PolyFit: Polynomial Fitting Approach

• PolyFit is a modeling strategy that approximates nonlinear functions using polynomials to capture underlying behavior and domain-specific constraints.
• It includes variants like least-squares regression, derivative-matching, and piecewise fits to enhance numerical stability and fit quality.
• Empirical studies show that PolyFit reduces computational effort and improves accuracy in applications from astronomy to quantum optimization.

Searching arXiv for recent and relevant papers on “PolyFit” and polynomial fitting to ground the article in current literature. {"query":"ti:PolyFit OR abs:PolyFit OR \"polynomial fitting\" arXiv", "max_results": 10, "sort_by": "submittedDate"} I found multiple relevant arXiv papers spanning distinct uses of “PolyFit,” including activation-function approximation (Chiang, 2022), numerically stable Vandermonde fitting (Brubeck et al., 2019), approximate query processing (Li et al., 2020), peg-in-hole assembly (Lee et al., 2023), global 21 cm signal fitting (Liu et al., 2023), radar-guided depth estimation (Rim et al., 21 Mar 2025), and quantum protein folding (Li et al., 11 Jul 2025). PolyFit denotes a class of methods in which polynomial representations are used to approximate functions, encode geometric or physical constraints, stabilize regression, or compress queryable data structures. Across arXiv, the term is not attached to a single canonical algorithm: it refers, among other things, to derivative-augmented activation approximation for homomorphic encryption, Arnoldi-stabilized polynomial interpolation and least-squares fitting, piecewise polynomial light-curve models, beam-aware spectral fitting, scene-adaptive depth correction, polynomial learned indexes, and Chebyshev-fitted constraint penalties in quantum optimization (Chiang, 2022, Brubeck et al., 2019, Kochoska et al., 2017, Liu et al., 2023, Rim et al., 21 Mar 2025, Li et al., 2020, Li et al., 11 Jul 2025). The unifying motif is the replacement of a difficult target object—non-polynomial nonlinearity, noisy data, sparse metric cues, discrete constraints, or large datasets—by a compact polynomial surrogate whose coefficients are estimated from data and then exploited analytically or computationally.

1. Terminological scope and major variants

In the literature, “PolyFit” can describe ordinary least-squares polynomial regression, weighted or derivative-constrained fitting, piecewise polynomial chains, polynomial transforms of cumulative functions, local jet fitting on surface patches, or polynomial penalties embedded into larger optimization systems. The term therefore functions more as a methodological family name than as a uniquely standardized procedure.

Variant	Core polynomial object	Representative paper
Activation approximation	Low-degree polynomial $p_n(x)$ for $f(x)$	(Chiang, 2022)
Stabilized least squares/interpolation	Orthogonal basis replacing monomials	(Brubeck et al., 2019)
Light-curve geometry modeling	Piecewise quadratic polynomial chain	(Kochoska et al., 2017)
Approximate range aggregation	Piecewise polynomial index over cumulative functions	(Li et al., 2020)
Metric-depth correction	Scene-adaptive polynomial map $\hat d=\sum_i \hat c_i z^i$	(Rim et al., 21 Mar 2025)
Quantum constraint encoding	Chebyshev polynomial penalty $P_d(D)$	(Li et al., 11 Jul 2025)
Surface deformation modeling	Local jet $z(u,v)$ on patches	(Chen et al., 8 Jan 2026)

What links these variants is not a shared software interface or common benchmark, but a recurring design choice: polynomial coefficients act as the low-dimensional state of the problem. In some settings those coefficients parameterize a curve or surface directly; in others they parameterize an index, a calibration transform, or a Hamiltonian penalty.

2. Mathematical formulations

The baseline PolyFit formulation is the classical approximation problem: given a continuous target $f(x)$ on $[a,b]$, find a polynomial $p_n(x)$ of degree at most $n$ minimizing

$\min_{p_n}\int_a^b (f(x)-p_n(x))^2\,dx.$

This is the ordinary least-squares criterion underlying standard polyfit workflows (Chiang, 2022). A matricized least-squares version for sampled data $f(x)$0 writes $f(x)$1, minimizes $f(x)$2, and derives the normal equations $f(x)$3, where $f(x)$4 and $f(x)$5 contains the mixed moments $f(x)$6 (Dasgupta, 2015).

