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REVIEW 4 major objections 6 minor 38 references

General Coded Computing: Adversarial Settings

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper establishes a general coded-computing scheme that tolerates any sublinear number of adversarial servers, with worst-case average error decaying as $N^{6/5(a-1)}$ when $\gamma=\mathcal{O}(N^a)$.

desk verdict New spline-based coded computing idea with a provable sublinear-adversary rate and a matching impossibility result, but the preprint's proof chain has fixable gaps in Lemma 6, Eq. (70), and Lemma 8. read the letter →

arxiv 2502.08058 v1 pith:DB3WYHEX submitted 2025-02-12 cs.DC cs.LG

Parsa Moradi , Hanzaleh Akbarinodehi , Mohammad Ali Maddah-Ali This is my paper · ORCID
classification cs.DCcs.LG MSC 68M1441A1562G08
keywords codedcomputingadversarialserversByzantineworkerssmoothingsplinesSobolevspacesreproducingkernelHilbertdistributedinferenceapproximationerror
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

Conventional coded computing only works for structured jobs like polynomial evaluation or matrix multiplication, because it leans on algebraic codes. This paper argues that the same reliability goal can be achieved for essentially arbitrary computations—anything with bounded first and second derivatives—by letting the encoder and decoder be smooth functions chosen from second-order Sobolev spaces. For a system of $N$ workers with at most $\mathcal{O}(N^a)$ adversarial servers, $a<1$, the scheme drives the worst-case average approximation error, taken over all adversarial strategies, to zero at rate $N^{6/5(a-1)}$. The paper also proves a limit: if the adversary controls a constant fraction of servers, no encoder–decoder pair in this framework can make the error vanish. Together these claims make a case that general, adversarial-robust coded computing is possible exactly in the sublinear regime.

What carries the argument

The load-bearing machinery is the smoothing-spline decoder: $u_d$ is the minimizer of a regularized least-squares problem in the second-order Sobolev RKHS, so it is a linear smoother of the worker outputs with weight function $G_{N,\lambda}$. The proof uses an equivalent-kernel approximation (Lemma 6) that bounds $|G_{N,\lambda}-K_\lambda|$ uniformly by a term of order $1/N$ plus an exponentially small term in $\lambda^{-1/4}$, where $K_\lambda$ is an explicit kernel of bandwidth $\lambda^{1/4}$. That bound is what converts an adversary's corrupted outputs into a controlled additive error of order $M\gamma/N$, which balances against the spline approximation error to yield the rate.

What would settle it

Compute the sup-norm difference $\sup_{x,t\in[0,1]} |G_{N,\lambda}(x,t)-K_\lambda(x,t)|$ numerically for equidistant $\beta_i=i/N$ over a sweep of $\lambda$ with $N\lambda^{1/4}$ above the stated threshold, and check whether it decays as $\mu_3/N$ plus $\mu_4\lambda^{-1/4}\exp(-\lambda^{-1/4}/\sqrt{2})$; a slower decay in either $N$ or $\lambda$ would falsify Lemma 6 and with it the rate in Corollary 1. A complementary check: run the scheme with $\gamma=\mu N$ for any $\mu>0$ and verify the average error stays bounded away from zero, as Theorem 1 predicts.

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Extended reading notes

Core claim

Using second-order Sobolev spaces as the search space for both encoder $u_e$ and decoder $u_d$, the master encodes the $K$ input points into a smooth curve, sends its samples at $N$ fixed points to the workers, and reconstructs the answers by fitting a smoothing spline to the received (possibly corrupted) outputs. The central result is that when $\gamma=\mathcal{O}(N^a)$ with $a\in[0,1)$, the supremum of the average squared error over all adversarial strategies is $\mathcal{O}(N^{6/5(a-1)})$, provided the decoder points are equidistant and the smoothing parameter is set to $\lambda_d=\mathcal{O}(N^{8/5(a-1)})$. The companion impossibility result shows that for $\gamma=\mu N$ with $0<\mu<1$ there is a function $f$ with bounded first and second derivatives for which no choice of $u_e,u_d\in H^2(\Omega)$ yields error tending to zero, so the sublinear adversarial regime is optimal within this framework.

Load-bearing premise

Everything rests on the imported statistical fact (Lemma 6) that the smoothing-spline weight function is exponentially close to an explicit kernel; if that approximation is not as tight as stated, or if the needed conditions (equidistant decoder points and $N\lambda_d^{1/4}$ bounded below) are violated, the adversarial error bound and the $N^{6/5(a-1)}$ rate collapse.

