Quadratic Transform: Methods & Applications

• Quadratic Transform is a collection of mathematical techniques that reformulate nonconvex problems by decoupling ratios via auxiliary variables, providing closed-form updates and convergence guarantees.
• The technique extends to Fourier-type transforms with quadratic phase modulation, enabling advanced time–frequency analysis and localization in signal processing.
• In quantum optimization, quadratisation reduces high-degree polynomial problems to quadratic form (QUBO), offering significant speedups and efficient hardware implementation.

The term "Quadratic Transform (QT)" encompasses a broad collection of mathematical concepts and techniques, spanning optimization, signal processing, integral transforms, and invertible polynomial maps. In contemporary research, Quadratic Transforms are fundamental to (1) fractional programming in signal processing and machine learning, (2) time–frequency analysis via quadratic-phase Fourier-type transforms, and (3) the quadratisation of high-degree pseudo-Boolean functions for quantum optimization toolchains. The notion also appears in algebraic geometry via invertible quadratic mappings. This article systematically delineates principal forms, analytical frameworks, and applications of Quadratic Transforms, with rigorous links to the recent arXiv literature.

1. Quadratic Transform in Fractional Programming and Optimization

General Framework

Quadratic Transform methods are a core finite-dimensional technique for reformulating nonconvex fractional programming (FP) problems, particularly those with sum-of-ratios objectives ubiquitous in signal processing and machine learning. FP problems typically take forms such as

$$\max_{x\in\mathcal{X}}\;F(x)=\sum_{i=1}^n\frac{A_i(x)}{B_i(x)},$$

with $A_i(x)\geq 0$, $B_i(x)>0$, and $\mathcal{X}\subseteq\mathbb{R}^d$ convex. The QT approach introduces auxiliary variables to decouple numerators and denominators, yielding a surrogate function jointly quadratic in decision and auxiliary variables (Shen et al., 13 Mar 2025, Shen et al., 2023). For scalar cases: $$\frac{A(x)}{B(x)} = \max_{y\in\mathbb{R}}\left(2y\sqrt{A(x)} - y^2 B(x)\right)$$ with closed-form solution $y^* = \sqrt{A(x)}/B(x)$. This reformulation supports block-coordinate MM-type algorithms with global convergence to stationary points under mild regularity (Shen et al., 13 Mar 2025).

Matrix Extensions and Algorithmic Structure

For matrix-valued objectives, e.g., $$\mathrm{Tr}\left[\sqrt{A(x)}^{H} B(x)^{-1} \sqrt{A(x)}\right]$$, QT generalizes to introduce matrix auxiliary variables, enabling efficient MM or block coordinate-descent algorithms. Each quadratic surrogate admits closed-form auxiliary updating, while the $x$-step typically solves a convex or structured quadratic problem per iteration (Shen et al., 13 Mar 2025).

Acceleration and Convergence

Algorithmic analysis demonstrates $O(1/k)$ convergence for standard QT and $O(1/k^2)$ convergence for Nesterov-accelerated variants, the latter exploiting a gradient-projection viewpoint and extrapolation (Shen et al., 2023). For non-smooth or high-dimensional scenarios, this yields competitive runtimes and substantial practical acceleration compared to classical Dinkelbach or Charnes-Cooper methods.

Principal Applications

Key applications include:

• SINR maximization in wireless networks,
• Cramér–Rao Bound minimization in radar,
• Normalized-cut graph clustering,
• Margin maximization in SVMs.

Algorithms admit efficient closed-form or low-complexity updates per iteration, consistently outperforming naive convex optimization approaches (Shen et al., 13 Mar 2025, Shen et al., 2023).

2. Quadratic Transform (Quadratisation) in Quantum Optimization

Quadratic Transforms also refer to "quadratisation," or reduction of higher-degree polynomial unconstrained binary optimization (PUBO) problems to quadratic unconstrained binary optimization (QUBO) form, which is required for current quantum and annealing hardware (Schmidbauer et al., 2024). Formally, given $f(x)$ of degree $d>2$, one introduces auxiliary binary variables $y$ such that

$$f(x) = \min_{y\in\{0,1\}^m} f'(x,y), \quad \deg(f')=2.$$

The graph-based LSR quadratisation scheme constructs a gadget $p(x_i,x_j,y_h)=3y_h + x_ix_j - 2x_iy_h - 2x_jy_h$ to systematically replace pairwise products, exploiting the structure of multi-edges in a monomial dependency graph $G_f$. This two-stage algorithm achieves order(s) of magnitude speedup over monomial-based methods, admits fine-grained tradeoff between auxiliary variable count and QUBO density, and preserves the global minimum (Schmidbauer et al., 2024).

3. Quadratic Phase Fourier and Related Integral Transforms

Quadratic Transforms are also foundational in Fourier-type integral transforms with quadratic phase modulation. The most general is the Quadratic-Phase Fourier Transform (QPFT), parameterized by five real constants $(a, b, c, d, e)$: $$(Q_{\Lambda}f)(u) = \int_{-\infty}^\infty f(x) \frac{1}{\sqrt{2\pi |b|}} e^{-i[a x^2 + b x u + c u^2 + d x + e u]} dx,$$ which subsumes the ordinary Fourier Transform (FT), fractional FT, and linear canonical transform as special cases (Gupta et al., 2024, Varghese et al., 5 May 2025). Core properties include unitarity (up to a normalization), explicit inversion, and a generalized convolution theorem with chirp-augmented convolution kernels.

