On An Interpretation Of Spectral Clustering Via Heat Equation And Finite Elements Theory

Spectral clustering methods use eigenvectors of a matrix, called Gaussian affinity matrix, in order to define a low-dimensional space in which data points can be clustered. This matrix is widely used and depends on a free parameter σ. It is usually interpreted as some discretization of the Heat Equation Green kernel. Combining tools from Partial Differential Equations and Finite Elements theory, we propose an interpretation of this spectral method which offers an alternative point of view on how spectral clustering works. This approach develops some particular geometrical properties inherent to eigenfunctions of some specific partial differential equation problem. We analyze experimentally how this geometrical property is recovered in the eigenvectors of the affinity matrix and we also study the influence of the parameter σ.