RANDOM SECTIONS OF ELLIPSOIDS AND THE POWER OF RANDOM INFORMATION

We study the circumradius of the intersection of an rn-dimensional ellipsoid epsilon with semi-axes sigma(1) >= ... >= sigma(m) with random subspaces of codimension n, where n can be much smaller than m. We find that, under certain assumptions on a, this random radius R-n = R-n (sigma) is of the same order as the minimal such radius sigma(n+1) with high probability. In other situations R-n is close to the maximum sigma(1). The random variable R-n naturally corresponds to the worst-case error of the best algorithm based on random information for L-2-approximation of functions from a compactly embedded Hilbert space H with unit ball epsilon. In particular, sigma(k) is the kth largest singular value of the embedding H hooked right arrow L-2. In this formulation, one can also consider the case m = infinity and we prove that random information behaves very differently depending on whether a sigma is an element of l(2) or not. For a sigma is not an element of l(2) we get E[R-n] = sigma(1) and random information is completely useless. For sigma is an element of l(2) the expected radius tends to zero at least at rate o(1/root n,) as n -> infinity. In the important case sigma(k) asymptotic to k(-alpha) ln(-beta) (k + 1), where alpha > 0 and beta is an element of R (which corresponds to various Sobolev embeddings we prove E[R-n(sigma)] asymptotic to {sigma 1 if alpha < 1/2 or beta <= alpha = 1/2, sigma(n+1) root ln(n+1) if beta > alpha = 1/2, sigma(n+1) if alpha > 1/2. In the proofs we use a comparison result for Gaussian processes a la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices. The upper bound is constructive. It is proven for the worst case error of a least squares estimator.