnormal=0.9  leafsize=10  eps=1
# clumpiness: 42847 of 10^5 data bins are empty  av 1  max 21
# 0.4 sec to build KDtree of 100000 points
# 10.1 sec to query 100000 points
# distances to 2 nearest: av [ 0.07  0.08] max [ 0.15  0.18]

# kdtree:  dim=5  N=500000  nask=500000  nnear=2  rnormal=0.9  leafsize=10  eps=1
# clumpiness: 2562 of 10^5 data bins are empty  av 5  max 80
# 2.5 sec to build KDtree of 500000 points
# 60.1 sec to query 500000 points
# distances to 2 nearest: av [ 0.05  0.06] max [ 0.13  0.13]
# run: 17 Dec 2010 15:23  mac 10.4.11 ppc