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REVIEW 4 major objections 5 minor 23 references

A randomized progressive iterative regularization method for data fitting problems

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Randomized progressive iterative fitting converges in expectation to the regularized least-squares solution, and the paper derives a noise-based formula for the optimal smoothing parameter.

desk verdict A correct but routine convergence theory for regularized randomized PIA, with a self-consistent lambda-selection heuristic that is the real practical claim and the part that lacks proof. read the letter →

arxiv 2506.03526 v1 pith:VJPJHJ4J submitted 2025-06-04 math.NA cs.NA

Dakang Cen , Wenlong Zhang , Junbin Zhong This is my paper
classification math.NAcs.NA MSC 65D1065D1765F2265F10
keywords datafittingleastsquaresrandomizediterativealgorithmprogressiveapproximationTikhonovregularizationparameterselectionB-splinecurvesurface
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Data fitting with B-spline curves and surfaces is usually done by solving large least-squares systems that are costly and sensitive to measurement noise. This paper argues that a randomized, block-wise iterative method -- regularized progressive iterative approximation (RPIA) -- can do the same job while staying large-scale friendly: each step updates a randomly chosen block of control points using an augmented design matrix that carries a second-order difference penalty. The main theoretical claim is that, when the augmented collocation matrix has full column rank, the sequence of fits converges in expectation to the Tikhonov-regularized least-squares solution. The paper also proves a stochastic error bound whose dominant terms are a bias term proportional to the regularization strength and a variance term inversely proportional to $\lambda^{1/\alpha}$, and balances them into the parameter rule $\lambda^{1+1/\alpha} = O(\sigma^2 n^{-1})\|\Gamma \bar p\|_n^{-2}$. If these claims hold, noisy curve and surface fitting can be regularized without solving dense systems and with a principled way to choose how much smoothing to apply.

What carries the argument

The engine of the argument is the augmented collocation matrix $\hat A = [A; \sqrt{\lambda}\Gamma]$ (and its two-factor tensor form $\hat A P \hat B^T$ for surfaces), which converts Tikhonov regularization into an ordinary least-squares problem. The convergence proof rests on the conditional-expectation identity $\mathbb{E}[z^{(k+1)}] = (I - \hat A \hat A^T/\|\hat A\|_F^2) z^{(k)}$, where $z^{(k)} = \hat A(p^{(k)} - p^*)$; this identity turns the random block update into a deterministic contraction in expectation. For parameter selection, the load-bearing object is the eigenvalue decay assumption $\rho_k \le C k^{-\alpha}$ of $Q^T Q$, which converts the variance term into $\lambda^{-1/\alpha}$ and makes the bias-variance balance yield $\lambda^{1+1/\alpha} = O(\sigma^2 n^{-1})\|\Gamma \bar p\|_n^{-2}$. The self-consistent algorithm in Section 6 iterates that formula by substituting the observed residual ratio for the unknown true ratio.

What would settle it

A direct test is to run the self-consistent fixed-point algorithm on a synthetic B-spline problem with known ground truth $\bar p$, known noise variance $\sigma^2$, and a deliberately misspecified spectral decay rate $\alpha$ (for instance, off by a factor of two); if the converged $\lambda$ moves away from the empirically optimal value that minimizes prediction error, the claim that the algorithm selects the optimal regularization parameter without prior information is falsified.

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Extended reading notes

Core claim

On the paper's own terms, the contribution is twofold. First, the regularized RPIA update is shown to be a randomized block-coordinate iteration on the augmented least-squares problem $\min_p \|\hat A p - \hat q\|_F^2$ with $\hat A = [A^T, \sqrt{\lambda}\Gamma^T]^T$. Writing $p^*$ for the Tikhonov solution, the proof tracks $z^{(k)} = \hat A(p^{(k)} - p^*)$ and shows that conditional on the history, the expected next iterate is $(I - \hat A \hat A^T / \|\hat A\|_F^2) z^{(k)}$, a contraction whose spectral radius is below one when $\hat A$ has full column rank; hence $\mathbb{E}[p^{(k)}] \to p^*$ for curves, and a tensor-product analogue gives the same for surfaces. Second, for noisy data $q_e = A\bar p + \epsilon + e$ with i.i.d. noise of variance $\sigma^2$ and eigenvalue decay $\rho_k \le C k^{-\alpha}$, the paper proves the mean squared prediction error is bounded by $C\lambda\|\Gamma \bar p\|^2 + C(\sigma^2 + \sum_i \epsilon_i^2)/\lambda^{1/\alpha}$, and balancing these terms yields the optimal-parameter formula. The same formula then motivates a self-consistent fixed-point iteration that estimates $\lambda$ from the current fit rather than from known noise levels.