Several PolyFit variants alter the objective rather than the polynomial family. In activation approximation, the central extension is to match both function values and slopes: $f(x)$7 This “least squares plus derivative matching” formulation is explicitly motivated by the claim that a low-degree polynomial may minimize area-wise error while still failing to preserve the target activation’s shape or slope behavior (Chiang, 2022). The same paper introduces weights $f(x)$8 so that different intervals or derivative orders can be emphasized asymmetrically, and it generalizes further to matching higher-order derivatives of an intermediate polynomial surrogate.

Other formulations target discrete objects rather than continuous functions. In the learned-index setting, PolyFit approximates aggregate functions by a polynomial $f(x)$9 over an interval $\hat d=\sum_i \hat c_i z^i$0, with the fitting criterion

$\hat d=\sum_i \hat c_i z^i$1

This is converted to a linear program with an explicit $\hat d=\sum_i \hat c_i z^i$2-type error bound (Li et al., 2020). In distribution fitting, the target random variable is represented as

$\hat d=\sum_i \hat c_i z^i$3

with $\hat d=\sum_i \hat c_i z^i$4 chosen as a Weibull random variable so that substitution of the closed-form Weibull inverse CDF yields an analytic quantile approximation; coefficients are estimated either by percentile matching or by probability weighted moments (Xiao, 2015).

A further departure is piecewise fitting. The Gaia eclipsing-binary study defines PolyFit as an analytical “polynomial chain” on folded phase space, with segments joined at iteratively optimized knots. The chain is “connected and smoothly wrapped in phase space, but not necessarily differentiable at the knots,” allowing sharp ingress and egress behavior to be represented without enforcing global smoothness (Kochoska et al., 2017).

3. Numerical conditioning and computational realization

A central issue in polynomial fitting is basis conditioning. Direct monomial/Vandermonde formulations are well known to be fragile because powers $\hat d=\sum_i \hat c_i z^i$5 can vary exponentially in magnitude with $\hat d=\sum_i \hat c_i z^i$6, making the familiar polyfit/polyval pipeline ineffective at higher degrees (Brubeck et al., 2019). The “Vandermonde with Arnoldi” approach addresses this by treating monomials as a Krylov sequence and applying Arnoldi orthogonalization to construct an orthogonal basis $\hat d=\sum_i \hat c_i z^i$7 together with a lower-Hessenberg recurrence matrix $\hat d=\sum_i \hat c_i z^i$8 satisfying

$\hat d=\sum_i \hat c_i z^i$9

Fitting is then performed in the orthogonal basis by solving $P_d(D)$0 instead of $P_d(D)$1, while evaluation at new points is carried out by propagating the same recurrence on the evaluation nodes (Brubeck et al., 2019). The polynomial itself is unchanged; what changes is the coordinate system in which it is computed.

A different computational emphasis appears in the matricized least-squares formulation for large datasets. There the goal is not primarily conditioning but throughput: the sums $P_d(D)$2 and $P_d(D)$3 that populate $P_d(D)$4 and $P_d(D)$5 are embarrassingly parallel and map naturally to GPU kernels. The implementation reported in that work used CUDA on an NVIDIA Quadro 4000 with compute capability 2.0 and 256 cores, and claimed speedups of approximately $P_d(D)$6 on datasets with thousands of points (Dasgupta, 2015).

For repeated local polynomial fits, such as Savitzky–Golay filtering, another stabilization strategy replaces monomials by discrete orthogonal Chebyshev polynomials. The resulting smoothing matrix

$P_d(D)$7

and derivative matrix

$P_d(D)$8

inherit recurrence relations and strong symmetry, allowing accuracy-oriented and buffer-based algorithms that reduce memory usage and improve scaling with polynomial degree and window length (Rosso, 8 Apr 2026). This formulation is especially relevant where repeated fits dominate the total computational cost.

4. Structured and domain-adapted PolyFit formulations

Many PolyFit systems are distinguished less by the polynomial family itself than by how the polynomial is embedded into domain structure.

In astronomical light-curve analysis, PolyFit is a periodic piecewise model with four knots and four quadratic polynomials. Its non-differentiability at knots is not an incidental defect but a modeling choice that permits slope breaks at eclipse boundaries. Fitted models are sampled on $P_d(D)$9 equidistant phase points from $z(u,v)$0 to $z(u,v)$1, rescaled to $z(u,v)$2, and then passed to t-SNE and DBSCAN for morphology-based clustering (Kochoska et al., 2017).

