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K-Means Clustering

Note

K-Means and 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 \(X = \{ x_1= (x_{11},\ldots{},x_{1p} ), \ldots{}, x_n = (x_{n1},\ldots{},x_{np} ) \}\) of \(n\) \(p\)-dimensional feature vectors and a positive integer \(k\), the problem is to find a set \(C = \{ c_1, \ldots{} , c_k \}\) of \(k\) \(p\)-dimensional vectors that minimize the objective function (overall error)

\[{\text{Φ}}_{X}(C)=\sum _{{x}_{i\in X}}{d}^{2}({x}_{i},C)\]

where \({d}^{2}({x}_{i},C)\) is the distance from \(x_i\) to the closest center in \(C\), such as the Euclidean distance. The vectors \(c_1, \ldots{} , c_k\) 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 \(k\) feature vectors from the data set \(X\).

  • Random choice of \(k\) 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 \(x_i\) from \(X\) with equal probability.

    2. Excludes \(x_i\) from \(X\) and adds it to the current set of centers.

    3. Resumes from step 1 until the set of centers reaches the desired size \(k\).

  • K-Means++ algorithm [Arthur2007], which selects centers with the probability proportional to their contribution to the overall error \({\text{Φ}}_{X}(C)\) according to the following scheme:

    1. Chooses one of the feature vectors \(x_i\) from \(X\) with equal probability.

    2. Excludes \(x_i\) from \(X\) and adds it to the current set of centers \(C\).

    3. For each feature vector \(x_i\) in \(X\) calculates its minimal distance \({d}({x}_{i},C)\) from the current set of centers \(C\).

    4. Chooses one of the feature vectors \(x_i\) from \(X\) with the probability \(\frac{{d}^{2}({x}_{i},C)}{{\text{ Φ}}_{X}(C)}\).

    5. Resumes from step 2 until the set of centers \(C\) reaches the desired size \(k\).

  • Parallel K-Means++ algorithm [Bahmani2012] that does the following:

    1. Chooses one of the feature vectors \(x_i\) from \(X\) with equal probability.

    2. Excludes \(x_i\) from \(X\) and adds it to the current set of centers \(C\).

    3. Repeats \(nRounds\) times:

      1. For each feature vector \(x_i\) from \(X\) calculates its minimal distance \({d}({x}_{i},C)\) from the current set of centers \(C\).

      2. Chooses \(L = oversamplingFactor \cdot k\) feature vectors \(x_i\) from \(X\) with the probability \(\frac{{d}^{2}({x}_{i},C)}{{\text{ Φ}}_{X}(C)}\).

      3. Excludes \(x_i\) vectors chosen in the previous step from \(X\) and adds them to the current set of centers \(C\).

    4. For \(c_i \in C\) sets \(w_i\) to the ratings, the number of points in \(X\) closer to \(c_i\) than to any other point in \(C\).

    5. Applies K-Means++ algorithm with weights \(w_i\) to the points in \(C\), which means that the following probability is used in step:

      \[\frac{{w}_{i}{d}^{2}({x}_{i},C)}{\text{ }{\sum }_{{x}_{iϵX}}{w}_{i}{d}^{2}({x}_{i},C)}\]

    The algorithm parameters define the number of candidates \(L\) selected in each round and number of rounds:

    • Choose \(oversamplingFactor\) to make \(L = O(k)\).

    • Choose nRounds as \(O(\log({\text{Φ}}_{X}(C)))\), where \({\text{Φ}}_{X}(C)\) is the estimation of the goal function when the first center is chosen. [Bahmani2012] recommends to set \(nRounds\) to a constant value not greater than \(8\).

Computation

Computation of the goal function includes computation of the Euclidean distance between vectors \(||x_j - m_i||\). The algorithm uses the following modification of the Euclidean distance between feature vectors \(a\) and \(b\): \(d(a,b) = d_1(a,b) + d_2(a,b)\), where \(d_1\) is computed for continuous features as

\[{d}_{1}(a,b)=\sqrt{{\sum }_{k=1}^{{p}_{1}}{({a}_{k}-{b}_{k})}^{2}}\]

and \(d_2\) is computed for binary categorical features as

\[{d}_{2}(a,b)=\gamma \sqrt{\sum _{k=1}^{{p}_{2}}{({a}_{k}-{b}_{k})}^{2}}\]

In these equations, \(\gamma\) γ weighs the impact of binary categorical features on the clustering, \(p_1\) is the number of continuous features, and \(p_2\) 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 \(x_1, \ldots{} , x_k\), 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:

Note

Distributed mode is not available for oneAPI interfaces and for Python* with DPC++ support.

Examples

Performance Considerations

To get the best overall performance of the K-Means algorithm:

  • If input data is homogeneous, provide the input data and store results in homogeneous numeric tables of the same type as specified in the algorithmFPType class template parameter.

  • If input data is non-homogeneous, use AOS layout rather than SOA layout.

  • For the output assignments table, use a homogeneous numeric table of the int type.

Product and Performance Information

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex​.

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