Load-bearing premise

The load-bearing premise is that, inside the self-consistent parameter algorithm, the quantity $\|A p^{(k)} - q_{\text{noise}}\|_m^2 / (\|\Gamma p^{(k)}\|_n^2 \, n^{-1})$ can stand in for the true but unknown $\sigma^2 / \|\Gamma \bar p\|^2$, that the fixed point of this substitution is the minimizer of the true prediction error, and that the user knows the spectral decay rate $\alpha$; none of this is proved.

Editorial extensions

If this is right

  • For curve and surface fitting with noisy data, the randomized block iteration gives the Tikhonov solution in expectation, so standard regularization guarantees carry over while each step keeps the low per-iteration cost of RPIA.
  • The parameter rule $\lambda^{1+1/\alpha} = O(\sigma^2 n^{-1})\|\Gamma \bar p\|_n^{-2}$ predicts that optimal smoothing grows with noise level and shrinks with the number of data points, a trend the experiments confirm when the noise amplitude rises from 40 to 100.
  • In the surface case, regularization factorizes into separate $u$- and $v$-direction penalties and the method drops the $\lambda^2$ cross term; for the small $\lambda$ values reported, that discarded term is numerically harmless.
  • Because convergence is in expectation, averaging several independent runs reduces random-block variance and is part of the practical recipe the experiments use.
  • The self-consistent parameter iteration reaches fitting errors at or below the non-adaptive parameter choice in the reported examples, which suggests the residual-based estimate works as a usable discrepancy principle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not analyze the variance of the randomized estimator around its expectation; an unstated practical consequence is that the number of independent runs needed for a reliable fit should grow with the condition number of $\hat A$.
  • The fixed-point update can be read as a randomized discrepancy principle; a natural extension is to replace the in-sample residual with a cross-validated residual when deterministic model error dominates measurement noise, since the bound contains an extra $\sum_i \epsilon_i^2$ term the algorithm currently ignores.
  • The optimal-$\lambda$ formula depends on the spectral decay exponent $\alpha$, which the paper measures from the eigenvalue plot of $Q^T Q$; estimating $\alpha$ online from the same iterates would be needed to make the 'without prior information' claim literally true.
  • The parameter loop assumes the augmented least-squares problem is solved at each fixed-point step, while the fitting itself is iterative; how early stopping of the inner iteration changes the fixed point of $\lambda$ is left unexamined.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a randomized progressive iterative regularization method (regularized RPIA) for noisy curve and surface fitting. The method applies block-coordinate updates to the Tikhonov-augmented least-squares problem, and the authors prove that the resulting iteration converges in expectation to the augmented least-squares solution (Theorems 4.1 and 4.2). They then derive a stochastic bias-variance bound for the regularized estimate (Theorem 5.1) and use it to propose an order-optimal regularization parameter formula, leading to a self-consistent fixed-point algorithm (Section 6) intended to select λ without prior noise level information. Three numerical experiments (two curves, one surface) are reported to illustrate the theoretical analysis and the performance of the adaptive parameter selection.

Significance. If the parameter-selection algorithm were rigorously justified, the paper would provide a practical randomized method for large-scale noisy data fitting with an automatic regularization parameter choice. The convergence-in-expectation proof is a clean extension of the RPIA analysis and is the sound core of the paper. The bias-variance bound in Theorem 5.1 is a standard but useful result, and the optimal λ formula is derived from the bound rather than fitted to numerical error curves, which is a strength. The main weakness is that the self-consistent algorithm, advertised as a third contribution, is heuristic: no convergence or optimality statement is proved, and the 'without prior information' claim is overstated because the spectral decay rate α is required as an input.

