Analytic Regression via Lattice Reduction

• Analytic regression with lattice reduction is a method to recover exact analytic expressions from numerical data by detecting rational integer relations within a known function space.
• It leverages high-precision evaluations and lattice reduction algorithms such as LLL or PSLQ to construct and reduce an integer lattice that encodes the analytic formula.
• The approach is applied in Feynman integrals and high-dimensional linear regression, balancing trade-offs between precision, numerical conditioning, and function space constraints.

Analytic regression with lattice reduction is a method for recovering an exact analytic expression from numerical data when the target is expected to lie in a known function space and the sought coefficients are rational. In the formulation developed for Feynman integrals, one samples an unknown function $f$ at sufficiently many points and with sufficiently high precision, evaluates a prescribed basis $\mathcal{B}_i$ at the same points, and then uses lattice reduction to detect the integer relation encoding the rational coefficients of the analytic answer (Barrera et al., 23 Jul 2025). The same broad motif—replacing sparsity or generic smoothness assumptions by arithmetic structure and then invoking LLL or related integer-relation algorithms—also appears in high-dimensional linear regression, integer-relation detection, and lattice-hardness reductions (Gamarnik et al., 2018, Gamarnik et al., 2019, Gupte et al., 2024).

1. Definition and conceptual scope

In the sense introduced for Feynman integrals, analytic regression addresses problems where one can, in principle, produce data to arbitrary accuracy but seeks an exact analytic description rather than a numerical approximation. The method is designed for settings in which “the analytic answer is a rational linear combination of these functions,” with

$$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$

where the basis functions $\mathcal{B}_i$ are known and the coefficients $c_i$ are rational numbers (Barrera et al., 23 Jul 2025).

This construction differs from numerical symbolic regression as characterized in the source material: the latter is “not well-suited” when only approximations are produced, whereas analytic regression uses high-precision sampling together with prior analytic knowledge of the relevant function space (Barrera et al., 23 Jul 2025). In the Feynman-integral setting, that function space may consist of generalized polylogarithms up to fixed weight, and the method is presented as a “bottom-up approach” that “neatly complements the top-down Landau-bootstrap approach of trying to constrain the exact answer using the analytic structure alone” (Barrera et al., 23 Jul 2025).

A broader meaning of regression with lattice reduction appears in high-dimensional linear models. There, the objective is not reconstruction of a symbolic functional expression but exact inference of a coefficient vector under arithmetic constraints such as $Q$-rationality or mixed-support assumptions, using LLL and PSLQ rather than sparsity-promoting methods (Gamarnik et al., 2018, Gamarnik et al., 2019). This suggests a family resemblance: analytic regression with lattice reduction is unified less by a single application domain than by a shared algorithmic principle—exact recovery from approximate or undersampled data by exploiting arithmetic structure.

2. Mathematical formulation and lattice encoding

The central regression problem for analytic reconstruction is to determine the rational coefficients $c_i$ in a known basis from numerical samples. For $n$ basis functions and $p$ sampled points, the procedure forms an integer matrix by scaling and rounding the function evaluations: $$M=\text{round}\ 10^s \begin{pmatrix} f(\mathbf{x}_1) & \dots & f(\mathbf{x}_p) & 1 & 0 & \dots & 0 \ \mathcal{B}_1(\mathbf{x}_1) & \dots & \mathcal{B}_1(\mathbf{x}_p) & 0 & 1 & \dots & 0 \ \vdots & & \vdots & & & \ddots & \ \mathcal{B}_n(\mathbf{x}_1) & \dots & \mathcal{B}_n(\mathbf{x}_p) & 0 & 0 & \dots & 1 \end{pmatrix},$$ where $\mathcal{B}_i$0 is chosen so that the sampled numbers preserve their significant digits after multiplication by $\mathcal{B}_i$1 (Barrera et al., 23 Jul 2025).

The next step is lattice construction: the matrix of evaluations is concatenated with an identity matrix to form a lattice basis in higher dimension, and a lattice reduction algorithm such as LLL or L2 is applied to seek a short vector (Barrera et al., 23 Jul 2025). The short vector is expected to encode the integer relation

$\mathcal{B}_i$2

up to an overall common factor, yielding

$\mathcal{B}_i$3

to within $\mathcal{B}_i$4 error (Barrera et al., 23 Jul 2025).

The underlying rationale is the standard one from integer relation detection: given “nearly integer vectors” produced from high-precision sampled data, LLL finds short vectors that correspond to rational relations of modest coefficient size (Barrera et al., 23 Jul 2025). In a closely related regression literature, PSLQ is explicitly used for integer relation detection and is combined with LLL for high-dimensional linear inference under rationality or mixed-support assumptions (Gamarnik et al., 2019). The two strands share the same computational core, though their targets differ: basis coefficients of special functions in one case, regression coefficients in the other.

