Non-Homogeneous Complex Quadratic Transform

• Non-Homogeneous Complex Quadratic Transform is a framework in the complex domain that integrates quadratic forms with an additional linear term to improve optimization surrogate methods.
• It employs auxiliary variable reformulations and majorization-minimization techniques to translate complex Hermitian ratio problems into gradient projection updates.
• The transform spans multiple disciplines, appearing in fractional programming, discrete quadratic Fourier transforms, and affine birational maps related to integrable systems.

Searching arXiv for the exact phrase and the cited papers to ground the article in current arXiv records. “Non-Homogeneous Complex Quadratic Transform” does not denote a single uniformly standardized object across the arXiv literature. The closest exact match is the nonhomogeneous quadratic transform developed in the complex vector and matrix setting for fractional programming, where “nonhomogeneous” refers to a surrogate containing a linear term in addition to quadratic terms (Shen et al., 2023). In adjacent literatures, closely related meanings appear in two further forms: a quadratic discrete Fourier transform with a complex kernel containing quadratic, bilinear, linear, and constant phase terms (Kibler, 2010), and a generalized Manin transformation for a quadratic pencil formulated in affine coordinates and shown to be projectively equivalent to a QRT map (Kamp et al., 2018). Taken together, these works show that the phrase is best understood as a family resemblance rather than a single canonical construction: a complex-domain transform or birational map whose defining structure is quadratic but not purely homogeneous.

1. Terminological scope and principal meanings

Within optimization and communications, the strongest direct match is the paper “Accelerating Quadratic Transform and WMMSE,” which explicitly defines and analyzes a nonhomogeneous quadratic transform in the complex domain (Shen et al., 2023). In that setting, the standard quadratic transform rewrites complex Hermitian fractional terms of the form

$$s_i^H(\underline{x})G_i^{-1}(\underline{x})s_i(\underline{x})$$

by introducing auxiliary complex vectors $y_i$, and the nonhomogeneous variant arises after applying a further lower bound containing a linear term. The paper states that the bound is called nonhomogeneous “due to the linear term $2\Re\{x^H(L-K)z\}$” (Shen et al., 2023).

A second, mathematically close but structurally different meaning appears in the paper “Quadratic discrete Fourier transform and mutually unbiased bases,” which introduces a quadratic discrete Fourier transform whose kernel is a root-of-unity exponential with a quadratic phase in the input index together with linear and constant phase terms (Kibler, 2010). In that sense, the transform is not merely homogeneous in the phase variable; it is a discrete complex quadratic-phase transform with explicit nonhomogeneous terms.

A third usage arises in algebraic geometry and integrable systems. “Generalised Manin transformations and QRT maps” does not use the exact phrase “non-homogeneous complex quadratic transform,” but it gives the closest formal notion in that literature: a generalized Manin transformation for a quadratic pencil, written explicitly in affine coordinates $(u,v)$, preserving a quadratic pencil, and projectively equivalent to a QRT map (Kamp et al., 2018). Here “non-homogeneous” is naturally interpreted as the affine, non-projective coordinate presentation.

These usages are not interchangeable. A plausible implication is that the phrase functions as a cross-disciplinary descriptor rather than a settled term of art: in one literature it names an MM surrogate for complex FP, in another it describes a discrete quadratic-phase Fourier kernel, and in another it refers to affine birational maps preserving quadratic pencils.

2. Complex nonhomogeneous quadratic transform in fractional programming

The optimization-theoretic construction in (Shen et al., 2023) is formulated for a sum-of-weighted-ratios problem over complex vectors. Each ratio term is

$$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$

with $A_i\in\mathbb C^{\ell\times d}$, $B_{ij}\in\mathbb C^{\ell\times d}$, and $x_j\in\mathbb C^d$, and the objective is

$$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$

subject to $x_i\in\mathcal X_i$ (Shen et al., 2023). The same paper also treats a matrix-variable extension with $y_i$0 and objective $y_i$1 (Shen et al., 2023).

The standard quadratic transform in that paper is the exact reformulation

$y_i$2

for $y_i$3 and $y_i$4 (Shen et al., 2023). For the weighted-ratio model, the transformed objective becomes

$y_i$5

or equivalently

$y_i$6

(Shen et al., 2023). The optimal auxiliary update is

$y_i$7

The specifically nonhomogeneous step is obtained from the matrix inequality

$y_i$8

which yields a lower-bounded objective

$y_i$9

where

$2\Re\{x^H(L-K)z\}$0

The paper states that this bound is called nonhomogeneous because of the linear term $2\Re\{x^H(L-K)z\}$1 (Shen et al., 2023). In this usage, therefore, nonhomogeneity refers to the surrogate’s algebraic form rather than to arbitrary affine offsets in the original ratio.

