---
title: Non-Homogeneous Complex Quadratic Transform
url: https://www.emergentmind.com/topics/non-homogeneous-complex-quadratic-transform-qt
type: topic
---

# Non-Homogeneous Complex Quadratic Transform

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 [2312.05726]. 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 [1010.5964], and a **generalized Manin transformation for a quadratic pencil** formulated in affine coordinates and shown to be projectively equivalent to a QRT map [1806.05340]. 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** [2312.05726]. 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 \(2\Re\{x^H(L-K)z\}\)” [2312.05726].

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 [1010.5964]. 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 [1806.05340]. 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 [2312.05726] 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 \(x_i\in\mathcal X_i\) [2312.05726]. The same paper also treats a matrix-variable extension with \(X_i\in\mathbb C^{d\times m}\) and objective \(\sum_i \omega_i \operatorname{tr}(M_i(\underline{X}))\) [2312.05726].

The standard quadratic transform in that paper is the exact reformulation
\[
\sum_{i=1}^n s_i^H(\underline{x})G_i^{-1}(\underline{x})s_i(\underline{x})
\quad \Longleftrightarrow \quad
\sum_{i=1}^n \Big[2\Re\{s_i^H(\underline{x})y_i\}-y_i^H G_i(\underline{x}) y_i\Big],
\]
for \(s_i:\mathcal X\to\mathbb C^\ell\) and \(G_i:\mathcal X\to\mathbb S_{++}^{\ell\times\ell}\) [2312.05726]. For the weighted-ratio model, the transformed objective becomes
\[
f_q(\underline{x},\underline{y}) = \sum_{i=1}^n \omega_i\Bigg[ 2\Re\{x_i^H A_i^H y_i\} -\sum_{j=1}^n y_i^H B_{ij}x_jx_j^H B_{ij}^H y_i \Bigg],
\]
or equivalently
\[
f_q(\underline{x},\underline{y}) = \sum_{i=1}^n \Big[ 2\Re\{\omega_i x_i^H A_i^H y_i\} - x_i^H D_i x_i \Big],
\qquad
D_i=\sum_{j=1}^n \omega_j B_{ji}^H y_j y_j^H B_{ji}
\]
[2312.05726]. The optimal auxiliary update is
\[
y_i^\star = \Bigg(\sum_{j=1}^n B_{ij}x_jx_j^H B_{ij}^H\Bigg)^{-1}(A_i x_i).
\]

The specifically **nonhomogeneous** step is obtained from the matrix inequality
\[
x^H L x \le x^H K x + 2\Re\{x^H(L-K)z\} + z^H(K-L)z,
\qquad L\preceq K,
\]
which yields a lower-bounded objective
\[
f_q(\underline{x},\underline{y}) \ge f_t(\underline{x},\underline{y},\underline{z}),
\]
where
\[
\begin{aligned}
f_t(\underline{x},\underline{y},\underline{z}) = \sum_{i=1}^n \Big[ &2\Re\{\omega_i x_i^H A_i^H y_i + x_i^H(\lambda_i I-D_i)z_i\} \\
&+ z_i^H(D_i-\lambda_i I)z_i -\lambda_i x_i^H x_i \Big].
\end{aligned}
\]
The paper states that this bound is called **nonhomogeneous** because of the linear term \(2\Re\{x^H(L-K)z\}\) [2312.05726]. 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 [2312.05726], the standard quadratic transform is exact with respect to the auxiliary variables \(y_i\): optimizing over \(y_i\) recovers the original objective. By contrast, the variable \(z_i\) 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 \(z_i=x_i\), which yields the update
\[
z_i^\star = x_i
\]
and transforms the method into a tangent-surrogate scheme [2312.05726].

