Fast clustering in linear 1D subspaces: segmentation of microscopic image of unstained specimens.

Algorithms for subspace clustering (SC) are effective in terms of the accuracy but exhibit high computational complexity. We propose algorithm for SC of (highly) similar data points drawn from union of linear one-dimensional subspaces that are possibly dependent in the input data space. The algorithm finds a dictionary that represents data in reproducible kernel Hilbert space (RKHS). Afterwards, data are projected into RKHS by using empirical kernel map (EKM). Due to dimensionality expansion effect of the EKM one-dimensional subspaces become independent in RKHS. Segmentation into subspaces is realized by applying the max operator on projected data which yields the computational complexity of the algorithm that is linear in number of data points. We prove that for noise free data proposed approach yields exact clustering into subspaces. We also prove that EKM-based projection yields less correlated data points. Due to nonlinear projection, the proposed method can adopt to linearly nonseparable data points. We demonstrate accuracy and computational efficiency of the proposed algorithm on synthetic dataset as well as on segmentation of the image of unstained specimen in histopathology.