3. Workflow for exact analytic reconstruction

The method for Feynman integrals is organized into two modules: preparation and analytic regression via lattice reduction (Barrera et al., 23 Jul 2025).

The preparation stage begins with functional-space determination. The source material states that one uses a priori analytic knowledge, including “Landau singularity analysis, symbol bootstrap, or the software SOFIA,” to determine the basis functions relevant for the integral under study (Barrera et al., 23 Jul 2025). High-precision evaluation then produces numerical values for both $\mathcal{B}_i$5 and $\mathcal{B}_i$6, using “AMFlow (for the integral) and GiNaC or PolyLogTools (for GPLs)” (Barrera et al., 23 Jul 2025).

The regression stage consists of matrix construction, lattice construction, lattice reduction, and solution extraction. After scaling and rounding, one applies “a lattice reduction algorithm (e.g., LLL or L2)” and then reads off the non-trivial integer relation from the reduced basis to recover the analytic formula (Barrera et al., 23 Jul 2025). The method is exact in the sense emphasized by the paper: it recovers “analytic formulas with rational coefficients, not just floating-point fits” (Barrera et al., 23 Jul 2025).

This workflow depends critically on function-space awareness. The source emphasizes that the correct alphabet, weight, and possible branch structures matter; an over-complete basis “may increase computational cost but generally does not prevent regression” (Barrera et al., 23 Jul 2025). A plausible implication is that analytic insight is not auxiliary but constitutive: lattice reduction performs the final arithmetic decoding, but the feasible search space is supplied by physics or function theory.

4. Precision, conditioning, and sampling trade-offs

A defining feature of analytic regression with lattice reduction is the trade-off between the number of data points, the number of functional predicates, the numerical precision, and the available compute (Barrera et al., 23 Jul 2025). The paper gives both an empirical and theoretical relation between digits $\mathcal{B}_i$7 and sampling points $\mathcal{B}_i$8: $\mathcal{B}_i$9 where $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$0 is related to the coefficient size and $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$1 is a floor dictated by the ill-conditioning of the basis matrix (Barrera et al., 23 Jul 2025).

The condition number of the basis-function evaluation matrix controls numerical stability and therefore the required digit budget. The source notes that minimizing the condition number by a judicious choice of points can help, but only “up to a point,” because beyond a certain plateau additional points do not further reduce the necessary precision (Barrera et al., 23 Jul 2025). Random selection within a “reasonable” domain is described as robust, whereas clustering points too closely or sampling at widely varying scales can increase the condition number and digit requirements (Barrera et al., 23 Jul 2025).

These observations give the method a distinct computational profile. More sampling points may compensate for lower per-point precision, which is valuable when function evaluations are only moderately accurate (Barrera et al., 23 Jul 2025). This stands in contrast to direct linear-algebraic inversion, which the examples describe as requiring substantially higher precision for the same reconstruction task (Barrera et al., 23 Jul 2025). In a different regression setting, conditioning again plays a structural role: sparse linear regression hardness has been related to lattice condition number through the restricted eigenvalue constant, with “$$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$2” in the reduction described in the source material (Gupte et al., 2024). This suggests that conditioning is not merely a numerical nuisance but a bridge between statistical identifiability, algorithmic performance, and lattice geometry.

5. Representative examples and demonstrated use cases

The paper on Feynman integrals presents examples spanning toy relations, one-loop integrals, two-loop diagrams, triangle ladders, and an outer-mass double box (Barrera et al., 23 Jul 2025). A simple illustrative case compares PSLQ and LLL for the relation $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$3, where both methods recover the rational coefficients, but LLL is emphasized as the tool that generalizes to function regression (Barrera et al., 23 Jul 2025).

For the “1-Loop Massless Triangle,” the stated function space is a small dilogarithmic basis of approximately 17 functions. The reported result is that “using LLL, with 5 digits of precision and 20 points, the analytic form is recovered in all trials, outperforming naive matrix inversion (which requires $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$4 digits)” (Barrera et al., 23 Jul 2025). For “2-Loop Diagrams,” the basis size grows “up to 182 functions,” and the paper reports that assumptions such as uniform transcendental weight can reduce the basis and runtime significantly (Barrera et al., 23 Jul 2025).

At higher loop order, “Triangle Ladders and Outer-Mass Double Box” provide the large-scale demonstrations. The source states that “even for 3-loop diagrams (basis size $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$5), analytic regression succeeds with moderate digits and points by combining bootstrap constraints and LLL” (Barrera et al., 23 Jul 2025). For the double box, additional constraints from “Landau branch cut structure” and “Galois symmetry” are said to cut the basis dramatically and make regression tractable (Barrera et al., 23 Jul 2025).

