Abstract

We consider least-squares regression using a randomly generated subspace of finite dimension where is a function space of infinite dimension, e.g. is defined as the span of  random features that are linear combinations of the basis functions of F weighted by random Gaussian i.i.d. coefficients. In particular, we consider multi-resolution random combinations at all scales of a given mother function, such as a hat function or a wavelet. In this latter case, the resulting Gaussian objects are called scrambled wavelets and we show that they enable to approximate functions in Sobolev spaces As a result, given N data, the least-squares estimate gb built from P scrambled wavelets has excess risk for target functions of smoothness order An interesting aspect of the resulting bounds is that they do not depend on the distribution P from which the data are generated, which is important in a statistical regression setting considered here. Randomization enables to adapt to any possible distribution. We conclude by describing an efficient numerical implementation using lazy expansions with numerical complexity where d is the dimension of the input space.