Fréchet means in Wasserstein space

This work studies the problem of statistical inference for Fréchet means in the Wasserstein space of measures on Euclidean spaces, $\mathcal W_2 ( \mathbb R^d )$. This question arises naturally from the problem of separating amplitude and phase variation in point processes, analogous to a well-known problem in functional data analysis. We formulate the point process version of the problem, show that it is canonically equivalent to that of estimating Fréchet means in $\mathcal W_2 ( \mathbb R d )$, and carry out estimation by means of $M$-estimation. This approach allows to achieve consistency in a genuinely nonparametric framework, even in a sparse sampling regime. For Cox processes on the real line, consistency is supplemented by convergence rates and, in the dense sampling regime, $\sqrt n$-consistency and a central limit theorem. Computation of the Fréchet mean is challenging when the processes are multivariate, in which case our Fréchet mean estimator is only defined implicitly as the minimiser of an optimisation problem. To overcome this difficulty, we propose a steepest descent algorithm that approximates the minimiser, and show that it converges to a local minimum. Our techniques are specific to the Wasserstein space, because Hessian-type arguments that are commonly used for similar convergence proofs do not apply to that space. In addition, we discuss similarities with generalised Procrustes analysis. The key advantage of the algorithm is that it requires only the solution of pairwise transportation problems. The results in the preceding paragraphs require properties of Fréchet means in $\mathcal W_2 ( \mathbb R ^d )$ whose theory is developed, supplemented by some new results. We present the tangent bundle and exploit its relation to optimal maps in order to derive differentiability properties of the associated Fréchet functional, obtaining a characterisation of Karcher means. Additionally, we establish a new optimality criterion for local minima and prove a new stability result for the optimal maps that, enhanced with the established consistency of the Fréchet mean estimator, yields consistency of the optimal transportation maps.