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Gridge approximation and Radon compass

A 2000 Preprint by V. Maiorov, K. Oskolkov, and V. Temlyakov

  • 2000:09
  • Gridge approximation compiles greedy algorithms and ridge approximation. It is a class of algorithmic constructions of ridge functions - finite linear combinations of planar waves. The goal is to approximate a given target which is a multivariate function. On each step, a new planar wave is added to the preceding linear combination. This wave is selected greedily, i.e. optimally with regard to both the direction of propagation and the profile. In Mathematical Statistics, gridge approximation is known as projection pursuit regression. We consider gridge approximation in weighted Hilbert functional spaces on d-dimensional Euclidean space.

    The notion of Radon compass is introduced, which is a tool of search of the optimal direction of propagation on each step of the algorithm.

    The main quantitative result concerns error estimates for gridge processes in the norm of Hilbert space of functions supported on the unit ball, with regard to Lebesque measure. Fourier analysis of Radon transformation, in terms of Chebyshev - Gegenbauer polynomials, provides the crucial tool in such case.

    For a rather wide class of target functions whose polynomial approximations do not decrease "too rapidly", gridge approximation is equally efficient as classical algebraic polynomial. In particular, gridge approximation is not order-saturated.

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