Non-negative Tucker decomposition with graph regularization and smooth constraint for clustering

Non-negative Tucker decomposition (NTD) and its graph regularized extensions are the most popular techniques for representing high-dimensional non-negative data, which are typically found in a low-dimensional sub-manifold of ambient space, from a geometric perspective. Therefore, the performance of the graph-based NTD methods relies heavily on the low-dimensional representation of the original data. However, most existing approaches treat the last factor matrix in NTD as a low-dimensional representation of the original data. This treatment leads to the loss of the original data's multi-linear structure in the low-dimensional subspace. To remedy this defect, we propose a novel graph regularized Lp smooth NTD (GSNTD) method for high dimensional data representation by incorporating graph regularization and an Lp smoothing constraint into NTD. The new graph regularization term constructed by the product of the core tensor and the last factor matrix in NTD, and it is used to uncover hidden semantics while maintaining the intrinsic multi-linear geometric structure of the data. The addition of the Lp smoothing constraint to NTD may produce a more accurate and smoother solution to the optimization problem. The update rules and the convergence of the GSNTD method are proposed. In addition, a randomized variant of the GSNTD algorithm based on fiber sampling is proposed. Finally, the experimental results on four standard image databases show that the proposed method and its randomized variant have better performance than some other state-of-the-art graph-based regularization methods for image clustering.