3. Algorithmic structure, MM interpretation, and convergence

In (Shen et al., 2023), the standard quadratic transform is exact with respect to the auxiliary variables $2\Re\{x^H(L-K)z\}$2: optimizing over $2\Re\{x^H(L-K)z\}$3 recovers the original objective. By contrast, the variable $2\Re\{x^H(L-K)z\}$4 belongs to the lower-bounding step, so the nonhomogeneous construction is an MM surrogate rather than a second exact reformulation. Equality in the bound occurs at $2\Re\{x^H(L-K)z\}$5, which yields the update

$2\Re\{x^H(L-K)z\}$6

and transforms the method into a tangent-surrogate scheme (Shen et al., 2023).

For the $2\Re\{x^H(L-K)z\}$7-subproblem, the paper gives

$2\Re\{x^H(L-K)z\}$8

for the standard QT, and, after the nonhomogeneous bound,

$2\Re\{x^H(L-K)z\}$9

with $(u,v)$0 (Shen et al., 2023). The paper then shows that, after substituting optimal $(u,v)$1 and $(u,v)$2, the nonhomogeneous QT update becomes

$(u,v)$3

that is, a projected gradient step in the complex domain (Shen et al., 2023). This establishes the paper’s stated connection between quadratic transform and gradient projection.

The same work states that Algorithms 1 and 2, namely conventional QT and nonhomogeneous QT, are MM methods, and that Algorithms 1, 2, and 3, including extrapolated QT, all converge to some stationary point of problem $(u,v)$4 (Shen et al., 2023). For local rate analysis, the paper gives

$(u,v)$5

with $(u,v)$6 for QT and $(u,v)$7 for nonhomogeneous QT, and notes that both have $(u,v)$8 local objective error (Shen et al., 2023). For the extrapolated method, if $(u,v)$9 is $$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$0-Lipschitz and $$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$1, the paper states

$$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$2

that is, $$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$3 objective-value error (Shen et al., 2023).

These results delimit the most precise contemporary meaning of “non-homogeneous complex quadratic transform” on a complex-domain QT combined with a nonhomogeneous surrogate that yields a gradient-projection-compatible MM algorithm.

4. Discrete quadratic-phase transform with nonhomogeneous terms

A distinct but closely related construction is the quadratic discrete Fourier transform introduced in (Kibler, 2010). In the classical transform notation used there,

$$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$4

with

$$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$5

(Kibler, 2010). Expanding the phase gives

$$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$6

This kernel contains a quadratic term in $$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$7, a bilinear term $$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$8, a linear term in $$M_i(\underline{x}) = \big(A_ix_i\big)^H\Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}\big(A_ix_i\big),$$9, and a constant term. The paper therefore supports describing the transform as a complex quadratic-phase kernel with non-homogeneous terms (Kibler, 2010). It is not, however, the most general quadratic form in both indices, since the kernel as written has no $A_i\in\mathbb C^{\ell\times d}$0 term.

The transform is unitary: the paper states, “For $A_i\in\mathbb C^{\ell\times d}$1 arbitrary, the matrix $A_i\in\mathbb C^{\ell\times d}$2 is unitary” (Kibler, 2010). It also satisfies the Parseval–Plancherel identity

$A_i\in\mathbb C^{\ell\times d}$3

and it factorizes as

$A_i\in\mathbb C^{\ell\times d}$4

where $A_i\in\mathbb C^{\ell\times d}$5 is diagonal (Kibler, 2010). This shows that the transform is a chirp-modulated DFT.

In the paper’s terminology, it is a “two-parameter extension, with a quadratic term, of the usual discrete Fourier transform,” and when $A_i\in\mathbb C^{\ell\times d}$6 it reduces exactly to the ordinary DFT (Kibler, 2010). The basis vectors $A_i\in\mathbb C^{\ell\times d}$7 are orthonormal, unbiased with respect to the computational basis, and for prime $A_i\in\mathbb C^{\ell\times d}$8, the collection $A_i\in\mathbb C^{\ell\times d}$9 forms a complete set of $B_{ij}\in\mathbb C^{\ell\times d}$0 MUBs (Kibler, 2010). This usage of “quadratic transform” is thus spectral and unitary rather than variational.

5. Affine quadratic-pencil transformations and QRT equivalence

In algebraic geometry and integrable mappings, (Kamp et al., 2018) supplies a third interpretation. Its central object is the generalized Manin transformation preserving a pencil

$B_{ij}\in\mathbb C^{\ell\times d}$1

with $B_{ij}\in\mathbb C^{\ell\times d}$2 polynomials of total degree $B_{ij}\in\mathbb C^{\ell\times d}$3 (Kamp et al., 2018). For the quadratic case $B_{ij}\in\mathbb C^{\ell\times d}$4,

$B_{ij}\in\mathbb C^{\ell\times d}$5

and similarly for $B_{ij}\in\mathbb C^{\ell\times d}$6 (Kamp et al., 2018).