• Multidimensional generalization is achieved via separable products of 1D parameters.
• Parseval identities and full isometry hold in $L^2$ spaces (Varghese et al., 5 May 2025).
• Uncertainty principles (Heisenberg, Donoho–Stark, logarithmic, entropic) extend from the classical FT via appropriate parameter translation.

Windowed and Localized Variants

The Windowed Quadratic Phase Fourier Transform (WQPFT) further extends QPFT to time–frequency analysis, allowing nonstationary signals to be studied with joint quadratic-phase and window localization (Varghese et al., 5 Jul 2025). The translation and convolution theorems, range characterizations, and inversion formulas parallel the short-time Fourier and Gabor analysis but admit an additional quadratic-phase degree of freedom.

Quadratic-Phase Bessel and Dunkl Transforms

Generalizations to weighted and non-commuting settings include the quadratic-phase Fourier–Bessel transform (on $\mathbb{R}_+$, employing Bessel kernels) (Saoudi, 21 Jan 2026), and the quadratic-phase Dunkl transform (incorporating reflection-symmetric differential-difference kernels) (Saoudi, 26 Dec 2025). Each retains fundamental properties (continuity, inversion, Parseval, convolution), supports translation and convolution operators with explicit kernels, and admits generalized Heisenberg-type uncertainty bounds.

4. Discrete Quadratic Transforms

The quadratic discrete Fourier transform (QDFT) is a matrix transform of the form

$$F(\alpha, \beta)_{k,n} = \frac{1}{\sqrt{N}} \exp\left(2\pi i/N [\alpha n^2 + \beta n k]\right)$$

for $k, n=0,\dots,N-1$ and parameters $\alpha, \beta\in\mathbb{R}$ (Kibler, 2010). QDFTs are unitarily equivalent to the DFT for certain parameter choices and are key in constructing complete sets of mutually unbiased bases (MUBs) in prime dimension $N$. Their group-theoretic structure is embedded in the finite Heisenberg–Weyl/Pauli group.

5. Invertible Quadratic Transformations of Polynomial Maps

In algebraic geometry and affine algebraic automorphism, a quadratic (degree $\leq 2$) polynomial map $T\colon\mathbb{R}^2\rightarrow\mathbb{R}^2$ is, up to affine change of basis, invertible only if it is a (triangular) shear: $T(x, y) = (x + y^2, y)$ (Sharipov, 2015). All other possible quadratic polynomial forms fail global invertibility, as established via discriminant and Jacobian analysis. Such transformations are relevant in the equivalencing of certain Diophantine problems (e.g., perfect cuboid search) and integer lattice automorphisms.

6. Computational Complexity, Acceleration, and Implementation Considerations

For fractional programming, standard QT-based algorithms are $O(1/k)$ convergent, with each iteration involving closed-form auxiliary variable updates and a convex or otherwise efficiently structured subproblem in $x$ (Shen et al., 13 Mar 2025). Accelerated extrapolated QT methods, exploiting Nesterov's momentum and projected-gradient linkage, achieve $O(1/k^2)$ rates with nearly identical per-iteration cost (Shen et al., 2023).

In quantum quadratisation, the LSR graph-based algorithm is orders of magnitude faster than monomial-based heuristics, scalable to $n\sim100$ variables for dense degree-4 PUBOs, and tunable over coupling density/auxiliary tradeoffs (Schmidbauer et al., 2024). On the signal processing/numerical analysis front, fast numerical implementation, especially for localized transforms (windowed, Bessel, Dunkl), remains an ongoing research direction.

7. Applications Across Domains

Quadratic Transform Variant	Application Area(s)	Salient Features
Fractional Programming QT	SINR maximization, SVM, clustering	Block-wise closed-form updates
Quantum Quadratisation (LSR)	NISQ quantum annealing, QUBO embedding	Aux variable minimization, fast
QPFT, QDFT, WQPFT	Time–frequency & quantum information	Time–frequency localization, MUB
QPFBT, QPDT	Weighted/Fractional harmonic analysis	Bessel/Dunkl kernel convolution
Polynomial (Real-plane) Quadratic	Algebraic geometry, Diophantine analysis	Only triangular shear invertible

These transforms are core analytic, algebraic, and algorithmic tools in contemporary quantum information, communications, optimization, applied harmonic analysis, and computational mathematics.

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References:

• "Quadratic Transform for Fractional Programming in Signal Processing and Machine Learning" (Shen et al., 13 Mar 2025)
• "Accelerating Quadratic Transform and WMMSE" (Shen et al., 2023)
• "It's Quick to be Square: Fast Quadratisation for Quantum Toolchains" (Schmidbauer et al., 2024)
• "A mathematical survey on Fourier type integral transform and their offshoots" (Gupta et al., 2024)
• "The Multidimensional Quadratic Phase Fourier Transform" (Varghese et al., 5 May 2025)
• "The windowed quadratic phase Fourier transform" (Varghese et al., 5 Jul 2025)
• "Quadratic-Phase Fourier--Bessel Transform" (Saoudi, 21 Jan 2026)
• "Quadratic-Phase Dunkl Transform" (Saoudi, 26 Dec 2025)
• "Quadratic discrete Fourier transform and mutually unbiased bases" (Kibler, 2010)
• "A note on invertible quadratic transformations of the real plane" (Sharipov, 2015)