Bilinear Scheme: Theory & Applications

• Bilinear schemes are frameworks utilizing maps that are linear in each argument, unifying methods in algebraic geometry, quantum field theory, numerical analysis, and machine learning.
• They enable efficient computation through structured decompositions, facilitating moduli classifications, lattice renormalization, and matrix-free algorithms.
• Applications span from reducing complex dynamics in system identification and precise image interpolation to robust model compression in modern machine learning.

A bilinear scheme, as encountered in modern mathematical sciences, refers to conceptual, algebraic, or computational frameworks organized around bilinear maps—mappings linear in each argument separately. The term encompasses a broad spectrum spanning algebraic geometry, operator renormalization, numerical linear algebra, system identification, stochastic integration, optimization, and computational imaging. Bilinear schemes arise as organizing principles for efficient computation, geometric classification, robust algorithmic structures, and precise representation of processes involving products, interactions, or nonlinearities that separate cleanly across two modules or spaces.

1. Algebraic-Geometric Bilinear Schemes and Moduli Spaces

The bilinear scheme $$\mathrm{Bilin}_{d_1,d_2,d_3}^{r_1,r_2}(\mathbb{A}^n)$$, introduced in (Obcowska, 4 Jan 2026), formalizes the moduli of bilinear maps of locally free modules on algebraic varieties. Given $S=\mathbb{k}[x_1,\dots,x_n]$, and ranks $d_1,d_2,d_3$ with presentation ranks $r_1$ and $r_2$, $\mathrm{Bilin}$ is a functor:

$$\mathrm{Bilin}^{r_1, r_2}_{d_1, d_2, d_3}(\mathbb{A}^n)\colon \mathrm{Sch}_{/\mathbb{k}}^\mathrm{op} \to \mathrm{Set}$$

which classifies surjections $$p_1\colon \mathcal{O}_T \otimes S^{\oplus r_1}\to M_1$$, $$p_2\colon \mathcal{O}_T \otimes S^{\oplus r_2}\to M_2$$, and a bilinear map $\pi\colon M_1 \otimes M_2 \to M_3$ subject to $M_i$ being locally free of rank $d_i$, and identification up to kernel. $\mathrm{Bilin}$ generalizes the Quot scheme, and when $r_1 = r_2 = 1$, $d_1 = d_2 = d_3 = d$, recovers the Hilbert scheme of $d$ points.

There exists a closed embedding of $\mathrm{Bilin}$ as a subscheme in the product $Q_1 \times Q_2 \times Q_3$, where each $Q_i$ is an appropriate Quot scheme. The tangent space at a point $x$ in $\mathrm{Bilin}$ is cut out from the ambient product by linear constraints encoding bilinear compatibility via commutative diagrams (see (Obcowska, 4 Jan 2026), Thm. 3.8).

Two distinguished loci structure the geometry:

• Tuple-of-points locus $W$: Points where $M_i \cong S/I$ for $d$ distinct points, with "multiplication tensor" corresponding to minimal border rank and dimension $n d + (r_1 - 1) d + (r_2 - 1) d$.
• Totally degenerate locus $Z_m$: $M_i \cong (S/m)^{\oplus d}$ for a closed point $m$, with dimension $(r_1 - d)d + (r_2 - d)d + (d^2 - d)d$.

For $d \geq 3$, all $n$, and $r_i \geq d$, $\mathrm{Bilin}$ is reducible. In the detailed case $d=2$, $r_1=r_2=2$ in $\mathbb{A}^1$, the scheme is irreducible, with only mild singularities at the totally degenerate locus and those induced by non-cyclic modules (Obcowska, 4 Jan 2026).

2. Bilinear Renormalization Schemes in Quantum Field Theory

Bilinear renormalization schemes are central in lattice field theory for extracting multiplicative renormalization constants $Z_\Gamma$ of composite operators. The temporal-moment bilinear scheme (Ishikawa et al., 2020) implements renormalization by matching moments of two-point functions:

• Vacuum polarization: $$q^2 \Pi(q^2) = \int d^4 x\, e^{iqx}\, \langle0|T j_5(x) j_5(0)|0\rangle$$, with $$M_k = \left. \partial^k \Pi(q^2) / \partial(q^2)^k \right|_{q^2=0}$$.
• Lattice moments: $$G_n = \sum_t (t/a)^n\, G(t) = \left. \partial^k \Pi^{\mathrm{lat}}(q^2) / \partial(q^2)^k \right|_{q^2=0}$$, $n = 2k+2$.
• Renormalization condition: $$Z_\Gamma^{\mathrm{latt}}(\mu, a) = \sqrt{M_k^{\overline{\mathrm{MS}}}(\mu) / G_n^{\mathrm{lat}}(a)}$$

This protocol is non-perturbative, gauge-invariant, avoids gauge-fixing artifacts, and is robust to finite volume effects. The continuum limit is checked against independent X-space schemes, with discretization errors removed by $a^2 \to 0$ fits and truncation errors estimated via leading $\alpha_s^4$ bands.

Complementary approaches, notably RI/MOM and RI/SMOM intermediate schemes (He et al., 2021), impose momentum-space projection conditions with different kinematic setups. RI/SMOM improves infrared behavior for scalar/pseudoscalar operators and mitigates chiral symmetry breaking artifacts, with matching to $\overline{\mathrm{MS}}$ performed via perturbative factors. After continuum extrapolation, both schemes yield compatible $Z_\Gamma$, with systematic uncertainties dominated by lattice spacing and truncation error.