The generalized Manin involution $B_{ij}\in\mathbb C^{\ell\times d}$7 is constructed in affine coordinates by taking the line through $B_{ij}\in\mathbb C^{\ell\times d}$8 and an involution point $B_{ij}\in\mathbb C^{\ell\times d}$9,

$x_j\in\mathbb C^d$0

and choosing the second intersection with the same curve of the pencil, which is characterized by

$x_j\in\mathbb C^d$1

(Kamp et al., 2018). In the quadratic case the resulting affine formula is

$x_j\in\mathbb C^d$2

with

$x_j\in\mathbb C^d$3

and explicitly computable directional derivatives (Kamp et al., 2018).

The paper emphasizes a distinctive feature of the quadratic case: “For $x_j\in\mathbb C^d$4 we are free to choose the involution points $x_j\in\mathbb C^d$5 and there are no constraints on the pencil,” except that involution points are not base points (Kamp et al., 2018). The composition

$x_j\in\mathbb C^d$6

preserves the rational first integral

$x_j\in\mathbb C^d$7

and is measure-preserving with density

$x_j\in\mathbb C^d$8

where $x_j\in\mathbb C^d$9 is the line through the involution points (Kamp et al., 2018).

Most importantly for the relation to QRT theory, the paper proves that every generalized Manin transformation for $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$0 can be brought to QRT form by a projective collineation. If

$$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$1

then the fractional affine transformation

$$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$2

sends $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$3 and $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$4, and “brings the generalised Manin transformation into QRT form” (Kamp et al., 2018). In this literature, therefore, a “non-homogeneous complex quadratic transform” is most faithfully interpreted as an affine birational map on $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$5 preserving a quadratic pencil and conjugate to a QRT map.

6. Comparative interpretation and limitations

The three constructions share a quadratic core but differ in mathematical type, domain, and purpose.

Setting	Core object	Sense of “nonhomogeneous”
Fractional programming (Shen et al., 2023)	Complex auxiliary-variable reformulation and MM surrogate	Presence of the linear term $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$6
Discrete harmonic analysis (Kibler, 2010)	Unitary finite transform with quadratic-phase kernel	Linear and constant terms in the phase polynomial
Integrable birational maps (Kamp et al., 2018)	Affine generalized Manin transformation for a quadratic pencil	Affine, non-projective coordinate formulation

The optimization literature is the only one among these sources that explicitly uses the term nonhomogeneous quadratic transform (Shen et al., 2023). The discrete Fourier literature instead speaks of a quadratic discrete Fourier transform (Kibler, 2010), while the QRT-map literature speaks of generalized Manin transformations preserving quadratic pencils (Kamp et al., 2018). Consequently, identifying all three with a single canonical “Non-Homogeneous Complex Quadratic Transform” would overstate the consensus in the literature.

Several further limitations are explicit in the sources. In (Shen et al., 2023), the standard QT applies to terms of the form $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$7 with $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$8, nonempty convex $$f_o(\underline{x}) = \sum_{i=1}^n \omega_i M_i(\underline{x}),$$9, and differentiability assumptions for convergence theory. The paper notes that constant terms in numerators and denominators can be included by introducing dummy variables, but it does not provide a general transform for arbitrary expressions such as $x_i\in\mathcal X_i$0 (Shen et al., 2023). In (Kibler, 2010), the kernel is quadratic in one index and bilinear across indices, but it is not the most general quadratic form in both. In (Kamp et al., 2018), “quadratic” refers to the degree of the invariant pencil, not necessarily to the birational degree of the map; the paper remarks that $x_i\in\mathcal X_i$1 is generically represented by rational functions of degree $x_i\in\mathcal X_i$2 (Kamp et al., 2018).

A plausible synthesis is therefore the following. In current arXiv usage, “Non-Homogeneous Complex Quadratic Transform (QT)” is best treated as an umbrella expression for complex quadratic constructions with explicit nonhomogeneous structure, rather than as the name of one universally accepted transform. The most direct exact instance is the nonhomogeneous quadratic transform for complex FP and WMMSE-type problems (Shen et al., 2023). The most natural neighboring analogues are the discrete quadratic Fourier kernel with linear and constant phase terms (Kibler, 2010) and the affine quadratic-pencil Manin/QRT transformation on $x_i\in\mathcal X_i$3 (Kamp et al., 2018).