Scrambled Objects for Least-Squares Regression

We consider least-squares regression using a randomly generated subspace GP ⊂ F of finite dimension P, where F is a function space of infinite dimension, e.g. L2([0, 1]d). GP is defined as the span of P 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 Hs([0, l]d). As a result, given N data, the least-squares estimate ĝ built from P scrambled wavelets has excess risk ‖f* - ĝ‖2P = O(‖f*‖2Hs([0,1]d)(log N)/P + P(log N)/N) for target functions f* ∈ Hs ([0,1]d) of smoothness order s > d/2. 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 O(2dN3/2 log N + N2), where d is the dimension of the input space.