The approximation of probability measures on manifolds by measures supported in lower dimensions is a classical task in approximation and complexity theory with a wide range of applications. In this talk, we focus on measures supported on curves, where we highlight two approaches:

i) Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based on the conditional expectation and from the variational-calculus point of view as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. We show that principal curves in Rd can be computed as solutions of a system of ordinary differential equations and we provide several examples for principal curves related to the uniform distribution on certain domains, see [1].

ii) Discrepancy minimizing curves aim to minimize so-called discrepancies between measures. Besides proving optimal approximation rates in terms of the curve's length and Lipschitz constant, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 3-dimensional torus, the 2-sphere, the rotation group on R3 and the Grassmannian of all 2-dimensional linear subspaces of R3. Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds, see [2]. Finally, we are interested in the relation of our approach to Wasserstein gradient flows of discrepancies.