Scrambled Objects for Least-Squares Regression

Abstract

We consider least-squares regression using a randomly generated subspace G_P F of finite dimension P, where F is a function space of infinite dimension, e.g.~L_2([0,1]^d). G_P 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 H^s([0,1]^d). As a result, given N data, the least-squares estimate g built from P scrambled wavelets has excess risk ||f^* - g||_ ^2 = O(||f^*||^2_H^s([0,1]^d)( N)/P + P( N )/N) for target functions f^* H^s([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 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(2^d N^3/2 N + N^2), where d is the dimension of the input space.

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Reproductions