Wavelet decompositions of Random Forests-smoothness analysis, sparse approximation and applications

In this paper we introduce, in the setting of machine learning, a generalization of wavelet analysis which is a popular approach to low dimensional structured signal analysis. The wavelet decomposition of a Random Forest provides a sparse approximation of any regression or classification high dimensional function at various levels of detail, with a concrete ordering of the Random Forest nodes: from 'significant' elements to nodes capturing only 'insignificant' noise. Motivated by function space theory, we use the wavelet decomposition to compute numerically a 'weak-Type' smoothness index that captures the complexity of the underlying function. As we show through extensive experimentation, this sparse representation facilitates a variety of applications such as improved regression for difficult datasets, a novel approach to feature importance, resilience to noisy or irrelevant features, compression of ensembles, etc.