The breathing k-means algorithm

# The Breathing K-Means Algorithm

## A novel approximation algorithm for the k-means problem

Breathing k-means finds solutions which

• are usually better than solutions from k-means++
• are lower-bounded by k-means++ (the worst possible result is a solution from k-means++)
• require more computation time than k-means++ (ca 50%-300% depending on the problem)

The second claim is fulfilled, since the default seeding method of breathing k-means is k-means++. If no improvement is found by breathing k-means, it simply returns the k-means++ result used as seed.

The more interesting part about breathing k-means is probably its ability to improve upon k-means++ in many cases and to do so at moderate extra computational cost (considering that the k-means problem is NP-hard).

## Further Info

For a detailed motivation and description of the breathing k-means algorithm see the preprint https://arxiv.org/abs/2006.15666.

An extended software repo including the data sets used in the above preprint and jupyter notebooks to experiment with them can be found at https://github.com/gittar/breathing-k-means

## API

The included class BKMeans is subclassed from scikit-learn's KMeans class and has thus the same API, only extended by two parameters which, however, can be ignored for normal usage and thus simply left at their default values:

• m (breathing depth), default: 5, (m=0 is equivalent to running k-means++)
• theta (neighborhood freezing range), default: 1.1

## Installation

pip install bkmeans


## Example 1: running on simple random data set

Code:

import numpy as np
from bkmeans import BKMeans

# generate random data set
X=np.random.rand(1000,2)

# create BKMeans instance
bkm = BKMeans(n_clusters=100)

# run the algorithm
bkm.fit(X)

# print SSE (inertia in scikit-learn terms)
print(bkm.inertia_)


Output:

1.1775040547902602


## Example 2: comparison with k-means++ (multiple runs)

Code:

import numpy as np
from sklearn.cluster import KMeans
from bkmeans import BKMeans

# random 2D data set
X=np.random.rand(1000,2)

# number of centroids
k=100

for i in range(5):
# kmeans++
km = KMeans(n_clusters=k)
km.fit(X)

# breathing k-means
bkm = BKMeans(n_clusters=k)
bkm.fit(X)

# relative SSE improvement of bkm over km++
imp = 1 - bkm.inertia_/km.inertia_
print(f"SSE improvement over k-means++: {imp:.2%}")


Output:

SSE improvement over k-means++: 3.38%
SSE improvement over k-means++: 4.16%
SSE improvement over k-means++: 6.14%
SSE improvement over k-means++: 6.79%
SSE improvement over k-means++: 4.76%


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