Quadratic Formula-based Nonlinear Approximation

Abstract

This paper presents a quadratic formula-based nonlinear representation for a given single-variable function f(x), $-1 \leq x \leq 1$. First, we construct the explicit polynomial coefficient functions a(x), b(x), and c(x) using a least-squares approach. Then, f is reconstructed by solving the degree-2 polynomial equation a(x) f^2 - b(x) f - c(x)=0 for any $x \in [-1,1]$, where an index function is used to select the correct sign in the quadratic formula. The quadratic formula-based nonlinear approximation (degree-2 in f) outperforms classical orthogonal polynomial-based least-squares approximation (degree-0 in f) and rational approximation (degree-1 in f) for functions with sharp transitions or discontinuities. As a potential application, we apply the degree-2 representation to data denoising. Instead of relying on more complex "edge-preserving" metric-based optimization techniques, the smooth coefficient functions a(x), b(x), and c(x) enable effective least-squares-based denoising on the low-dimensional manifold described by the algebraic variety a(x) f^2 - b(x) f - c(x)=0. Denoising the index function, which determines the appropriate root to select, can be achieved using classical statistical or modern classification/clustering techniques. Numerical results and data denoising examples are provided to demonstrate the effectiveness of the degree-2 nonlinear approximation technique. The new nonlinear, quadratic formula-based representation also raises theoretical and numerical questions, including strategies for identifying numerically stable representations, developing optimal algorithms to construct the polynomial coefficient functions a(x), b(x), and c(x), and achieving economical representation and denoising of the index function.