We present the FuSSO, a functional analogue to the LASSO, that efficiently finds a sparse set of functional input covariates to regress a real-valued response against. The FuSSO does so in a semi-parametric fashion, making no parametric assumptions about the nature of input functional covariates and assuming a linear form to the mapping of functional covariates to the response. We provide a statistical backing for use of the FuSSO via proof of asymptotic sparsistency under various conditions. Furthermore, we observe good results on both synthetic and real-world data.
Lastly, it is worth noting that in our estimator we will have an additive linear model, @math where we search for @math in a broad, non-parametric family such that many @math are the zero function. Such a task is similar in nature to the SpAM estimator @cite_12 , in which one also has an additive model @math (in the dimensions of a real vector @math ) and searches for @math in a broad, non-parametric family such that many @math are the zero function. Note though, that in the SpAM model, the @math functions are applied to real covariates via a function evaluation. In the FuSSO model, @math are applied to functional covariates via an inner product; that is, FuSSO works over functional, not real-valued covariates, unlike SpAM.
@cite_12
2593996946
We present a new class of methods for high dimensional non-parametric regression and classification called sparse additive models. Our methods combine ideas from sparse linear modelling and additive non-parametric regression. We derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size. Sparse additive models are essentially a functional version of the grouped lasso of Yuan and Lin. They are also closely related to the COSSO model of Lin and Zhang but decouple smoothing and sparsity, enabling the use of arbitrary non-parametric smoothers. We give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data, showing that they can be effective in fitting sparse non-parametric models in high dimensional data. Copyright (c) 2009 Royal Statistical Society.