In this talk, we discuss a method for manifold learning that relies on a symmetric version of the optimal transport problem with a quadratic regularisation. We show that the solution of such a problem yields a sparse and adaptive affinity matrix that can be interpreted as a generalisation of the bistochastic kernel normalisation. We prove that the resulting kernel is consistent with a Laplace-type operator in the continuous limit, discuss geometric interpretations and establish robustness to heteroskedastic noise. The performance on certain simulated and real data examples will be shown. Some open questions will be raised across the talk.