Given a collection of $n$ records with a pairwise similarity measure, the goal of clustering is to assign a category label to each record so that similar records tend to get the same label. In contrast to multinomial logistic regression, clustering is an unsupervised learning problem with neither category assignments nor label interpretations given in advance. In $k$-means clustering, the records $x_1, x_2, \ldots, x_n$ are numerical feature vectors of $\dim x_i = m$ with the squared Euclidean distance $|x_i - x_{i'}|_2^2$ as the similarity measure. We want to partition $\{x_1, \ldots, x_n\}$ into $k$ clusters $\{S_1, \ldots, S_k\}$ so that the aggregated squared distance from records to their cluster means is minimized:
$$ \begin{equation} \textrm{WCSS},,=,, \sum_{i=1}^n ,\big|x_i - mean(S_j: x_i\in S_j)\big|_2^2 ,,\to,,\min \end{equation} $$
The aggregated distance measure in (1) is called the within-cluster sum of squares (WCSS). It can be viewed as a measure of residual variance that remains in the data after the clustering assignment, conceptually similar to the residual sum of squares (RSS) in linear regression. However, unlike for the RSS, the minimization of (1) is an NP-hard problem [AloiseDHP2009].
Rather than searching for the global optimum in (1), a heuristic algorithm called Lloyd’s algorithm is typically used. This iterative algorithm maintains and updates a set of $k$ centroids $\{c_1, \ldots, c_k\}$, one centroid per cluster. It defines each cluster $S_j$ as the set of all records closer to $c_j$ than to any other centroid. Each iteration of the algorithm reduces the WCSS in two steps:
After Step 1, the centroids are generally different from the cluster means, so we can compute another “within-cluster sum of squares” based on the centroids:
$$\textrm{WCSS_C},,=,, \sum_{i=1}^n ,\big|x_i - \mathop{\textrm{centroid}}(S_j: x_i\in S_j)\big|_2^2 \label{eqn:WCSS:C}$$
This WCSS_C after Step 1 is less than the means-based WCSS before Step 1 (or equal if convergence achieved), and in Step 2 the WCSS cannot exceed the WCSS_C for the same clustering; hence the WCSS reduction.
Exact convergence is reached when each record becomes closer to its cluster’s mean than to any other cluster’s mean, so there are no more re-assignments and the centroids coincide with the means. In practice, iterations may be stopped when the reduction in WCSS (or in WCSS_C) falls below a minimum threshold, or upon reaching the maximum number of iterations. The initialization of the centroids is also an important part of the algorithm. The smallest WCSS obtained by the algorithm is not the global minimum and varies depending on the initial centroids. We implement multiple parallel runs with different initial centroids and report the best result.
Scoring. Our scoring script evaluates the clustering output by comparing it with a known category assignment. Since cluster labels have no prior correspondence to the categories, we cannot count “correct” and “wrong” cluster assignments. Instead, we quantify them in two ways:
The number of categories and the number of clusters ($k$) do not have to be equal. A same-category pair of records clustered into the same cluster is viewed as a “true positive,” a different-category pair clustered together is a “false positive,” a same-category pair clustered apart is a “false negative” etc.
K-Means:
K-Means Prediction:
X: Location to read matrix $X$ with the input data records as rows
C: (default: "C.mtx") Location to store the output matrix with the best available cluster centroids as rows
k: Number of clusters (and centroids)
runs: (default: 10) Number of parallel runs, each run with different initial centroids
maxi: (default: 1000) Maximum number of iterations per run
tol: (default: 0.000001) Tolerance (epsilon) for single-iteration WCSS_C change ratio
samp: (default: 50) Average number of records per centroid in data samples used in the centroid initialization procedure
Y: (default: "Y.mtx") Location to store the one-column matrix $Y$ with the best available mapping of records to clusters (defined by the output centroids)
isY: (default: 0) 0 = do not write matrix $Y$, 1 = write $Y$
fmt: (default: "text") Matrix file output format, such as text, mm, or csv; see read/write functions in SystemML Language Reference for details.
verb: (default: 0) 0 = do not print per-iteration statistics for each run, 1 = print them (the “verbose” option)
X: (default: " ") Location to read matrix $X$ with the input data records as rows, optional when prY input is provided
C: (default: " ") Location to read matrix $C$ with cluster centroids as rows, optional when prY input is provided; NOTE: if both X and C are provided, prY is an output, not input
spY: (default: " ") Location to read a one-column matrix with the externally specified “true” assignment of records (rows) to categories, optional for prediction without scoring
prY: (default: " ") Location to read (or write, if X and C are present) a column-vector with the predicted assignment of rows to clusters; NOTE: No prior correspondence is assumed between the predicted cluster labels and the externally specified categories
fmt: (default: "text") Matrix file output format for prY, such as text, mm, or csv; see read/write functions in SystemML Language Reference for details.
0: (default: " ") Location to write the output statistics defined in Table 6, by default print them to the standard output
K-Means:
K-Means Prediction:
To predict Y given X and C:
To compare “actual” labels spY with “predicted” labels given X and C:
To compare “actual” labels spY with given “predicted” labels prY:
Our clustering script proceeds in 3 stages: centroid initialization, parallel $k$-means iterations, and the best-available output generation. Centroids are initialized at random from the input records (the rows of $X$), biased towards being chosen far apart from each other. The initialization method is based on the k-means++ heuristic from [ArthurVassilvitskii2007], with one important difference: to reduce the number of passes through $X$, we take a small sample of $X$ and run the k-means++ heuristic over this sample. Here is, conceptually, our centroid initialization algorithm for one clustering run:
The sampling of $X$ and the selection of centroids are performed independently and in parallel for each run of the $k$-means algorithm. When we sample the rows of $X$, rather than tossing a random coin for each row, we compute the number of rows to skip until the next sampled row as $\lceil \log(u) / \log(1 - p) \rceil$ where $u\in (0, 1)$ is uniformly random. This time-saving trick works because
$$Prob[k-1 < \log_{1-p}(u) < k] ,,=,, p(1-p)^{k-1} ,,=,, Prob[\textrm{skip $k-1$ rows}]$$
However, it requires us to estimate the maximum sample size, which we set near $ks + 10\sqrt{ks}$ to make it generous enough.
Once we selected the initial centroid sets, we start the $k$-means iterations independently in parallel for all clustering runs. The number of clustering runs is given as the runs input parameter. Each iteration of each clustering run performs the following steps:
When a termination condition is satisfied, we store the centroids and the WCSS value and exit this run. A run has to satisfy the WCSS convergence criterion to be considered successful. Upon the termination of all runs, we select the smallest WCSS value among the successful runs, and write out this run’s centroids. If requested, we also compute the cluster assignment of all records in $X$, using integers from 1 to $k$ as the cluster labels. The scoring script can then be used to compare the cluster assignment with an externally specified category assignment.
We output the $k$ centroids for the best available clustering, i. e. whose WCSS is the smallest of all successful runs. The centroids are written as the rows of the $k,{\times},m$-matrix into the output file whose path/name was provided as the C input argument. If the input parameter isY was set to 1, we also output the one-column matrix with the cluster assignment for all the records. This assignment is written into the file whose path/name was provided as the Y input argument. The best WCSS value, as well as some information about the performance of the other runs, is printed during the script execution. The scoring script Kmeans-predict.dml prints all its results in a self-explanatory manner, as defined in Table 6.