Quadratically Generated Projections

• Quadratically generated projections are precise mathematical mappings defined by quadratic forms, enabling analytic, algebraic, and optimization applications.
• They provide explicit representations for convex hulls, quadratic varieties, and operator algebras, unifying diverse approaches in convex and nonconvex analyses.
• These projections serve as key algorithmic tools in optimization, geometric analysis, and statistical physics, yielding near-optimal reductions and universal transport bounds.

A quadratically generated projection refers to the process by which projections—mathematical mappings, reductions, or lifts—are formed or characterized via quadratic structure. In both contemporary optimization and applied mathematics, these projections manifest through semialgebraic parametrizations, semidefinite programming relaxations, explicit analytic projection onto quadratic varieties (e.g., ellipsoids, paraboloids), and algebraic schemes (e.g., generation of $C^*$-algebras via projections). The unifying principle is that the geometry, feasibility, or algebraic generation is encoded by quadratic forms and their associated projections in function space, Hilbert space, or algebraic structures. Quadratically generated projections thus serve as a bridge between convex and nonconvex geometry, optimization, algebra, and mathematical physics—providing explicit analytic representations, algorithmic tools, and structure theorems critical in both theory and practice.

1. Semialgebraic Convex Hulls via SDP Projection

When a set $V \subset \mathbb{R}^m$ is parameterized by quadratic polynomials $f_1,\ldots,f_m$ over $x\in \mathbb{R}^n$ constrained to a quadratic domain $T$ (e.g., $q(x)\geq 0$ where $q$ is quadratic), the convex hull $\operatorname{conv}(V)$ can be exactly represented (up to closure) by projecting a spectrahedron defined through a “first-order moment” semidefinite program (SDP). This construction proceeds via:

• Lifting $x$ to a vector $y$ with coordinates corresponding to all monomials of degree $\leq 2$, including affine and quadratic products;
• Imposing the moment matrix $M_1(y)\succeq 0$ and corresponding localizing constraints for each quadratic;
• Enforcing $y_0=1$ (homogeneity) and lifting any further constraints;
• Projecting the feasible $y$ onto the original coordinate space via $\pi(y) = (a_i + b_i^T x + F_i\cdot X)_{i=1}^m$.

Nie’s main theorem establishes that for $T$ given by one quadratic (and compact), $\operatorname{conv}(V)=\pi(S)$, i.e., no further hierarchy is needed. When $T$ is noncompact, equality holds for closures. This first-order relaxation also handles certain multi-constraint and rational quadratic parametrizations via homogenization and an additional quadratic constraint (Nie, 2011).

Constraint set $T$	Moment formulation	Convex hull description
$q(x)\geq 0$/$q(x)=0$ (single quad.)	$M_1(y)\succeq 0$, $q$-localizing	$\operatorname{conv}(V)=\pi(S)$ (compact case)
Two homogeneous quadratics	$X\succeq 0$, $B_j\cdot X-c_j y_0 \geq 0$ ($j=1,2$)	$\operatorname{conv}(V)=\pi(S)$ under feasibility
Rational quadratics	Homogenization $+$ moment SDP	De-homogenize projection

This representation underlies convex geometry for semialgebraic images and provides exact semidefinite representations without recourse to infinite hierarchies (Nie, 2011).

2. Explicit Analytic Quadratic Projections: Quadratic Varieties

Projecting onto a quadratic hypersurface or variety with level set defined by $Q(x)=x^T A x + b^T x + c=0$ is reducible, via Lagrangian duality, to a one-dimensional nonlinear root-finding problem for a scalar Lagrange multiplier:

• The KKT conditions give $(I+\lambda A)x = y - \frac{\lambda}{2} b$;
• For each admissible $\lambda$, $x(\lambda)$ can be written explicitly;
• The unique feasible $\lambda^*$ is the root of a secular equation (strictly monotone in the admissible interval);
• The projection $x^* = x(\lambda^*)$ is the unique minimizer except for isolated degenerate situations, which can be computed separately and minimized over (Hoorebeeck et al., 2022).

Projection schemes for intersections with additional convex sets (e.g., box constraints) are handled via alternating projection and Douglas–Rachford splitting, embedded with exact or approximate quadratic projections per-iteration, yielding high computational efficiency relative to nonlinear programming solvers (Hoorebeeck et al., 2022).

Algebraic projections onto more complex quadratics, such as rectangular hyperbolic paraboloids in infinite-dimensional Hilbert space, reduce to solving either a quintic or cubic in the Lagrange multiplier. Existence, uniqueness, and multiplicity of projections are determined by monotonicity and signature of the associated equations (Bauschke et al., 2022).

For curve-like objects, projection onto the graph of a quadratic ($y=f(x)=ax^2+bx+c$) reduces to finding critical points of a quartic in $x$ or equivalently, solving a depressed cubic for the root. Analytic closed-form solutions via Cardano’s and trigonometric formulas allow explicit description, including higher-dimensional paraboloid analogues (Aragón-Artacho et al., 27 Dec 2025).

