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**Projection-based approximation and a duality with kernel methods.**
*(English)*
Zbl 0699.62067

Projection pursuit regression (PPR) and kernel regression (KR) are methods for estimating a smooth function of several variables from noisy data. The PPR method was introduced by J. H. Friedman and W. Stuetzle [J. Am. Stat. Assoc. 76, 817-823 (1981)] and the KR-method was discussed, for instance, by C. J. Stone [Ann. Stat. 5, 595-645 (1977; Zbl 0366.62051)] and G. Collomb [Int. Stat. Rev. 49, 75-93 (1981; Zbl 0471.62039)]. The KR-method is essentially a method based on local averaging and it is known that the performance of this method is poor in high dimensions. Earlier examples have suggested that the PPR approach performs better than the KR approach.

Here the authors investigate the following questions: For what sorts of regression functions is this true? When and by how much do PPR methods reduce the curse of dimensionality? They consider a two-dimensional problem and show that both the methods are complementary in nature. For a given function, if one method offers a dimensionality reduction, the other does not.

Here the authors investigate the following questions: For what sorts of regression functions is this true? When and by how much do PPR methods reduce the curse of dimensionality? They consider a two-dimensional problem and show that both the methods are complementary in nature. For a given function, if one method offers a dimensionality reduction, the other does not.

Reviewer: B.L.S.Prakasa Rao

### MSC:

62J02 | General nonlinear regression |

62G99 | Nonparametric inference |

41A10 | Approximation by polynomials |

62H99 | Multivariate analysis |

41A25 | Rate of convergence, degree of approximation |

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |