Approximation of sign-regular kernels

Abstract

Parameterized integrals, qy=∫Kx,ydPx, are common in economic applications. To optimize a parameterized integral, or to relate one such integral to others, it is helpful to reduce the rank of the kernel K while controlling approximation error. A bound on the uniform approximation error of Kx,y≈∑i=1nfixgiy also bounds the uniform approximation error of qy≈∑i=1ncigiy=∑i=1n∫fixdPxgiy over all y and all signed measures P whose total variation does not exceed a fixed bound (such as probability measures), which may be useful in optimizing q or in relating q to other parameterized integrals. Bounding the mean squared error or the local error in approximating K generally does not bound the uniform approximation error of q. Many economically interesting kernels of parameterized integrals, including expcxy, the Gaussian probability density function, and a wide class of Green’s functions, satisfy a condition known as strict sign-regularity.For Kx,y strictly sign-regular on a rectangular domain x∈xL,xU,y∈yL,yU, I introduce a new method to efficiently compute a lower bound on the uniform error achievable by any rank-n approximation. I also provide a novel method to construct a rank-n approximation that numerically achieves the lower bound in every example I have examined, so in each such example my new method solves, to within rounding error, (1)inffi,gii=1nsupx∈xL,xU,y∈yL,yUKx,y−∑i=1nfixgiy.My approach uses tools from the literature on n-widths in approximation theory (as summarized by Pinkus (1985)). I show that my new method’s uniform error can be orders of magnitude smaller than that of a Taylor series with the same rank. It also outperforms singular function approximations and the Chebfun2 approach of Townsend and Trefethen (2013) in uniform error, typically by wide margins.I describe several applications that demonstrate the practical utility of my approximation method.