Quadratic Formula-based Nonlinear Approximation

TL;DR

This work introduces a degree-2 nonlinear approximation for single-variable functions on $[-1,1]$ by embedding the function on an algebraic variety $a(x) f^2 - b(x) f - c(x) = 0$ and selecting the appropriate root with an index function $\zeta(x)$. Coefficients $a(x)$, $b(x)$, and $c(x)$ are learned via least squares using Legendre bases, enabling a flexible representation that captures multi-branch and discontinuous behavior through the quadratic formula. The approach yields faster convergence for discontinuities and sharp transitions, supports data denoising by decoupling the manifold from root selection, and is backed by greedy and rank-revealing QR construction methods. The paper also discusses stability concerns, numerical experiments across discontinuous, oscillatory, and sharp-transition cases, and outlines extensions to higher-degree and multivariate settings for broader impact in numerical analysis and signal processing.

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.

Figures (9)

• Figure 1: $f(x)$: Blue. Root 1: Orange, Root 2: Green. Top Left: $f(x)$ with one discontinuity. Bottom Left: well-separated branches of the manifold. Top Right: $f(x)$ with two jumps. Bottom Right: more complex branches and index function for degree-2 representation.
• Figure 1: Convergence of degree-0, degree-1, and degree-2 representations.
• Figure 1: Small error $\varepsilon$ is added to the function: $\tilde{f}=f+\varepsilon$ with $\varepsilon=30\cdot\tt{randn}$. Top: original data $\tilde{f}$ (blue solid line) and recovered $f$ (red dashed line). Bottom: estimated noise $\tilde{f}-f$.
• Figure 2: The $L_2$ convergence of the Legendre approximation to $f(x)=\sin(10\pi x)$ as the polynomial degree $N$ increases. After reaching a resolution threshold ($N\approx 30$), the error decays exponentially to machine precision.
• Figure 2: White noise added to the manifold: $(\tilde{f}-25)(\tilde{f}-255)=\varepsilon$ with $\varepsilon=5000\cdot\tt{randn}$. Top: noisy data $\tilde{f}$ (blue solid line) and recovered $f$ (red dashed line). Bottom: estimated noise $\tilde{f}-f$.

Theorems & Definitions (3)

• Definition 2.1: Degree-0 representation
• Definition 2.2: Degree-1 representation
• Definition 3.1: Degree-2 representation