In weak-lensing PSF interpolation, Polyfit is a global bivariate trend surface

$z(u,v)$3

The paper frames it as a global, approximate, deterministic interpolator that is simple and often effective for smooth large-scale variation, but limited when the PSF exhibits localized or turbulent structure. The authors note that second- and third-order polynomials are common in the literature, while a fifth-degree polynomial gave the best Polyfit results in their GREAT10 pipeline, with higher orders degrading the fit (Gentile et al., 2012).

The Vari-Zeroth-Order Polynomial method for global 21 cm detection modifies the usual log-polynomial foreground model by importing a measured 24-hour averaged beam directly into the fitting equation. In matrix form,

$z(u,v)$4

so the zeroth-order term becomes frequency dependent through beam-weighted sky averaging rather than remaining a constant offset (Liu et al., 2023). This is still polynomial fitting, but the polynomial now models residual spectral behavior after explicit beam coupling has been incorporated.

In radar-guided depth estimation, PolyFit is a scene-adaptive polynomial transformation of a frozen monocular depth estimate $z(u,v)$5: $z(u,v)$6 The coefficients $z(u,v)$7 are predicted from fused radar and depth features, and the mapping is regularized through its derivative

$z(u,v)$8

which is encouraged to remain close to $z(u,v)$9 in order to preserve local depth ordering while allowing non-uniform global correction (Rim et al., 21 Mar 2025).

In geometry processing, PolyFit becomes a local surface representation. Each oriented patch is modeled as a height graph

$f(x)$0

The polynomial coefficients $f(x)$1, together with patch orientation parameters, serve as the deformation state, so downstream optimization is carried out in coefficient space rather than over per-vertex degrees of freedom (Chen et al., 8 Jan 2026).

5. Empirical behavior across application domains

Reported performance varies substantially with the problem class, but a recurring pattern is that polynomial fitting is used to trade off fidelity, robustness, and compactness.

In medical image analysis, the HS-PCL model for tooth-root segmentation fits a smooth polynomial curve through detected edge landmarks and uses a maximum number of the shortest distances algorithm to reduce the influence of wrongly detected landmarks. Its numerical experiments reported reductions of Hausdorff95 by $f(x)$2 and Average Surface Distance by $f(x)$3 relative to the state-of-the-art method (Li et al., 2021).

In global 21 cm signal extraction, VZOP reportedly reduces residuals to about $f(x)$4 mK for the three main antenna designs when the beam is accurately known, and it can recover the absorption feature even when structured beam errors reach $f(x)$5. In the most extreme scenario of completely random beam errors, the method is reported to be at least not worse than common polynomial fitting (Liu et al., 2023).

In metric depth estimation, PolyRad reports state-of-the-art results on nuScenes, ZJU-4DRadarCam, and View-of-Delft, with an overall improvement of $f(x)$6 in MAE and $f(x)$7 in RMSE, and with best performance occurring for moderate polynomial degrees, improving up to about degree $f(x)$8 and slightly worsening at degree $f(x)$9 (Rim et al., 21 Mar 2025).

In data systems, the learned-index PolyFit supports COUNT, SUM, MIN, and MAX with guaranteed absolute and relative error bounds. Experiments reported that PolyFit was about $[a,b]$0 to $[a,b]$1 faster than existing learned indexes for one-key COUNT, often at least one order of magnitude faster than aR-tree and MRTree for two-key COUNT, at least $[a,b]$2 faster for MAX, and on the $[a,b]$3M-record OSM dataset could use a $[a,b]$4 MB index while delivering about $[a,b]$5 response time for 2D COUNT (Li et al., 2020).

In astronomy, the Gaia eclipsing-binary study concluded that about $[a,b]$6 of Kepler eclipsing binaries are potentially detectable by Gaia when folded with the orbital period. The PolyFit-based clustering produced five classes, and class 0, comprising $[a,b]$7 sources or $[a,b]$8 of the sample, contained mainly bad fits; about $[a,b]$9 of visually flagged poor fits fell into that class while only about $p_n(x)$0 of good-quality fits did (Kochoska et al., 2017).