major comments (4)
  1. [Section 6, Tables 6.1/6.2] The self-consistent fixed-point algorithm is presented as the paper's third contribution ('self-consistent algorithms without prior information'), but no theorem establishes that the sequence λ_k converges, that a fixed point exists or is unique, or that the fixed point is close to the minimizer of the true prediction error. The update rule replaces the unknown noise variance σ^2 by the empirical residual norm ||Ap^{(k)}−q_noise||_m^2, which contains a regularization-bias term, and replaces ||Γpbar||_n by the systematically smaller ||Γp^{(k)}||_n; these substitutions are not justified. The algorithm also requires the spectral decay rate α as an input, contradicting the 'without prior information' claim. The numerical evidence in Section 7.3 consists of three examples with means only, no error bars, and does not settle the convergence or optimality question.
  2. [Section 5, Theorem 5.1 proof, Eqs. (5.4)-(5.5)] The proof contains a normalization inconsistency with Assumption 1. Assumption 1 states that the eigenvectors φ_k form an orthonormal basis, but the proof sets (φ_k, φ_k) = 1/ρ_k. Under true orthonormality one would have (u,u) = Σ (u,φ_k)^2 and (Qu,Qu) = Σ ρ_k (u,φ_k)^2, not the formulas in (5.4)-(5.5). The subsequent derivation requires a rescaled basis satisfying (Qφ_k,Qφ_k)=1 and (φ_k,φ_k)=1/ρ_k, which is not the basis described in the assumption. The assumption and proof must be reconciled, for example by explicitly stating that the basis is orthogonal with those norms.
  3. [Abstract and Theorem 4.1] The abstract and introduction claim that the method 'converges to the least-squares solution'; Theorem 4.1 actually proves convergence in expectation to the least-squares solution of the augmented system (Âp = q̂), i.e., to the Tikhonov-regularized solution for a fixed λ. This is not the unregularized least-squares solution of the original problem. In addition, the phrase 'under a certain level of noise' is not connected to any hypothesis in Theorem 4.1, which holds for any λ>0. The wording should be corrected to avoid overclaiming.
  4. [Remark 5.1 and Section 6] The optimal parameter formula λ^{1+1/α}=O(σ^2 n^{-1})||Γpbar||_n^{-2} is obtained by balancing the two terms of an upper bound, so the proportionality constant is not determined. The self-consistent algorithm in Tables 6.1 and 6.2 fixes the constant to 1 without justification. Furthermore, the claim that the estimated λ is 'very close' to the empirical optimum is supported only by visual inspection of Figures 2(d), 4(d), 5(d), and 7(d); for a quantitative parameter-selection claim, the paper should provide repeated-trial statistics (e.g., standard errors) or a theoretical error bound on the selected λ.
minor comments (5)
  1. [Throughout] There are several typos and grammatical slips, e.g., 'arrvies' in the proof of Theorem 5.1 and 'It achieves that' in the proof of Theorem 4.2; these should be corrected in a revision.
  2. [Theorem 4.2 proof] In the denominator after the conditional expectation for surfaces, the term '||B̂V_{sk}||_F^2' and later '||B̂^T||_F^2' should be consistently written as '||B̂||_F^2'.
  3. [Section 3.2 and Table 6.2] The surface regularization problem includes a λ^2 term, but the parameter-estimation update in Table 6.2 omits it; the text in Section 7.2 states that the high-order term is discarded, but no justification is given beyond the observation that λ is small. A comment in Section 6 would clarify this approximation.
  4. [Section 7.3] The results of the self-consistent algorithm are reported as arithmetic means of 10 (curves) or 3 (surfaces) runs, but no standard deviations or confidence intervals are given, making it difficult to judge the reliability of the reported improvements.
  5. [Section 7, fitting error definition] In the fitting error definitions Ek(λ) = ||Ap∗−A p̄||_F^2 / ||A p̄||_F^2 (and the surface analogue), the notation p∗ refers to the regularized solution; this should be stated explicitly to avoid confusion with the unregularized least-squares solution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence proof and the optimal-regularization-parameter bound are self-contained; the unproven fixed-point lambda iteration is a missing-proof concern, not a circular reduction.

