Quadratic Formula-Based Nonlinear Representation

• Quadratic formula-based nonlinear representation is a modeling framework that expresses functions as roots of parameterized quadratic equations, capturing nonlinear behaviors with algebraic precision.
• The approach employs least-squares fitting and robust numerical solvers to estimate smooth coefficient functions, effectively handling sharp transitions and nonsmooth data.
• Empirical results indicate exponential convergence and improved computational efficiency compared to linear and rational models, making it advantageous for denoising and system identification tasks.

A quadratic formula-based nonlinear representation is a class of nonlinear modeling frameworks in which the target quantity or function is identified as a root (usually via the quadratic formula) of a degree-2 polynomial equation whose coefficients are parameterized—often smoothly—in an independent variable or as functions of data features. The formulation leverages explicit degree-2 algebraic relationships to achieve efficient approximation, system identification, denoising, dimensionality reduction, or control synthesis, typically providing advantages in accuracy and computational tractability over classical (linear, affine, or rational) representations for functions or systems exhibiting nonsmooth, sharply transitioning, or fundamentally nonlinear behavior.

1. Algebraic Formulation and Reconstruction Principle

The defining characteristic of the approach is the explicit representation

$a(x)\, f(x)^2 - b(x)\, f(x) - c(x) = 0$

for a scalar function $f(x)$ over a domain $x \in D \subset \mathbb{R}$ (or multi-dimensional analogues), where $a(x), b(x), c(x)$ are smooth (often polynomial) coefficient functions. The function value is reconstructed pointwise by the quadratic formula: $$f(x) = \frac{b(x) \pm \sqrt{b(x)^2 + 4 a(x) c(x)}}{2 a(x)}$$ with the correct sign determined by an index function $\zeta(x)$, customarily defined to select the root closest to observed data or ground truth (He et al., 6 Dec 2025). This construction unifies and extends classical, degree-0 (linear) and degree-1 (rational) manifold approximation strategies, enabling the description and recovery of function classes not well-captured by standard orthogonal expansions.

2. Numerical and Algorithmic Construction

Given pointwise observations of $f(x)$ on a discrete grid, the quadratic representation is constructed by solving a least-squares problem for the coefficient functions: $$\min_{a,b,c} \int_D \left[a(x) f(x)^2 - b(x) f(x) - c(x)\right]^2 dx$$ with $a(x), b(x), c(x)$ parameterized as polynomials or other convenient bases, e.g., normalized Legendre polynomials. The optimal coefficients are obtained via discretized normal equations or robust solvers such as QR or rank-revealing QR (He et al., 6 Dec 2025). Algorithmic steps involve:

• Forming a dictionary matrix with columns $[\phi_n(x),\, f(x)\phi_n(x),\, f(x)^2\phi_n(x)]$, where $\phi_n$ are basis functions,
• Solving the resulting least-squares problem for concatenated coefficient vectors,
• Reassembling $a(x), b(x), c(x)$,
• Determining the index function $\zeta(x)$ for root selection, via minimal distance, voting among neighbors, or classification.

This method is directly extensible to higher-degree polynomial relations and to multivariate data domains.

3. Approximation Properties and Comparative Performance

Empirical evaluations have demonstrated that quadratic formula-based representations can achieve exponential convergence in $L^2$ error for functions with sharp transitions, discontinuities, or pronounced nonlinearities, outperforming both degree-0 polynomial and degree-1 rational least-squares approximations. For oscillatory functions, the quadratic algebraic variety can be precisely aligned with periodic structure, while for step-like or sigmoid functions, adaptive degree-2 fits via greedy or rank-revealing QR achieve rapid convergence with minimal basis size (He et al., 6 Dec 2025). Error scaling with basis count $K$ decisively favors the degree-2 manifold representation over traditional approaches, especially for piecewise-analytic or nonsmooth targets.

The generalization to denoising and regularization tasks exploits the separation of the smooth coefficient manifold $(a,b,c)$ from the typically low-dimensional, possibly discontinuous index function $\zeta(x)$. The main algorithmic pipeline is:

• Fit $(a,b,c)$ to noisy data via least-squares on the algebraic variety,
• Denoise the index function using proximity, $k$-nearest neighbors, or statistical classification,
• Optionally enforce orthogonality/moment-matching constraints to debias for known noise statistics.

This yields an explicit, iteration-free edge-preserving regularization scheme, avoiding the nonconvexity and complexity of total variation or $L^1$-based methods, as illustrated across a series of test cases (He et al., 6 Dec 2025).

4. Theoretical Properties and Generalizations

A core theoretical question is the characterization of the function classes for which a quadratic formula-based representation exists or is optimal. For differentially algebraic ($\mathrm{DA}$) functions, it is known that every such function admits a quadratic fixed-initial-state constant-input dynamical system (Q-FISCIDS) representation, i.e., can be realized exactly as the output after unit time of a quadratic input-affine ODE with constant input (Ohtsuka, 3 Dec 2025). The construction proceeds by introducing an augmented state encoding the function and its partial derivatives, then closing under monomial multiplication to ensure at most quadratic right-hand sides. This framework extends beyond DA functions: certain transcendental and even non-DA analytic functions are representable by quadratic FISCIDS, but a full characterization remains open.

For multivariate or higher-degree algebraic approximation, the formalism generalizes to $d$-degree polynomial varieties: $\sum_{j=0}^d a_j(x)\, f(x)^j \approx 0$ with $d=2$ as the quadratic case and $d>2$ applicable to more complex settings. This approach is under exploration for multivariate denoising, image or video data, and higher-dimensional manifold regularization (He et al., 6 Dec 2025).

5. Practical and Computational Considerations

The numerical stability of the quadratic formula-based scheme requires careful management of the discriminant $b(x)^2 + 4a(x)c(x)$ and the conditioning of $a(x)$. Instabilities may arise when $a(x)$ approaches zero or for near-coincident roots. Adaptive basis-selection, balanced degree allocation among $a$, $b$, $c$, and greedy strategies for adding basis functions are active research directions. Economical representations of the index function $\zeta(x)$, potentially using tree or wavelet structures for multijump scenarios, are required for scalable implementation in high-dimensional or multimodal contexts (He et al., 6 Dec 2025).

The method is inherently parallelizable and well-suited for modern computational architectures. For data-aligned tasks (e.g., denoising), the simplicity of least-squares fitting and local classification in $\zeta(x)$ enables near-real-time edge-preserving filtering.

6. Open Problems and Future Directions

Several research directions remain unresolved:

• Systematic convergence theory for piecewise-analytic or branch-singular functions under the manifold viewpoint,
• Algorithmic strategies for numerically stable, adaptively organized basis construction,
• Tight lower bounds on the minimal dimension or degree required for accurate representation,
• Multidimensional and higher-degree generalizations, both for functional approximation and for data-regularization/denoising,
• Optimal regularization of $\zeta(x)$ in the presence of noise or ill-posed index selection,
• Extension to image, video, or time series data domains with spatial/temporal polynomial parameterization (He et al., 6 Dec 2025).

A plausible implication is that quadratic formula-based nonlinear representation, undergirded by the explicit algebraic structure of degree-2 varieties, will play an increasingly central role in nonlinear approximation, denoising, and system identification—especially in domains where conventional linear or rational models are inadequate, and where sharp transitions or discontinuities dominate the signal landscape.