In quantum protein structure prediction on the FCC lattice, PolyFit approximates the overlap penalty with a Chebyshev polynomial $p_n(x)$1, avoiding slack variables but producing a much larger Hamiltonian. For the $p_n(x)$2 peptide KLVFFA, the PolyFit Hamiltonian had $p_n(x)$3 terms versus $p_n(x)$4 for VQEC. Nevertheless, PolyFit recovered ground-state conformations on both Eagle R3 and Heron R2 hardware, with the reported ground-state probability increasing from about $p_n(x)$5 on ibm_cleveland to about $p_n(x)$6 on ibm_kingston, described as a nearly $p_n(x)$7-fold improvement (Li et al., 11 Jul 2025).

In patch-based deformation modeling, PolyFit underlies two applications. For Shape-from-template, PolySfT achieved depth RMSE $p_n(x)$8 on Kinect-Paper, compared with $p_n(x)$9 for SfT, $n$0 for DeepSfT, and $n$1 for TD-SfT, with runtime about $n$2 s/frame versus about $n$3 s/frame for $n$4-SfT. For garment draping, OneFit reported runtime $n$5 ms, compared with $n$6 ms for SNUG, $n$7 ms for HOOD, and $n$8 ms for GAPS (Chen et al., 8 Jan 2026).

6. Limitations, common misconceptions, and interpretive issues

A common misconception is that PolyFit is synonymous with ordinary least squares in a monomial basis. The surveyed literature contradicts that view. Some methods augment least squares with derivative matching, some change the basis through Arnoldi or discrete orthogonal polynomials, some fit piecewise periodic chains that are intentionally non-differentiable at knots, and some use polynomial coefficients as compact latent variables inside larger systems (Chiang, 2022, Brubeck et al., 2019, Kochoska et al., 2017, Rosso, 8 Apr 2026).

A second misconception is that increasing degree automatically improves performance. Multiple papers state the opposite under realistic constraints. High-order bivariate polynomials can oscillate and produce edge artifacts in PSF interpolation; higher polynomial degree in PolyRad can induce oscillatory behavior, which is why derivative regularization is required; and direct Vandermonde formulations degrade numerically as degree rises (Gentile et al., 2012, Rim et al., 21 Mar 2025, Brubeck et al., 2019). In query processing, higher degree can also increase evaluation cost or complicate root finding for MAX queries, so small degrees are explicitly recommended (Li et al., 2020).

A third issue is that the polynomial is often only one component of the total system. VZOP depends on beam measurement quality, PolyRad assumes the frozen monocular depth backbone already provides reasonably good local structure, the quantum FCC PolyFit saves qubits only by paying with a rapid growth in Hamiltonian terms, and the patch-based surface model assumes each patch is representable as a single-valued height function, which can fail under extreme wrinkles, large bulges, or self-occlusions (Liu et al., 2023, Rim et al., 21 Mar 2025, Li et al., 11 Jul 2025, Chen et al., 8 Jan 2026).

Finally, there is a terminological complication: in some papers the name “PolyFit” functions as a framework label even when ordinary polynomial curve fitting is not the dominant operation. The peg-in-hole assembly framework called PolyFit is a Force/Torque-based supervised learning system for $n$9-DoF pose estimation and sim-to-real adaptation, achieving $\min_{p_n}\int_a^b (f(x)-p_n(x))^2\,dx.$0 and $\min_{p_n}\int_a^b (f(x)-p_n(x))^2\,dx.$1 success rates for seen and unseen shapes in simulation and $\min_{p_n}\int_a^b (f(x)-p_n(x))^2\,dx.$2 and $\min_{p_n}\int_a^b (f(x)-p_n(x))^2\,dx.$3 in real-world experiments (Lee et al., 2023). This suggests that, in current usage, “PolyFit” should be read contextually: sometimes it names a specific polynomial approximation scheme, and sometimes it names a broader pipeline in which polynomial fitting is only one historical or conceptual antecedent.

Taken together, these works suggest that PolyFit is best understood not as a single algorithm but as a recurrent modeling strategy: encode the target phenomenon in a polynomial basis that is sufficiently expressive for the task, sufficiently structured for the domain, and sufficiently stable for the numerical regime. The decisive technical questions are therefore basis choice, objective design, degree control, structural constraints, and the way polynomial coefficients interface with the surrounding system.