full rationale

The paper's main derivation chain is self-contained. Theorem 4.1 proves convergence in expectation of the regularized RPIA by analyzing the recurrence E[z^(k+1)] = (I - Ahat Ahat^T / ||Ahat||_F^2) E[z^(k)] and using only the positive definiteness of Ahat^T Ahat; no fitted quantity or prior result is needed for the key step, and the surface case in Theorem 4.2 is the same argument with Kronecker products. The optimal-regularization-parameter formula in Remark 5.1 is obtained by balancing the two terms of the derived mean-squared-error bound, not by fitting the numerical error curves: the bound has the form C n lambda/m ||Gamma pbar||_n^2 + C(sigma^2 + sum epsilon_i^2)/(m lambda^(1/alpha)), and equating the leading terms gives lambda^(1+1/alpha) = O(sigma^2 n^{-1}) ||Gamma pbar||_n^{-2}. This is a bias-variance tradeoff derived from the stochastic estimate, so the claimed optimal parameter is not equivalent to the input data by construction. Section 6's self-consistent fixed-point iteration is a heuristic replacement of the unknown sigma^2 and ||Gamma pbar||_n by the current residual and ||Gamma p^(k)||_n; no theorem proves that its fixed point coincides with the minimizer of the true expected error, but that absence is a correctness or rigor gap, not circularity, because the update is not asserted to be equivalent to the derived optimum. The paper does not contain load-bearing self-citation: the cited RPIA work [18] is the base algorithm, while the convergence and parameter estimates are proved in the present paper. The numerical validation compares the formula with true error curves on synthetic examples with known ground truth; even though alpha is estimated from the spectrum of Q, the reported lambda values are not, by the paper's own account, fitted to the error minima. Overall, the central claims are independent of their inputs, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central convergence claim needs only full-rank assumptions and the standard randomized-Kaczmarz argument. The optimal-parameter claim rests on a source condition (Assumption 1), an i.i.d. noise model, and a hand-estimated alpha; the self-consistent algorithm adds a hand-chosen initial lambda and an unproven fixed-point assumption.

free parameters (3)
  • spectral decay rate alpha = 4.1315 (Ex 7.1), 4.1317 (Ex 7.2), 2.0929 (Ex 7.3)
    Estimated from the eigenvalue decay of Q^T Q (Figures 1, 3, 6) and used in the optimal-lambda formula and self-consistent algorithm; it is an input, not derived from first principles.
  • regularization matrix scale C = C = 1600 in curve examples, C = 91 in surface example
    The second-order difference matrix Gamma is scaled by hand-chosen C; the effective smoothing strength depends on C, yet the paper gives no criterion for choosing it.
  • initial regularization parameter lambda_1 = lambda_1^(1+1/alpha) = n^{-1}
    Hand-chosen starting value for the self-consistent iteration; the final fixed point may depend on it since no convergence proof is given.
assumptions (5)
  • domain assumption Assumption 1: eigenvalues rho_k of (Q phi, Q u) = rho(phi, u) decay as rho_k <= C k^{-alpha}
    Source condition on the design matrix that is necessary for the error bound in Theorem 5.1; no verification is provided beyond empirical eigenvalue plots.
  • domain assumption Noise model: q_e = q + e with e_i i.i.d., E[e_i]=0, E[e_i^2]=sigma^2
    Used in Theorem 5.1 to compute the expectation of the noise term; real noise may not be i.i.d. or may be heteroscedastic.
  • ad hoc to paper Small model error: q = A pbar + epsilon with epsilon very small relative to e
    The paper says this is 'an intuitively feasible idea'; the optimal-lambda formula assumes the model bias epsilon is negligible compared with the measurement noise.
  • standard math Full-column rank of A and of Ahat = [A; sqrt(lambda)Gamma]
    Needed for the spectral radius bound in Theorem 4.1; standard in least-squares analysis.
  • standard math Gamma is invertible and B = Gamma^{-1} exists
    Needed for the change of variables Q = AB in Section 5; the displayed second-order difference matrix is invertible, though the paper incorrectly calls it positive definite.

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Cite this review

Pith. "Pith review of A randomized progressive iterative regularization method for data fitting problems." pith.science (2026). https://pith.science/paper/VJPJHJ4J

@misc{pith2026250603526,
  author       = {Pith},
  title        = {Pith review of: A randomized progressive iterative regularization method for data fitting problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJPJHJ4J}},
  note         = {Machine review of arXiv:2506.03526}
}
read the original abstract

In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares solution. Furthermore, we present an optimal estimation for the regularization parameter, which inspires the construction of self-consistent algorithms without prior information. The numerical results confirm the theoretical analysis and show the performance in curve and surface fittings.