For the \(x_i\)-subproblem, the paper gives
\[
x_i^\star = \arg\min_{x_i\in\mathcal X_i} \left\| D_i^{1/2}\Big(x_i-\omega_i D_i^{-1}A_i^H y_i\Big) \right\|_2
\]
for the standard QT, and, after the nonhomogeneous bound,
\[
x_i^\star = \mathcal P_{\mathcal X_i} \left( z_i + \frac{1}{\lambda_i}\big(\omega_i A_i^H y_i - D_i z_i\big) \right)
\]
with \(\lambda_i\ge \lambda_{\max}(D_i)\) [2312.05726]. The paper then shows that, after substituting optimal \(y^k\) and \(z^k=x^{k-1}\), the nonhomogeneous QT update becomes
\[
x_i^k = \mathcal P_{\mathcal X_i} \left( x_i^{k-1} + \frac{1}{2\lambda_i^k} \frac{\partial f_o(\underline{x}^{k-1})}{\partial x_i^c} \right),
\]
that is, a projected gradient step in the complex domain [2312.05726]. 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 \((3)\) [2312.05726]. For local rate analysis, the paper gives
\[
f_o(\underline{x}^\ast)-f_o(\underline{x}^{k}) \le \frac{2\Lambda R^2 + 2LR^3/3}{k+3},
\qquad k\ge 2,
\]
with \(\Lambda=\Lambda_q\) for QT and \(\Lambda=\Lambda_t\) for nonhomogeneous QT, and notes that both have \(O(1/k)\) local objective error [2312.05726]. For the extrapolated method, if \(\nabla f_o\) is \(C\)-Lipschitz and \(\lambda_i^k = 1/(2C)\), the paper states
\[
f(\underline{x}^\ast)-f(\underline{x}) \le \frac{2C\cdot [f(\underline{x}^\ast)-f(\underline{x}^0)]}{(k+1)^2},
\qquad k\ge 1,
\]
that is, \(O(1/k^2)\) objective-value error [2312.05726].

These results delimit the most precise contemporary meaning of “non-homogeneous complex quadratic transform” on arXiv: 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 [1010.5964]. In the classical transform notation used there,
\[
y_{n} = \sum_{m = 0}^{d-1} \left( {\bf F}_{ra} \right)_{m n}  \, x_{m},
\]
with
\[
({\bf F}_{ra})_{n m} = \frac{1}{\sqrt{d}} q^{n(d -n) a/2 + (d-1)^2 r / 4 + n[m -(d-1)r/2]},
\qquad
q=\exp\left(\frac{2\pi i}{d}\right)
\]
[1010.5964]. Expanding the phase gives
\[
({\bf F}_{ra})_{nm} = \frac{1}{\sqrt d} q^{-\frac a2 n^2 + mn + \left(\frac{ad}{2}-\frac{(d-1)r}{2}\right)n + \frac{(d-1)^2r}{4}}.
\]

This kernel contains a quadratic term in \(n\), a bilinear term \(mn\), a linear term in \(n\), and a constant term. The paper therefore supports describing the transform as a **complex quadratic-phase kernel with non-homogeneous terms** [1010.5964]. It is not, however, the most general quadratic form in both indices, since the kernel as written has no \(m^2\) term.

The transform is unitary: the paper states, “For \(d\) arbitrary, the matrix \({\bf F}_{ra}\) is unitary” [1010.5964]. It also satisfies the Parseval–Plancherel identity
\[
\sum_{n = 0}^{d-1} \overline{y_{n}} \, y'_{n} = \sum_{m = 0}^{d-1} \overline{x_{m}} \, x'_{m},
\]
and it factorizes as
\[
{\bf F}_{ra} = {\bf D}_{ra} {\bf F},
\qquad {\bf F} = {\bf F}_{00},
\]
where \({\bf D}_{ra}\) is diagonal [1010.5964]. 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 \(a=r=0\) it reduces exactly to the ordinary DFT [1010.5964]. The basis vectors \(B_{ra}=\{|a\alpha;r\rangle\}\) are orthonormal, unbiased with respect to the computational basis, and for prime \(d=p\), the collection \(B_{r0},B_{r1},\ldots,B_{r,p-1},B_p\) forms a complete set of \(p+1\) MUBs [1010.5964]. 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, [1806.05340] supplies a third interpretation. Its central object is the generalized Manin transformation preserving a pencil
\[
P_{\alpha,\beta}(u,v):=\alpha F_a(u,v)+\beta F_b(u,v),
\]
with \(F_a,F_b\) polynomials of total degree \(N\) [1806.05340]. For the quadratic case \(N=2\),
\[
F_a(u,v):=a_1 + a_2 u + a_3 v + a_4 u^2 + a_5 u v + a_6 v^2,
\]
and similarly for \(F_b\) [1806.05340].

