Discrete Basis Parametrization

• Discrete basis parametrization is the explicit representation of complex models using finite discrete basis functions, offering clear interpretability and reduced dimensionality.
• It employs strategies like spectral grid sampling and group-theoretic labeling to achieve orthogonality, sparsity, and numerical efficiency in computations.
• Applications in quantum physics, imaging, and coding theory demonstrate its ability to improve stability, convergence, and algorithmic tractability.

Discrete basis parametrization refers to the explicit expansion, representation, or solution of complex physical, mathematical, or computational models in terms of a set of basis functions, vectors, or elements defined on a discrete domain. Such representations are central across computational physics, geometry, coding theory, statistical inference, numerical linear algebra, and quantum theory, where they yield sparse or orthogonal formulations, algorithmic tractability, improved numerical conditioning, or physically interpretable decompositions.

1. Foundational Concepts and Mathematical Definition

At its core, discrete basis parametrization entails expressing objects (functions, operators, data, solutions) as finite or countable linear combinations of basis elements $(f = \sum_{n} c_n \phi_n),$ where the $\{\phi_n\}$ are basis functions/vectors sampled or constructed at discrete locations, indices, or parameter-sets, and $\{c_n\}$ are the expansion coefficients. These bases might be multivariate polynomials (Legendre, Bessel, Fourier), hierarchical details adapted to a problem structure, or physical modes (sinograms, oscillator functions). Discreteness is imposed either by quadrature or sampling (eg. Legendre polynomials evaluated at Gauss nodes (Celeghini et al., 2023)), by combinatorial labeling (spin networks (Freidel et al., 2013)), or by explicit construction in finite fields or lattices (toric codes (Baran et al., 2018), DFT eigenfunctions (0808.3214)).

Orthogonality, completeness, and localization are recurrent themes: for instance, the Discrete Variable Representation (DVR) basis in quantum physics utilizes orthonormal sinc functions at regularly spaced locations (Bulgac et al., 2013), ensuring sparse operator structure and efficient diagonalization (see section 3). In hierarchical bases for RBF interpolation (Castrillon-Candas et al., 2011), basis vectors are constructed to be orthogonal to all polynomials up to a specified degree, yielding decoupling and well-conditioned linear systems.

Parametrization also refers to representing the problem-data (eg. geometric deformation fields, group elements, probability densities) by the discrete coefficients in these expansions, often enabling model reduction, statistical estimation, or fast numerical solves.

2. Algorithmic Construction and Parametrization Strategies

Discrete basis construction commonly proceeds by:

• Spectral grid sampling: Legendre, Fourier, Bessel, or other orthogonal polynomial bases are evaluated at discrete quadrature nodes, producing numerically orthogonal finite-dimensional bases (eg. generalized Legendre basis on rectangle (Celeghini et al., 2023); Fourier-sine modes in S-matrix bootstrap (Cordoba, 14 Nov 2025); DVR sinc functions (Bulgac et al., 2013)).
• Group-theoretic labeling: Eigenfunctions are indexed by irreducible representations, character labels, or symmetry-adapted data. E.g., DFT eigenbasis via Weil representation and maximal torus character decomposition (0808.3214); SU(2) intertwiner bases indexed by pairs of coupling labels (S, T) (Freidel et al., 2013).
• Hierarchical/localization mechanisms: Basis vectors are recursively 'lifted' to annihilate lower-order polynomial moments within each spatial region/cube, yielding sparse multiresolution structure (hierarchical basis solver for RBF interpolation (Castrillon-Candas et al., 2011)).
• Algebraic kernel solution: In toric code parametrization, the discrete basis of a lattice ideal is algorithmically constructed from the integer kernel of an augmented block matrix involving the parameterization matrix and grading-lattice basis (Baran et al., 2018).

The discrete parametrization is explicit: once the construction is performed, each coefficient (basis label) corresponds to a physically or algorithmically meaningful component, such as the attenuation of a material along a ray in spectral CT (Gao et al., 2023), a geometric deformation field in reduced-order modeling (Stabile et al., 2019), or a mode index in the celestial holography basis (Freidel et al., 2022).

3. Sparse Operator Structure and Computational Advantages

A principal outcome of discrete basis parametrization is operator sparsity and computational efficiency. When the basis functions are localized (as in DVR (Bulgac et al., 2013) or hierarchical basis (Castrillon-Candas et al., 2011)), local operators (like potential energy, RBF kernels, mesh motion matrices) become (block-)diagonal or near-diagonal, drastically accelerating matrix-vector products and iterative solves.

For example:

• DVR Basis: Potential operators are diagonal; kinetic energy operators are known analytically with only O(N) off-diagonal elements in high dimensions (Bulgac et al., 2013). For fixed ultraviolet (UV) cutoff and box length, the DVR basis requires substantially fewer elements than traditional harmonic oscillator bases.
• Hierarchical Basis for RBF Interpolation: The polynomial part is decoupled; the remaining system is block-sparse, enabling O(N log N) solution complexity and remedying ill-conditioning associated with the polynomial part (Castrillon-Candas et al., 2011).
• Toric Codes: The lattice basis via integer kernel calculation appends basis elements in a single step, bypassing Gröbner basis computation and enabling direct determination of code length, dimension, and minimum distance (Baran et al., 2018).

Consequently, discrete bases are preferred in applications where operator sparsity, localized computation, and explicit parameter mapping are required (quantum few-body problems, coding theory, adaptive mesh interpolation, etc).

