Kernelyze has developed a novel approximation of two-variable functions that achieves the smallest possible worst-case error among all rank-n approximations.
The Kernelyze technology is able to approximate any strictly sign-regular kernel, as defined by Karlin (1968), Total Positivity Volume 1. A strictly sign-regular kernel can be characterized as one with all order n minors having the same sign. Integrating a function with m sign changes against a strictly sign regular kernel produces, subject to weak regularity conditions, a function with no more than m sign changes. (This is a “variation-diminishing” property.)
Examples include the discounting kernel e-rt, the normal distribution’s probability density function (along with any other probability density function of a one-parameter exponential family of distributions, such as the gamma distribution), the Black-Scholes option pricing formula, the non-central t-distribution’s density, Green’s function of an even-order differential operator (subject to regularity conditions and appropriate boundary conditions), the generalized convolution of any two strictly sign-regular kernels, and the solution of the heat equation (via the Gaussian density).
KernelyzeBase is a library, written in Fortran and with bindings to OCaml and C, that computes low-rank approximations of two-variable functions (”kernels”).
KernelyzePro increases the functionality of KernelyzeBase by computing approximations for the products of multiple kernels and providing a thread-safe store of approximations (suitable for use in a multithreaded application).
This is early work that explores approximations of the kernel ecxy where c is a constant and x and y are variables ranging from -1 to 1.
Approximation of Sign-regular Kernels
This paper introduces, for any sign-regular kernel, a new method to efficiently compute a lower bound on any rank-n approximation’s uniform error. It also provides a novel method to construct a rank-n approximation that numerically achieves this lower bound in every example that has been examined.