The generalized Manin involution \(\iota_p\) is constructed in affine coordinates by taking the line through \((u,v)\) and an involution point \(p=(c,d)\),
\[
(x,y)=(u+(c-u)z,\;v+(d-v)z),
\]
and choosing the second intersection with the same curve of the pencil, which is characterized by
\[
F_a(x,y)F_b(u,v)=F_a(u,v)F_b(x,y)
\]
[1806.05340]. In the quadratic case the resulting affine formula is
\[
\iota_p(u,v)=(u,v)+z(c-u,d-v),
\]
with
\[
z = -2\, \frac{F_a(0)F_b^{(z)}(0)-F_a^{(z)}(0)F_b(0)} {F_a(0)F_b^{(z,z)}(0)-F_a^{(z,z)}(0)F_b(0)}
\]
and explicitly computable directional derivatives [1806.05340].

The paper emphasizes a distinctive feature of the quadratic case: “For \(N=2\) we are free to choose the involution points \(p,q\) and there are no constraints on the pencil,” except that involution points are not base points [1806.05340]. The composition
\[
\tau_{p,q}=\iota_q\circ \iota_p
\]
preserves the rational first integral
\[
I(u,v)=\frac{F_a(u,v)}{F_b(u,v)}
\]
and is measure-preserving with density
\[
\rho(u,v)=\frac{1}{L(u,v)F_a(u,v)},
\]
where \(L(u,v)=0\) is the line through the involution points [1806.05340].

Most importantly for the relation to QRT theory, the paper proves that every generalized Manin transformation for \(N=2,3,4\) can be brought to QRT form by a projective collineation. If
\[
L(u,v)=(d-f)(u-e)-(c-e)(v-f),
\]
then the fractional affine transformation
\[
(u,v)\rightarrow \left(\frac{A(u-e)+B(v-f)}{L},\frac{C(u-c)+D(v-d)}{L}\right)
\]
sends \(p\mapsto(\infty,0)\) and \(q\mapsto(0,\infty)\), and “brings the generalised Manin transformation into QRT form” [1806.05340]. In this literature, therefore, a “non-homogeneous complex quadratic transform” is most faithfully interpreted as an affine birational map on \(\mathbb C^2\) 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 [2312.05726] | Complex auxiliary-variable reformulation and MM surrogate | Presence of the linear term \(2\Re\{x^H(L-K)z\}\) |
| Discrete harmonic analysis [1010.5964] | Unitary finite transform with quadratic-phase kernel | Linear and constant terms in the phase polynomial |
| Integrable birational maps [1806.05340] | 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** [2312.05726]. The discrete Fourier literature instead speaks of a **quadratic discrete Fourier transform** [1010.5964], while the QRT-map literature speaks of **generalized Manin transformations** preserving **quadratic pencils** [1806.05340]. 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 [2312.05726], the standard QT applies to terms of the form \(s_i^H G_i^{-1}s_i\) with \(G_i(\underline{x})\in \mathbb S_{++}^{\ell\times\ell}\), nonempty convex \(\mathcal X_i\), 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 \(\frac{|a(x)|^2+c(x)}{b(x)}\) [2312.05726]. In [1010.5964], the kernel is quadratic in one index and bilinear across indices, but it is not the most general quadratic form in both. In [1806.05340], “quadratic” refers to the degree of the invariant pencil, not necessarily to the birational degree of the map; the paper remarks that \(\iota_p\) is generically represented by rational functions of degree \(3\) [1806.05340].

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 [2312.05726]. The most natural neighboring analogues are the discrete quadratic Fourier kernel with linear and constant phase terms [1010.5964] and the affine quadratic-pencil Manin/QRT transformation on \(\mathbb C^2\) [1806.05340].

Source: https://www.emergentmind.com/topics/non-homogeneous-complex-quadratic-transform-qt