4. Physical Interpretability and Model Reduction

Discrete bases often correspond directly to physical variables, observable modes, or symmetry labels. Examples include:

• MSCT Basis Sinograms: Each unknown vector $x_j$ encodes line integrals of basis material attenuation along ray $j$; the nonlinear discrete system manifests the material decomposition problem, with stability and uniqueness established under straightforward spectral/matrix conditions (Gao et al., 2023).
• Gauge Theory Bootstrap Discrete Expansion: S-matrix and spectral densities expanded in real sine-modes furnish explicit semidefinite constraints, allowing fast convex optimization; poles of the physical S-matrix (resonances) are extracted by analytic continuation in the disk via the discrete expansion (Cordoba, 14 Nov 2025).
• Celestial Holography Discrete Basis: Discrete conformal primary basis with boost weights $\Delta\in\mathbb{Z}$ supplies orthogonal towers of memory/Goldstone operators, matching observables and signal reconstruction in gravitational wave analysis, with completeness and orthogonality proven via the Klein–Gordon inner product (Freidel et al., 2022).
• Stochastic Model Reduction (NARMAX): Memory kernels and unresolved tendencies in nonlinear dynamical systems are represented via finite sums over discrete, physically motivated basis functions, leading to efficient parameter estimation and accurate reproduction of long-term system statistics (Chorin et al., 2015).

In all cases, discrete parametrization makes direct use of the problem’s structural symmetries, measurement protocols, or the physical domain’s discretization.

5. Convergence, Stability, and Error Analysis

Convergence properties of discrete basis parametrizations derive from the completeness and orthogonality of the basis functions:

• DVR Basis: UV and IR convergence rates are exponential in the cutoff and grid size; the Hilbert-space dimension required for given accuracy is sharply reduced compared to non-sparse bases (Bulgac et al., 2013).
• Discrete Newton Schemes: For well-posed nonlinear inverse problems (eg. sinogram inversion in MSCT (Gao et al., 2023)), the Newton–Raphson method converges super-exponentially under guaranteed invertibility, with the relative error dropping to machine precision in tens of iterations. Stability bounds are computable via principal-minor analysis of the Jacobian.
• S-Matrix Bootstrap Discrete Expansion: As the truncation order $N$ increases, boundary values and physical predictions stabilize, with the residual tail of the expansion decaying uniformly. Semidefinite constraints and analytic continuation are systematically improvable, with computational cost scaling polynomially in the basis size (Cordoba, 14 Nov 2025).
• Reduced Order Modeling: In the finite-volume reduced-order framework, the basis on an "average mesh" together with Discrete Empirical Interpolation (DEIM) controls the approximation error of geometric deformation and non-affine operators, preserving fidelity across parameter sets (Stabile et al., 2019).
• Hierarchical Basis: The detail levels ensure that high-frequency or localized features are captured, suppressing ill-conditioning and permitting fast iterative solves even for very large $N$ (Castrillon-Candas et al., 2011).

Guidelines for basis size selection and error control are typically furnished by explicit bounds tied to physical problem parameters and desired numerical tolerance.

6. Applications Across Mathematical and Physical Sciences

Discrete basis parametrization finds widespread application:

• Quantum physics/nuclear structure: DVR and Jacobi-coordinate tensor product bases enable efficient and sparse handling of many-body quantum Hamiltonians (Bulgac et al., 2013).
• Tomographic imaging: Data-domain decomposition and sinogram inversion in multispectral CT is tractable only when discrete basis sinogram and image construction is used, yielding robust and stable reconstructions (Gao et al., 2023).
• Coding theory: Parameterized toric codes in algebraic geometry require explicit lattice basis computation over finite fields; the described algorithm yields both code construction and fundamental invariants instantly (Baran et al., 2018).
• Celestial holography/gravity: Discrete conformal primary basis organizes gravitational memory and Goldstone modes with clarity, enabling algebraic and signal-theoretic analyses (Freidel et al., 2022).
• Representation theory/Fourier analysis: Canonical eigenbasis for the DFT on $Z/pZ$, rooted in the Weil representation, provides both structural insight and fast transform algorithms (0808.3214).
• Nonlinear reduced-order modeling and stochastic parametrization: Discrete basis expansions underlie model reduction, data-driven forecast, and stability in physical systems from climate to turbulence (Chorin et al., 2015).

These diverse implementations share the principles of computational sparsity, physical interpretability, and systematic parametrization, each leveraging discrete basis construction to overcome technical and algorithmic hurdles inherent in their domains.

7. Outlook and Generalizations

Ongoing and prospective developments include:

• Advanced optimization in S-matrix and gauge theory bootstraps using higher-dimensional or multi-channel discrete bases (Cordoba, 14 Nov 2025).
• Extensions of discrete variable representation to more general quadratures and complex geometries, further optimizing convergence and sparsity (Bulgac et al., 2013).
• Unifying discrete parametrization across stochastic modeling and data assimilation, particularly in multiscale and nonstationary systems (Chorin et al., 2015).
• Discrete basis methods in celestial holography, soft theorems, and loop algebras (Freidel et al., 2022).
• Systematic adaptation of discrete hierarchical and DEIM-type bases to real-time and parametric computational simulation (Stabile et al., 2019).
• Explicit basis construction in lattice algebraic coding and computational algebraic geometry, with implications for code design and cryptography (Baran et al., 2018).

Discrete basis parametrization forms one of the unifying frameworks in modern computational, physical, and applied mathematics, offering analytic transparency, computational efficiency, rigorous convergence, and direct physical meaning across a broad array of problems.