3. Projections in the Context of Geometric and Physical Structures

Stereographic and other geometric projections adapted to quadrics, such as the ellipsoid and elliptic paraboloid, are constructed by mapping from a distinguished “north pole” onto a reference plane. The mapping, its inverse, and the induced metric/geometric invariants are all expressed in terms of rational functions of the pre-image variables. This delivers explicit parameterizations, allowing computation of first/second fundamental forms, Gaussian curvature, and geometric quantities (area, arc length, eccentricity of sections) purely in terms of the quadratic generators of the surface and projection (Barboza et al., 9 Jun 2025). In the paraboloid case, noncompactness induces singularities corresponding to “points at infinity” in the parameter domain.

4. Algorithmic Quadratic Projections in Optimization

Quadratically generated projections arise as explicit variable reduction tools in high-dimensional quadratic programming (QP), where a (linear) projection matrix $P$ is constructed to map from a high-dimensional $x\in\mathbb{R}^N$ to a lower-dimensional $y\in\mathbb{R}^K$. The matrix $P$ is itself generated instance-specifically by a graph neural network trained via bilevel optimization:

• The inner problem solves the low-dimensional (quadratically parametrized) QP for each instance;
• The outer problem trains the generator $f_\theta$ to output projections $P_\theta$ that minimize the expected objective of the lifted solutions;
• Derivatives are computed efficiently via the envelope theorem, bypassing solver backpropagation;
• This yields strong theoretical generalization bounds, controlled by covering numbers of the network class and sensitivity of the original QP to changes in $P$ (Iwata et al., 30 Oct 2025).

This “quadratically generated projection” mechanism exploits problem structure, leading to instance-specific reductions that provably maintain near-optimality.

5. Quadratic Generation in Algebraic and Operator-Theoretic Contexts

In the theory of $C^*$-algebras, a powerful result due to Hu and Xue establishes that for a $C^*$-algebra $A$ generated by $n$ elements, its matrix algebra $M_k(\widetilde{A})$ is generated by $k$ projections that are mutually unitarily equivalent and almost mutually orthogonal if and only if $(k-1)(k-2)\geq 2n$:

• Each projection is constructed as a quadratic (with small perturbation) in the algebra’s matrix elements;
• The full matrix algebra is recovered as the spectrum of polynomials generated by these projections;
• Reduction to three projections for sufficiently large matrix algebras is achieved via matrix amplification and single-generation results;
• The approach extends to purely infinite/simple, and AF algebras, with the minimal number of projections determined by divisibility properties of the $K_0$ class of the unit (Hu et al., 2012).

This construction formalizes the algebraic sense in which noncommutative or nonclassical objects may be “quadratically generated” via projections and matrix dissection.

6. Quadratic Projections in Physics: Hydrodynamic and Information-Theoretic Contexts

In nonequilibrium statistical physics, especially in hydrodynamic projection theory, projections onto spaces of charges and currents of quadratic extensivity deliver exact and universal lower bounds for transport coefficients such as diffusion and superdiffusion exponents:

• The diffusive Hilbert space is constructed by factoring out translations in space and time, and pairing via the Green–Kubo expression;
• Bilinear (“quadratically extensive”) charges represent the second-order (covariant) derivative of the manifold of maximum entropy states, with projection onto such directions controlling the contribution to diffusion from scattering;
• Projecting currents onto the subspace spanned by these bilinear charges gives lower bounds on Onsager coefficients;
• Fractionally extensive charges and their projections yield universal lower bounds on superdiffusive exponents, recovering predictions such as the KPZ and Lévy exponents (Doyon, 2019).

This reveals a direct connection between the underlying quadratic (bilinear) structure of conserved quantities and universal features of nonequilibrium transport, accessed purely through projection geometry in operator/Hilbert space.

7. Synthesis and Outlook

Quadratically generated projections unify several central themes in computational mathematics, optimization, operator theory, geometry, and physical modeling:

• They deliver exact or nearly-exact representations of convex hulls of nonlinear algebraic images without recourse to infinite hierarchies (Nie, 2011).
• They support explicit analytic computations for projections onto, or along, quadratic manifolds, critical for applications in signal processing, power systems, and geometry (Hoorebeeck et al., 2022, Bauschke et al., 2022, Aragón-Artacho et al., 27 Dec 2025).
• They drive modern algorithmic reductions in quadratic optimization via data-driven learning (Iwata et al., 30 Oct 2025).
• They play a central role in the generation and analysis of operator algebras and transport properties in statistical physics (Hu et al., 2012, Doyon, 2019).

A plausible implication is that advances in the analytic tractability and algebraic generation of projections via quadratic forms will continue to enable more efficient optimization algorithms, deeper understanding of quantum and statistical systems, and refined geometric analysis across mathematics and its applications.