## Developer Guide and Reference

• 2021.6
• 04/11/2022
• Public Content
Contents

# K-Means Clustering

K-Means ans K-Means initialization are also available with oneAPI interfaces:
K-Means is among the most popular and simplest clustering methods. It is intended to partition a data set into a small number of clusters such that feature vectors within a cluster have greater similarity with one another than with feature vectors from other clusters. Each cluster is characterized by a representative point, called a centroid, and a cluster radius.
In other words, the clustering methods enable reducing the problem of analysis of the entire data set to the analysis of clusters.
There are numerous ways to define the measure of similarity and centroids. For K-Means, the centroid is defined as the mean of feature vectors within the cluster.

## Details

Given the set of -dimensional feature vectors and a positive integer , the problem is to find a set of -dimensional vectors that minimize the objective function (overall error)
where is the distance from to the closest center in , such as the Euclidean distance. The vectors are called centroids. To start computations, the algorithm requires initial values of centroids.
Centroid Initialization
Centroids initialization can be done using these methods:
• Choice of first feature vectors from the data set .
• Random choice of feature vectors from the data set using the following simple random sampling draw-by-draw algorithm. The algorithm does the following:
1. Chooses one of the feature vectors from with equal probability.
2. Excludes from and adds it to the current set of centers.
3. Resumes from step 1 until the set of centers reaches the desired size .
• K-Means++ algorithm [Arthur2007], which selects centers with the probability proportional to their contribution to the overall error according to the following scheme:
1. Chooses one of the feature vectors from with equal probability.
2. Excludes from and adds it to the current set of centers .
3. For each feature vector in calculates its minimal distance from the current set of centers .
4. Chooses one of the feature vectors from with the probability .
5. Resumes from step 2 until the set of centers reaches the desired size .
• Parallel K-Means++ algorithm [Bahmani2012] that does the following:
1. Chooses one of the feature vectors from with equal probability.
2. Excludes from and adds it to the current set of centers .
3. Repeats times:
1. For each feature vector from calculates its minimal distance from the current set of centers .
2. Chooses feature vectors from with the probability .
3. Excludes vectors chosen in the previous step from and adds them to the current set of centers .
4. For sets to the ratings, the number of points in closer to than to any other point in .
5. Applies K-Means++ algorithm with weights to the points in , which means that the following probability is used in step:
The algorithm parameters define the number of candidates selected in each round and number of rounds:
• Choose to make .
• Choose nRounds as , where is the estimation of the goal function when the first center is chosen. [Bahmani2012] recommends to set to a constant value not greater than .
Computation
Computation of the goal function includes computation of the Euclidean distance between vectors . The algorithm uses the following modification of the Euclidean distance between feature vectors and : , where is computed for continuous features as
and is computed for binary categorical features as
In these equations, γ weighs the impact of binary categorical features on the clustering, is the number of continuous features, and is the number of binary categorical features. Note that the algorithm does not support non-binary categorical features.
The K-Means clustering algorithm computes centroids using Lloyd’s method [Lloyd82]. For each feature vector , you can also compute the index of the cluster that contains the feature vector.
In some cases, if no vectors are assigned to some clusters on a particular iteration, the iteration produces an empty cluster. It may occur due to bad initialization of centroids or the dataset structure. In this case, the algorithm uses the following strategy to replace the empty cluster centers and decrease the value of the overall goal function:
• Feature vectors, most distant from their assigned centroids, are selected as the new cluster centers. Information about these vectors is gathered automatically during the algorithm execution.
• In the distributed processing mode, most distant vectors from the local nodes are computed (Step 1), stored in
PartialResult
, and collected on the master node (Step 2). For more details, see the
PartialResult
description at Step 1 [Tan2005].

## Initialization

The K-Means clustering algorithm requires initialization of centroids as an explicit step. Initialization flow depends on the computation mode. Skip this step if you already calculated initial centroids.
For initialization, the following computation modes are available:

## Computation

The following computation modes are available:
Distributed mode is not available for oneAPI interfaces and for Python* with DPC++ support.

## Examples

oneAPI DPC++
Batch Processing:
oneAPI C++
Batch Processing:
C++ (CPU)
Batch Processing:
Distributed Processing:
Java*
There is no support for Java on GPU.
Batch Processing:
Distributed Processing
Python* with DPC++ support
Batch Processing:
Python*
Batch Processing:
Distributed Processing