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$k$ k -means method using Lloyd heuristic is a traditional clustering method which has played a key role in multiple downstream tasks of machine learning because of its simplicity. However, Lloyd heuristic always finds a bad local minimum, i.e., the bad local minimum makes objective function value not small enough, which limits the performance of $k$ k -means. In this paper, we use coordinate descent (CD) method to solve the problem. First, we show that the $k$ k -means minimization problem can be reformulated as a trace maximization problem, then a simple and efficient coordinate descent scheme is proposed to solve the maximization problem. Two interesting findings through theory are that Lloyd cannot decrease the objective function value of $k$ k -means produced by our CD further, and our proposed method CD to solve $k$ k -means problem can avoid produce empty clusters. In addition, according to the computational complexity analysis, it is verified CD has the same time complexity with original $k$ k -means method. Extensive experiments including statistical hypothesis testing, on several real-world datasets with varying number of clusters, varying number of samples and varying number of dimensions show that CD performs better compared to Lloyd, i.e., lower objective value, better local minimum and fewer iterations. And CD is more robust to initialization than Lloyd whether the initialization strategy is random or initialization of $k$ k -means++. [ABSTRACT FROM AUTHOR]

Process discovery is the learning task that entails the construction of process models from event logs of information systems. Typically, these event logs are large data sets that contain the process executions by registering what activity has taken place at a certain moment in time. By far the most arduous challenge for process discovery algorithms consists of tackling the problem of accurate and comprehensible knowledge discovery from highly flexible environments. Event logs from such flexible systems often contain a large variety of process executions which makes the application of process mining most interesting. However, simply applying existing process discovery techniques will often yield highly incomprehensible process models because of their inaccuracy and complexity. With respect to resolving this problem, trace clustering is one very interesting approach since it allows to split up an existing event log so as to facilitate the knowledge discovery process. In this paper, we propose a novel trace clustering technique that significantly differs from previous approaches. Above all, it starts from the observation that currently available techniques suffer from a large divergence between the clustering bias and the evaluation bias. By employing an active learning inspired approach, this bias divergence is solved. In an assessment using four complex, real-life event logs, it is shown that our technique significantly outperforms currently available trace clustering techniques. [ABSTRACT FROM PUBLISHER]