Editorial extensions

If this is right

  • Any computing task whose input–output map has bounded first and second derivatives can be distributed with coded redundancy; no polynomial or algebraic structure is required.
  • The master can tolerate any sublinear number of colluding adversarial workers, and the worst-case error shrinks polynomially as $N$ grows, with faster decay when the adversarial fraction is smaller.
  • A constant fraction of adversarial workers is a hard wall in this framework: for $\gamma=\mu N$, some admissible function $f$ makes the error bounded away from zero for every $N$.
  • The implemented scheme achieves the predicted behavior on a scalar function and on LeNet5 inference, with empirical convergence at least as fast as the theoretical upper bound.

Reading between the lines

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

  • An implication the authors leave implicit is that the decoder's linear-smoother structure should combine with the straggler-resilience analysis of the same framework, yielding one scheme robust to both stragglers and adversaries simultaneously.
  • A testable extension is to replace non-smooth activation functions in deep networks with smooth ones, which would make the bounded-second-derivative assumption hold and bring the theoretical rate directly to neural-network inference.
  • The optimality result is framework-relative: it rules out smooth encoder–decoder schemes that tolerate a linear fraction of adversaries for general tasks, but it does not foreclose algebraic codes that tolerate linear adversaries for the specific structured computations they are built for.
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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 / 6 minor

Summary. The paper proposes a general coded computing framework for the distributed evaluation of functions belonging to second-order Sobolev spaces, using smoothing splines as encoder and decoder. With N workers, of which at most γ = O(N^a) are adversarial, it claims a worst-case average approximation error decaying as O(N^{(6/5)(a-1)}) for a ∈ [0,1), and an impossibility result for γ = Ω(N). The proof decomposes the error into an encoder term, a no-adversary decoder generalization term, and an adversarial decoder perturbation term; the last is bounded through an equivalent-kernel approximation of the spline smoother.

Significance. If correct, the paper would widen coded computing from structured algebraic tasks to a broad class of smooth functions and would provide a quantitative trade-off between the adversarial fraction and the approximation error. The formulation in terms of Sobolev/RKHS spaces, the explicit error decomposition in (2) and (10), and the concrete, falsifiable rate O(N^{6/5(a-1)}) are appealing; the numerical experiments on a one-dimensional function and on LeNet5 are consistent with the claimed rate. However, several load-bearing steps of the proof are not valid as printed, so the significance depends on whether those gaps can be repaired without changing the stated rate.

major comments (4)
  1. [Appendix C, Eq. (70)] The displayed triangle inequality for |u_d(z) - u_d^o(z)| is not an upper bound as written. Expanding the difference (1/N)Σ G_N,λ y_i - (1/N)Σ G_N,λ \bar y_i and inserting ±(1/N)Σ K_λ y_i and ±(1/N)Σ K_λ \bar y_i yields three nonzero terms: one proportional to (y_i - \bar y_i)(G_N,λ - K_λ), one to (y_i - \bar y_i)K_λ, and one to \bar y_i(K_λ - G_N,λ). The last term, which is bounded by M sup|G_N,λ - K_λ| over all N indices, is omitted in (70). Since γ may be much smaller than N, retaining only the two 2Mγ/N terms makes the bound too small by a factor of about N/γ. This occurs exactly in the term whose optimization produces the N^{6/5(a-1)} rate, so the formal derivation of L_a^dec must be repaired or the rate re-derived.
  2. [Appendix B, Lemma 6 and Section IV, Theorem 2] The proof of Lemma 6 contains a sign error: in Eq. (50) the second triangle-inequality term is written as |G_λ + K_λ|, whereas the required term is |G_λ - K_λ|. In addition, Theorem 2 in Section IV displays the exponential factor e^{+√2 λ_d^{-1/4}}, while Appendix D and the proof of Theorem 3 use e^{-√2 λ_d^{-1/4}}. If the positive sign were taken literally, the C2 term would diverge as λ_d → 0 and Corollary 1 would fail. Since these signs appear in the exponential term that is optimized to obtain the exponent 6/5(a-1), they must be corrected consistently in the theorem statement and in the proof.
  3. [Section VI, Theorem 1 proof] The impossibility proof contains a scaling error in the definition of the adversarial interval. With β_i = i/N, the set V = {i : β_i ∈ [α − γ/2, α + γ/2]} has about γN = μN² elements, not at most μ as claimed. The interpolation argument requires the adversary to control the specified points near α; as written, the proof does not show that this set is small enough or that the adversary can manipulate it as intended. The construction presumably needs an interval of length about γ/N rather than γ; this must be fixed for the claimed optimal robustness result to stand.
  4. [Appendix C, Lemma 8] The proof that h has a zero is not valid as written. From h(t) > 0 for all t and β* minimizing h(β_v), it does not follow that \bar u_d^o(β_v) - f(u_e(β_v)) ≥ 0 for every v; this difference equals h(β_v) - u_d^o(β*), which can be negative. Consequently the inequality labeled (a) in Eq. (62) is unjustified. Since Lemma 8 is used in the appendix to derive the L_g^dec bound, the self-contained proof is incomplete; a correct argument can be obtained by subtracting min_v h(β_v) instead of u_d^o(β*), or the authors should rely explicitly on the cited result [6].
minor comments (6)
  1. [Section III-B] There is a duplicated 'where where λ_e' in the encoder design paragraph; please fix the typo.
  2. [Section II] The sentence describing the adversary's knowledge contains 'and and the scheme'; please correct the duplicated conjunction.
  3. [Section VI and Appendix B, Lemma 3] The proof sketch states that |K_λ| is bounded by τ λ_d^{1/4} exp(|x−β|λ_d^{-1/4}), but Lemma 3 proves the correct bound τ λ_d^{-1/4}. These statements should be reconciled.
  4. [Appendix C, Eq. (70)] The notation y_i versus \bar y_i (or y_i versus y_i) is confusing; please define the received values and the honest values f(u_e(β_i)) explicitly before the equation.
  5. [Abstract] The abstract says 'maximum approximation error', while the objective in (1) is the supremum of the average approximation error; please align the terminology.
  6. [Theorem 4] Part (ii) is delegated to '[6, Proposition 1]' without even a proof sketch; for a self-contained submission, please provide at least the main steps.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the adversarial-rate result is derived by optimizing an upper bound, and the self-citations to prior work are independent support rather than re-used conclusions.

