Reconstruction and subgaussian operators in asymptotic geometric analysis

We present a randomized method to approximate any vector v from a set T subset of R-n. The data one is given is the set T, vectors (X-i)(i=1)(k) where (X-i)(i=1)(k) are i.i.d. isotropic of R-n and k scalar products (< X-i,v >)(i=1)(k) subgaussian random vectors in R-n, and k << n. We show that with high probability, any y epsilon T for which (< X-i, y >)(i=1)(k) is close to the data vector ( Xi, v)) 1 will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {-1,1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {-1,1}-polytope; we show that a k-dimensional random {-1,1}-polytope with n vertices is m-neighborly for m <= ck/log(c'n/k). The proofs are based on new estimates on the behavior of the empirical process SUPf epsilon F vertical bar k(-1) Sigma(k)(i=1) f(2)(X-i) - Ef(2)vertical bar when F is a subset of the L-2 sphere. The estimates are given in terms of the gamma(2) functional with respect to the psi(2) metric on F, and hold both in exponential probability and in expectation.