Figures

Figures reproduced from arXiv: 2506.03526 by the authors.

Figure 1
Figure 1. Spectrum of QTQ for Example 7.1 (a) Example 7.1 Real Curve (b) Regularized RPIA Fitted Curve (c) Regularized RPIA Fitted Curve(λ = 0) (d) Fitting Error [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The fitting curves given by Regularized RPIA for Example [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Spectrum of QTQ for Example 7.2 It can also be seen from [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The fitting curves given by Regularized RPIA for Example [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: The fitting surface given by Regularized RPIA for Example [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Spectrum of QTQ for Example 7.3 21 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: The fitting surface given by Regularized RPIA for Example [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Adaptive algorithm results for Example 7.1 23 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: Iteration Step for Example 7.1 (a) Fitted Curve for Example 7.2 (b) Fitting error [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Adaptive algorithm results for Example 7.2 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Iteration Step for Example 7.2 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Adaptive algorithm results for Example 7.3 with noise level parameter a = 40 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Iteration Step for Example 7.3 with noise level parameter a = 40 (a) Fitted Surface for Example 7.3 (b) Fitting Error [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: Adaptive algorithm results for Example 7.3 with noise level parameter a = 100 25 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: Iteration Step for Example 7.3 with noise level parameter a = 100 non-adaptive algorithm. It only requires a small number of iterations and is closer to the optimal regularization point (see Figures 14 and 15). The above curve results are obtained by running the regul…

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Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    A variational non-linear constrained model for the inversion of fdem data

    Alessandro Buccini and Patricia Diaz de Alba. A variational non-linear constrained model for the inversion of fdem data. Inverse Problems, 38, 2021

    work page 2021

  2. [2]

    Distributed tikhonov regularization for ill- posed inverse problems from a bayesian perspective

    Daniela Calvetti and Erkki Somersalo. Distributed tikhonov regularization for ill- posed inverse problems from a bayesian perspective. Computational Optimization and Applications, 2025. 26

    work page 2025

  3. [3]

    Carnicer, J

    J.M. Carnicer, J. Delgado, and J.M. Pe˜ na. Richardson method and totally non- negative linear systems. Linear Algebra and its Applications , 433(11):2010–2017, 2010

    work page 2010

  4. [4]

    How does agee’s smoothing method work? In Proceedings of the 1979 Army Numerical Analysis and Computers Conference , ARO Report 79-3, pages 299–302

    de Boor C. How does agee’s smoothing method work? In Proceedings of the 1979 Army Numerical Analysis and Computers Conference , ARO Report 79-3, pages 299–302. Army Research Office, 1979

    work page 1979

  5. [5]

    Progressive and iterative approximation for least squares b-spline curve and surface fitting

    Chongyang Deng and Hongwei Lin. Progressive and iterative approximation for least squares b-spline curve and surface fitting. Computer-Aided Design, 47:32–44, 2014

    work page 2014

  6. [6]

    Ebrahimi and G.B

    A. Ebrahimi and G.B. Loghmani. A composite iterative procedure with fast con- vergence rate for the progressive-iteration approximation of curves. Journal of Computational and Applied Mathematics , 359:1–15, 2019

    work page 2019

  7. [7]

    On a progressive and iterative approximation method with memory for least square fitting

    Zheng-Da Huang and Hui-Di Wang. On a progressive and iterative approximation method with memory for least square fitting. Computer Aided Geometric Design , 82:101931, 2020

    work page 2020

  8. [8]

    Totally positive bases and pro- gressive iteration approximation

    Hong-Wei Lin, Hu-Jun Bao, and Guo-Jin Wang. Totally positive bases and pro- gressive iteration approximation. Computers & Mathematics with Applications , 50(3):575–586, 2005

    work page 2005

Show all 23 references
  1. [9]

    Local progressive-iterative approximation format for blending curves and patches

    Hongwei Lin. Local progressive-iterative approximation format for blending curves and patches. Computer Aided Geometric Design , 27(4):322–339, 2010

    2010

  2. [10]

    Survey on geometric iter- ative methods and their applications

    Hongwei Lin, Takashi Maekawa, and Chongyang Deng. Survey on geometric iter- ative methods and their applications. Computer-Aided Design, 95:40–51, 2018

