Non-linear independent component analysis with diffusion maps

We introduce intrinsic, non-linearly invariant, parameterizations of empirical data, generated by a non-linear transformation of independent variables. This is achieved through anisotropic diffusion kernels on observable data manifolds that approximate a Laplacian on the inaccessible independent variable domain. The key idea is a symmetrized second-order approximation of the unknown distances in the independent variable domain, using the metric distortion induced by the Jacobian of the unknown mapping from variables to data. This distortion is estimated using local principal component analysis. Thus, the non-linear independent component analysis problem is solved whenever the generation of the data enables the estimation of the Jacobian. In particular, we obtain the non-linear independent components of stochastic Itô processes and indicate other possible applications.