Extended Researches On Convex Clustering

Objective-based clustering is a class of important clustering analysis techniques,however these methods are easily beset by local minima and are sensitive to the provided initialization due to the non-convexity of their objective functions involved,as a result,impacting final clustering performance.Recently,a convex clustering method has been on the spot light and enjoys the global optimality and independence on the initialization.In this paper,we improve the performance of the convex clustering from the aspects of semi-supervised learning and robustness by combining the specific structure information of data into its objective.The contributions obtained are as follows:1.It has been proven that clustering performance can be improved effectively by combining useful auxiliary information(typically must-links and/or cannot-links)obtained from reality with the corresponding objective.To the best of our knowledge,all such semi-supervised objective function-based clustering algorithms are based on non-convex objective,semi-supervised convex clustering has not been proposed yet.Thus,we attempt to combine pairwise constraints with convex clustering.However,the existing methods usually make the original convex objectives lose their convexity by adding constraint penalty terms to the objective function.In order to deal with such problem,we introduce a novel semi-supervised convex clustering model by using the weakly-supervised information.In particular,the key idea is to change distance metric instead of adding constraint penalty terms to the objective function.As a result,the proposed method not only maintains the advantages of convex clustering,but also improves the performance of convex clustering.Finally,the effectiveness of the proposed method is verified by numerical experiments on synthetic and real datasets.2.Convex clustering is non-robust to the frequent presence of outlying data in real scenarios,leading to a deviation of the clustering results.In order to improve its robustness,in this paper,an outlier-aware robust convex clustering algorithm,called as RCC,is proposed.Specifically,RCC extends the CC by modeling the contaminated data as the sum of the clean data and the sparse outliers and then adding a Lasso-type regularization term to the objective of the CC to reflect the sparsity of outliers.In this way,RCC can both resist the outliers to great extent and still maintain the advantages of CC,including the convexity of the objective.Further a block coordinate descent approach with the convergence guarantee is developed and finding that RCC can usually converge just in a few iterations.Finally,the effectiveness and robustness of RCC are empirically corroborated by numerical experiments both on synthetic and real datasets.