Multiple kernel clustering (MKC) algorithms optimally combine a group of pre-specified base kernel matrices to improve clustering performance. However, existing MKC algorithms cannot efficiently address the situation where some rows and columns of base kernel matrices are absent. This paper proposes two simple yet effective algorithms to address this issue. Different from existing approaches where incomplete kernel matrices are first imputed and a standard MKC algorithm is applied to the imputed kernel matrices, our first algorithm integrates imputation and clustering into a unified learning procedure. Specifically, we perform multiple kernel clustering directly with the presence of incomplete kernel matrices, which are treated as auxiliary variables to be jointly optimized. Our algorithm does not require that there be at least one complete base kernel matrix over all the samples. Also, it adaptively imputes incomplete kernel matrices and combines them to best serve clustering. Moreover, we further improve this algorithm by encouraging these incomplete kernel matrices to mutually complete each other. The three-step iterative algorithm is designed to solve the resultant optimization problems. After that, we theoretically study the generalization bound of the proposed algorithms. Extensive experiments are conducted on 13 benchmark data sets to compare the proposed algorithms with existing imputation-based methods. Our algorithms consistently achieve superior performance and the improvement becomes more significant with increasing missing ratio, verifying the effectiveness and advantages of the proposed joint imputation and clustering.

中文翻译：

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具有不完整内核的多内核 k 均值


多核聚类（MKC）算法以最佳方式组合一组预先指定的基核矩阵，以提高聚类性能。然而，现有的MKC算法无法有效解决基核矩阵某些行和列缺失的情况。本文提出了两种简单而有效的算法来解决这个问题。与首先插补不完整核矩阵并将标准 MKC 算法应用于插补核矩阵的现有方法不同，我们的第一个算法将插补和聚类集成到统一的学习过程中。具体来说，我们在存在不完整核矩阵的情况下直接执行多核聚类，将其视为辅助变量进行联合优化。我们的算法不要求所有样本至少有一个完整的基核矩阵。此外，它还自适应地估算不完整的核矩阵并将它们组合起来以最好地服务于聚类。此外，我们通过鼓励这些不完整的核矩阵相互完成来进一步改进该算法。三步迭代算法旨在解决由此产生的优化问题。之后，我们从理论上研究了所提出算法的泛化界限。在 13 个基准数据集上进行了大量实验，以将所提出的算法与现有的基于插补的方法进行比较。我们的算法始终如一地实现了卓越的性能，并且随着缺失率的增加，改进变得更加显着，验证了所提出的联合插补和聚类的有效性和优势。