论文信息 - On Nonlinear Functions of Linear Combinations

On Nonlinear Functions of Linear Combinations

Projection pursuit algorithms approximate a function of p variables by a sum of nonlinear functions of linear combinations: \[ (1)\qquad f\left( {x_1 , \cdots ,x_p } \right) \doteq \sum_{i = 1}^n {g_i \left( {a_{i1} x_1 + \cdots + a_{ip} x_p } \right)} . \] We develop some approximation theory, give a necessary and sufficient condition for equality in (1), and discuss nonuniqueness of the representation.

P. Diaconis | M. Shahshahani

[1] Walter Gautschi,et al. On inverses of Vandermonde and confluent Vandermonde matrices. II , 1963 .

[2] G. Lorentz. Approximation of Functions , 1966 .

[3] Decomposability of Positive Functions on R n , 1968 .

[4] J. Barros-Neto. An introduction to the theory of distributions , 1973 .

[5] Michael Taylor,et al. Pseudo differential operators , 1974 .

[6] John W. Tukey,et al. A Projection Pursuit Algorithm for Exploratory Data Analysis , 1974, IEEE Transactions on Computers.

[7] Henry C. Thacher,et al. Applied and Computational Complex Analysis. , 1988 .

[8] B. Logan. The uncertainty principle in reconstructing functions from projections , 1975 .

[9] B. Logan,et al. Optimal reconstruction of a function from its projections , 1975 .

[10] Studies in harmonic analysis , 1979 .

[11] Peter J Huber. Density Estimation and Projection Pursuit Methods. , 1981 .

[12] J. Friedman,et al. Projection Pursuit Regression , 1981 .

[13] J. Friedman,et al. Multidimensional Additive Spline Approximation , 1983 .