    2018

  3. [11]

    An extended iterative format for the progressive- iteration approximation

    Hongwei Lin and Zhiyu Zhang. An extended iterative format for the progressive- iteration approximation. Computers & Graphics , 35(5):967–975, 2011

    2011

  4. [12]

    Preconditioned progressive iterative approximation for triangular b´ ezier patches and its application

    Chengzhi Liu, Xuli Han, and Juncheng Li. Preconditioned progressive iterative approximation for triangular b´ ezier patches and its application. Journal of Com- putational and Applied Mathematics , 366:112389, 2020

    2020

  5. [13]

    Weighted progressive iteration approximation and convergence anal- ysis

    Lizheng Lu. Weighted progressive iteration approximation and convergence anal- ysis. Computer Aided Geometric Design , 27(2):129–137, 2010

    2010

  6. [14]

    A regularization approach for sur- face reconstruction from point clouds

    Hebert Montegranario and Jairo Espinosa. A regularization approach for sur- face reconstruction from point clouds. Applied Mathematics and Computation , 188(1):583–595, 2007

    2007

  7. [15]

    Uniform b-spline curve interpolation with prescribed tan- gent and curvature vectors

    Shoichi Okaniwa, Ahmad Nasri, Hongwei Lin, Abdulwahed Abbas, Yuki Kineri, and Takashi Maekawa. Uniform b-spline curve interpolation with prescribed tan- gent and curvature vectors. IEEE Transactions on Visualization and Computer Graphics, 18(9):1474–1487, 2012

    2012

  8. [16]

    D. Qi, Z. Tian, Y. Zhang, and J. Feng. The method of numeric polish in curve fitting. ACTA MATHEMATICA SINICA, 18(3):173–184, 1975

    1975

  9. [17]

    Lspia, (stochastic) gradient descent, and parameter correction

    Dany Rios and Bert J¨ uttler. Lspia, (stochastic) gradient descent, and parameter correction. Journal of Computational and Applied Mathematics , 406:113921, 2022

    2022

  10. [18]

    Randomized progressive iterative approximation for b-spline curve and surface fittings

    Nian-Ci Wu and Cheng-Zhi Liu. Randomized progressive iterative approximation for b-spline curve and surface fittings. Applied Mathematics and Computation , 473:128669, 2024

    2024

  11. [19]

    On the triple-parameter least squares progressive iterative approximation and its convergence analysis

    Nian-Ci Wu, Chengzhi Liu, and Juncheng Li. On the triple-parameter least squares progressive iterative approximation and its convergence analysis. Computer Aided Geometric Design, 119:102439, 2025

    2025

  12. [20]

    Curve fitting of the user barrage emotional change based on the hybrid kernel pso lssvm model

    Fulian Yin, Xiaoli Feng, Fangyuan Ju, and Yanyan Wang. Curve fitting of the user barrage emotional change based on the hybrid kernel pso lssvm model. Proceedings of the 2020 3rd International Conference on Algorithms, Computing and Artificial Intelligence, 2020. 27

    2020

  13. [21]

    Impulse noise image restoration using nonconvex variational model and dif- ference of convex functions algorithm

    Benxin Zhang, Guopu Zhu, Zhibin Zhu, Hongli Zhang, Yicong Zhou, and Sam Kwong. Impulse noise image restoration using nonconvex variational model and dif- ference of convex functions algorithm. IEEE Transactions on Cybernetics, 54:2257– 2270, 2022

    2022

  14. [22]

    Generalized b-splines’ geomet- ric iterative fitting method with mutually different weights

    Li Zhang, Jieqing Tan, Xianyu Ge, and Guo Zheng. Generalized b-splines’ geomet- ric iterative fitting method with mutually different weights. Journal of Computa- tional and Applied Mathematics , 329:331–343, 2018. The International Conference on Information and Computational S...

    2018

  15. [23]

    Parametric surface reconstruction from 3d point data using partial differential equation and bilinearly blended coons patch

    Zaiping Zhu, Shuangbu Wang, Lihua You, and Jianjun Zhang. Parametric surface reconstruction from 3d point data using partial differential equation and bilinearly blended coons patch. Journal of Computational Physics , 519:113436, 2024. 28

    2024

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