基于高斯核的非线性逼近
Nonlinear Approximation Using Gaussian Kernels
摘要：
众所周知，非线性逼近比线性方案具有优势，因为它提供了与线性方案相当的逼近速率，但是提供了更大类的逼近。这是为样条逼近和小波逼近建立的，最近由DeVore和Ron为齐次径向基函数（表面样条）逼近建立的。然而，没有这样的结果是已知的高斯函数，在机器学习和几个工程问题的首选内核。本文介绍并分析了一种使用具有不同张力参数的高斯函数平移的逼近函数的新算法。它的核心是采用DeVore和Ron的非线性逼近策略，但它选择核的方法并不简单。困难的关键在于需要根据被逼近对象的局部信息在空间上改变高斯函数中的张力参数：具有变化张力的高斯近似方案的误差分析大体上是逼近器的一个难以捉摸的目标。我们表明，我们的算法是适当的最佳意义上，它提供了类似于其他建立的非线性方法，如样条和小波逼近的近似率。正如预期和期望的那样，近似率可以根据需要而高，并且基本上仅由近似被积的平滑度饱和。
英文摘要：
It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore and Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.
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