Fast Large-Scale Spectral Clustering via Explicit Feature Mapping
We propose an efficient spectral clustering method for large-scale data. The main idea in our method consists of employing random Fourier features to explicitly represent data in kernel space. The complexity of spectral clustering thus is shown lower than existing Nyström approximations on large-scale data. With <inline-formula> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> training points from a total of <inline-formula> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> data points, Nyström method requires <inline-formula> <tex-math notation="LaTeX">${O(nmd+m^{3}+nm^{2})}$ </tex-math></inline-formula> operations, where <inline-formula> <tex-math notation="LaTeX">${d}$ </tex-math></inline-formula> is the input dimension. In contrast, our proposed method requires <inline-formula> <tex-math notation="LaTeX">${O(nDd+D^{3}+n'D^{2})}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">${n}'$ </tex-math></inline-formula> is the number of data points needed until convergence and <inline-formula> <tex-math notation="LaTeX">${D}$ </tex-math></inline-formula> is the kernel mapped dimension. In large-scale datasets where <inline-formula> <tex-math notation="LaTeX">${n' \ll n}$ </tex-math></inline-formula> hold true, our explicitly mapping method can significantly speed up eigenvector approximation and benefit prediction speed in spectral clustering. For instance, on MNIST (60 000 data points), the proposed method is similar in clustering accuracy to Nyström methods while its speed is twice as fast as Nyström.