Stochastic algorithms for computing means of probability measures (Q424479)

Consider a probability measure \(\mu\) on a regular geodesic ball of a Riemannian manifold \(M\) with distance \(\rho\). For \(p\geq1\), a stochastic gradient descent algorithm converging almost surely to the (unique) \(p\)-mean \(e_p\) of \(\mu\) is described (\(e_p\) minimises \(x\mapsto\int_M\rho^p(x,y)\mu(dy)\)). More precisely, a time inhomogeneous Markov chain \((X_k)\) is introduced explicitly, and it is proved that \(X_k\) converges almost surely and in \(L^2\) to \(e_p\). The speed of convergence is estimated, and an invariance principle type result is proved. The advantage with respect to a deterministic gradient descent algorithm is that it is easier to implement.