Vertical Federated K-Means for Multi-View Data Guided by a K-Means Cost Bound after Projection

Multi-view data is widely present in the real world. Multi-view clustering is an unsupervised method for capturing the grouping structure of such data. However, multi-view clustering struggles to meet the requirements of real-world scenarios, such as distributed storage of different views and data protection needs. These requirements align with the setting of vertical federated clustering. However, vertical federated clustering still faces two challenges: (1) Under the constraints of privacy protection mechanisms, how to theoretically analyze the clustering consistency between the data uploaded by clients to the server and the original client data is challenging. (2) The feature space differences among different clients make cross-view information sharing and fusion difficult. To address the first challenge, we provide a theoretical analysis of the upper bound of the loss of k-means for transformation matrix mapping, revealing the relationship between the k-means loss of the transformed data and the original data. We then propose a vertical federated clustering method (V-HDKM). In this method, clients handle the second challenge by transposing the feature matrix. Guided by the projected k-means loss bound, we expand the feature space and perform k-means clustering to obtain feature cluster centers, which are then uploaded to the server. The server aggregates the global centers and feeds back the optimized results, achieving cross-view knowledge fusion through iterative interactions. Experimental results show that V-HDKM significantly improves local clustering performance and performances better than other seven vertical federated mthods on 20 multi-view datasets. Furthermore, sensitivity analysis on 8 UCI datasets with respect to the number of clients demonstrates the stability of the method. The code is available at https://github.com/jiangjh/V-HDKM.