3. Bilinear Interpolation and Geometry Mapping in Numerical Analysis

Bilinear interpolation is a foundational mapping technique in numerical geometry, image processing, and finite element methods. The standard isoparametric bilinear mapping for quadrilaterals (Diab, 2019) uses four Lagrange shape functions:

$$N_i(\xi, \eta) = \frac{1}{4}(1+\xi_i\xi)(1+\eta_i\eta), \quad i = 1,2,3,4$$

to map reference ($\xi$, $\eta$) domains to physical coordinates via

$$x(\xi,\eta) = \sum_{i=1}^4 N_i(\xi,\eta)\,x_i, \quad y(\xi,\eta) = \sum_{i=1}^4 N_i(\xi,\eta)\,y_i$$

The Jacobian $J(\xi,\eta)$ allows computation of area and moments of inertia. For straight-edged quadrilaterals, all area and low-order moment integrals can be evaluated exactly, and computational cost is minimal. Limitations include inability to handle curved edges and underprediction of higher plate-bending modes.

Extensions employing complete Pascal polynomial bases (quadratic, 6-term) enable accurate geometry mapping beyond bilinear, at the cost of greater algebraic complexity and pole-finding (Diab, 2019).

In quantum computing contexts, bilinear interpolation translates to quantum circuits for FRQI images, where modularized assignments, increments, and weighted color computations yield up-scaling and down-scaling circuits with demonstrable improvements in PSNR and SSIM compared to nearest-neighbor methods (Yan et al., 2020).

4. Bilinear Schemes for Identification, Reduction, and Approximate Inference

System identification theory frequently employs bilinear models for efficient representation and reduction of dynamics. The Loewner framework with Volterra series (Karachalios et al., 2020) provides a nonintrusive method to identify reduced-order bilinear models:

• Volterra expansion: $y(t) = \sum_{n=1}^N y_n(t)$ via symmetric multilinear kernels $h_n$.
• Carleman bilinearization: Embeds nonlinear dynamics into bilinear ODE forms $\dot x = A x + \sum N_i x u_i + B u$.
• Loewner reduction: Rank-revealing SVD of matrix pencils recovers the minimal linear realization $A_r$.
• Bilinear operator recovery: Second-order kernel measurements fit the $N_{i,r}$ matrices, yielding the full bilinear surrogate.

Accuracy hinges on the number of Volterra kernels considered, with weakly nonlinear systems often needing only first and second order, and the Loewner framework providing robust reduction even for large noisy datasets.

5. Bilinear Schemes in Optimization and Numerical Integration

The representation of second-order structure via bilinear forms is central in high-dimensional optimization. The bilinear Hessian framework (Carlsson et al., 5 Feb 2025) avoids explicit matrix formation by only contracting the Hessian with vectors:

• Bilinear Hessian: $H_x[u,v] = d^2f|_x(u,v)$, computable via polarization or directly via operator equations $\mathcal H_x(u)$.
• Algorithms: Hessian-vector products, Newton's method, Daniel's CG rule, and truncated CG are all implemented matrix-free using only operator actions and inner products.
• Complexity: Each step requires only $O(n\log n)$ (e.g., for Poisson likelihoods with FFT forward/adjoint), enabling application to imaging problems with $n\sim10^8$–$10^9$ without $O(n^2)$ storage.

First-order weak balanced schemes for bilinear SDEs (Mardones et al., 2014) also exploit bilinear structure to ensure stability and order-optimal accuracy:

$$Z_{n+1} = Z_n + BZ_n\Delta + H(\Delta) (Z_{n+1} - Z_n)\Delta + \sum_{k=1}^m\sigma^k Z_n \sqrt{\Delta} \xi_n^k$$

with stabilization weights $H(\Delta)$ chosen by heuristic or optimization over Lyapunov exponents, guaranteeing $O(\Delta)$ weak error and long-time stability.

6. Bilinear Decomposition in Machine Learning and Message Passing

Bilinear schemes are increasingly used in machine learning for inductive bias and model compression. Bilinear Value Networks (Hong et al., 2022) decompose the $Q$-function as

$Q(s, a, g) = f(s, a)^\top \phi(s, g)$

where $f$ encodes local dynamics and $\phi$ encodes global goal geometry. This low-rank structure yields superior generalization to out-of-distribution goals and improved data-efficiency in off-policy reinforcement learning.

In probabilistic inference, parametric bilinear G-AMP (Parker et al., 2015) generalizes AMP to bilinear forms:

$$z_m = \sum_{i,j} b_i z_m^{(i,j)} c_j + \text{noise}$$

By deploying AMP simplifications under large random tensor assumptions and Onsager corrections adapted to the bilinear context, it supports tractable inference for blind deconvolution, compressive sensing, and matrix uncertainty. The schema is flexible to various priors and scales, converging rapidly to near-MMSE performance in practice.

7. Structural Themes and Extensions

A recurring theme across the spectrum of bilinear schemes is their structural leverage—by exploiting separability or factorization intrinsic to the bilinear map, one achieves dramatic simplification, computational scalability, or geometric clarity. In moduli problems, this yields irreducible components, tangent spaces, and singularities; in signal processing and learning, it enables robust generalization and efficient inference; in numerical analysis, it achieves exact quadrature among polynomials and stable time-stepping for SDEs.

Extensions include replacement by multilinear (e.g., tensor) schemes, incorporating border rank and secant varieties, and cross-fertilization between algebraic geometry and computational numerical methods. The flexibility and scalability of bilinear schemes continues to drive research in representation theory, numerical analysis, and applied algorithmic sciences.