These examples establish the method’s intended operating regime: the analytic form is not guessed from a generic symbolic-search space but reconstructed within a sharply constrained functional ansatz. This suggests that lattice reduction is most effective when combined with strong analytic priors rather than used as a stand-alone discovery procedure.

6. Relation to lattice-based regression in high dimensions

A separate line of work applies lattice reduction directly to high-dimensional linear regression, replacing sparsity assumptions with arithmetic structure. In “High Dimensional Linear Regression using Lattice Basis Reduction,” the assumed structure is $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$6-rationality: each entry of the unknown vector satisfies $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$7, with $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$8 (Gamarnik et al., 2018). The paper proposes a polynomial-time algorithm based on LLL and states that, under the $$f(\mathbf{x}_j)=\sum_{i=1}^n c_i \mathcal{B}_i(\mathbf{x}_j),$$9-rationality assumption, it “recovers exactly the vector $\mathcal{B}_i$0” for a large class of distributions for the iid entries of $\mathcal{B}_i$1, even with “only one observation ($\mathcal{B}_i$2),” provided the noise is sufficiently small (Gamarnik et al., 2018).

The later work on “Inference in High-Dimensional Linear Regression via Lattice Basis Reduction and Integer Relation Detection” extends this approach by combining PSLQ with LLL and introducing a mixed-support assumption in which entries may be $\mathcal{B}_i$3-rational or belong to a known rationally independent set (Gamarnik et al., 2019). The source states that exact recovery is possible in polynomial time, again even in the single-sample regime, and that these methods expose an algorithmic connection among linear regression, integer relation detection, randomized subset-sum, and shortest vector problems (Gamarnik et al., 2019).

These developments are not identical to the Feynman-integral program, but they clarify the broader meaning of regression with lattice methods. In one direction, lattice reduction recovers analytic coefficients of basis functions from high-precision evaluations (Barrera et al., 23 Jul 2025). In the other, it recovers coefficient vectors in linear models from highly undersampled measurements by exploiting rational or mixed arithmetic structure (Gamarnik et al., 2018, Gamarnik et al., 2019). A plausible unifying description is that lattice reduction acts as an arithmetic decoder whenever the unknown object is embedded into integer relations or short vectors.

7. Limitations, hardness, and related lattice algorithms

The principal limitations reported for analytic regression are basis specification, numerical conditioning, and silent failure modes. The method “requires substantial a priori knowledge of the function space,” and an overlarge or incomplete basis “can impair efficiency and reliability” (Barrera et al., 23 Jul 2025). For large or ill-conditioned basis sets, “the required precision and computational effort may become prohibitive,” and if “the true functional relation is not in the basis, or the coefficients are unexpectedly large, regression may fail silently” (Barrera et al., 23 Jul 2025).

Hardness results from sparse linear regression place these computational issues in a wider context. The paper “Sparse Linear Regression and Lattice Problems” gives “evidence of average-case hardness of SLR w.r.t. all efficient algorithms assuming the worst-case hardness of lattice problems,” via a reduction from a variant of bounded distance decoding to sparse linear regression (Gupte et al., 2024). The source further states that for ill-conditioned design matrices, the statistical-computational gap mirrors the failure of Babai’s algorithm on ill-conditioned lattice bases, while for well-conditioned essentially isotropic Gaussian designs, hardness persists in an “unidentifiable regime where there are many solutions” (Gupte et al., 2024). Although this is a distinct problem class, it reinforces the general point that lattice geometry and regression complexity are tightly coupled.

Related work on lattice reduction itself also matters for implementation. “Lattice Reduction over Imaginary Quadratic Fields” extends Gauss and LLL reduction to algebraic lattices over rings of imaginary quadratic integers and states that correct analogues of the reduction conditions require the ring to be Euclidean (Lyu et al., 2018). The source reports numerical time advantages for algebraic LLL in wireless communications and cryptography, together with the observation that generalized lattice reduction can be useful when algebraic structure must be preserved (Lyu et al., 2018). In communication-theoretic decoding, LR-aided regularized lattice sphere decoding with timeout achieves a “vanishing gap” to exact lattice decoding at subexponential complexity, highlighting lattice reduction as a “pivotal role” in complexity reduction (Singh et al., 2011). This suggests that the success of analytic regression methods depends not only on the existence of a lattice formulation but also on the quality of the reduction algorithm and the structural compatibility between the lattice and the ambient algebraic setting.