Approximate Shifted Laplacian Reconstruction for Multiple Kernel Clustering

Multiple kernel clustering (MKC) has demonstrated promising performance for handing non-linear data clustering. Positively, it can integrate complementary information of multiple base kernels and avoid kernel function selection. However, negatively, the main challenging is that the kernel matrix with the size n x n leads to O(n2) memory complexity and O(n3) computational complexity. To mitigate such a challenging, taking graph Laplacian as breakthrough, this paper proposes a novel and simple MKC method, dubbed as approximate shifted Laplacian reconstruction (ASLR). For each base kernel, we propose the r-rank shifted Laplacian reconstruction scheme by considering the energy losing of Laplacian reconstruction and the clustering information preserving of Laplacian decompose simultaneously. Then, by analyzing the eigenvectors of the reconstructed Laplacian, we impose some constrains to tame its solution within a Fantope. Accordingly, the byproduct (i.e. the most informative eigenvectors) contains the main clustering information, such that the clustering assignments can be obtained relying on simple k-means algorithm. Owe to the Laplacian reconstruction scheme, the memory and computational complexity can be reduced to O(n) and O<(n^2)$, respectively. As experimentally demonstrated on eight challenging MKC benchmark datasets, the results verify the effectiveness and efficiency of ASLR.