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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]

Sparse unmixing has become a popular tool for hyperspectral imagery interpretation. It refers to finding the optimal subset of a spectral library to reconstruct the image data and further estimate the proportions of different materials. Recently, multi-objective based sparse unmixing methods have presented promising performance because of their advantages in addressing combinatorial problems. A spectral and multi-objective based sparse unmixing (SMoSU) algorithm was proposed in our previous work, which solves the decision-making problem well. However, it does not show outstanding advantages in strong noise cases. To solve the problem, in this paper, SMoSU is improved based on the estimation of distribution algorithms (EDAs). The machine learning based EDAs have been a reliable approach in solving multi-objective problems. However, most of them are for special problems and relatively weak in theoretical foundations. Thus, it is unreliable to extend it directly to sparse unmixing. Here, we improve EDA on the basis of classification and propose a classification-based model for individual generating under the framework of SMoSU (CM-MoSU). In CM-MoSU, the whole population is divided to be positive and negative. Then, the macroinformation of positive individuals is used to guide the generation of new individuals. Therefore, the optimization task could pay more attention to the feasible space with high quality. Moreover, some theoretical analyses are presented to prove the reliability of CM-MoSU. In experiments, several state-of-the-art sparse unmixing algorithms are compared. Both synthetic and real-world experiments demonstrate the effectiveness of CM-MoSU. [ABSTRACT FROM AUTHOR]