full rationale

The claimed rate O(N^{6/5(a-1)}) is not fitted or definitionally forced. Theorem 2 gives an upper bound on R(\hat f) as a sum of L_enc, L^g_dec, and L^a_dec; L^a_dec is controlled via the equivalent-kernel approximation (Lemmas 3-6, proven/imported from the statistics literature), and L_enc is controlled by the Lipschitz property. The encoder is then chosen to minimize this derived upper bound, and in Appendix E the smoothing parameter lambda_d is optimized analytically to balance the terms N^{2a-2} lambda_d^{-1/2} and lambda_d^{3/4}, yielding lambda_d^* = O(N^{8/5(a-1)}) and rate O(N^{6/5(a-1)}). This is a derivation, not a renaming or an in-sample fit. The paper does import several lemmas from the authors' prior work [6] (e.g., Lemma 1, no-adversary decoder bound, and norm-equivalence lemmas), and these are load-bearing; however, they are stated results with assumptions that do not include the present adversarial claim, and they are not re-fit to the paper's data or to the claimed rate. Thus the self-citations are independent evidence rather than circular justification. The appendix does contain a likely proof gap in Lemma 6's proof, where the triangle inequality is applied to |G_lambda + K_lambda| instead of |G_lambda - K_lambda| (Eq. (50)), and Eq. (70) appears to drop honest-data terms; these are correctness risks, not instances of circularity, because even if the displayed inequalities are wrong the argument is not equivalent to its inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central rate depends on the smoothness assumptions on f, the restriction of encoder and decoder to H^2, equidistant decoder points, and a collection of classical spline and kernel approximation results from the statistics literature. The scheme has two design hyperparameters, lambda_d and lambda_e, whose asymptotic scalings are prescribed but whose constants are left unspecified. No new physical entities are introduced.

free parameters (2)
  • lambda_d (decoder smoothing parameter) = Theta(N^(8/5(a-1)))
    Design parameter balancing decoder fit and smoothness; the proof optimizes it to obtain the rate, but the constant J is unspecified and not fitted to data.
  • lambda_e (encoder regularization parameter) = max{C1 M^2 gamma^2/N^4, C2' M^2 gamma^2/N^2 lambda_d^(-1/2), (nu^2+eta^2)(C4 lambda_d^(3/4)+C5 N^(-3))}
    Introduced in the proof of Theorem 3 to upper bound the encoder's contribution; depends on unspecified constants and is chosen by hand.
assumptions (6)
  • domain assumption The computing function f lies in W^(2,infinity) and W^(2,2) with ||f'||_{L_infinity} <= nu and ||f''||_{L_infinity} <= eta, and takes values in [-M,M]^m.
    Used throughout Theorems 2-4 to bound Lenc and Lemma 7 on composition; the whole scheme targets general functions under this smoothness assumption.
  • domain assumption The encoder and decoder are restricted to second-order Sobolev (RKHS) spaces H^2 on Omega=[0,1].
    This restriction defines the scheme and is what enables the representer theorem and spline solutions; the impossibility result is only within this class.
  • domain assumption The decoder points beta_i = i/N are equidistant and N lambda_d^(1/4) > c0.
    Required to apply the equivalent-kernel approximation lemmas from references [27]-[30], which supply the bound on L^a_dec.
  • standard math Classical spline error bounds (Theorem 6 from [36]) and equivalent-kernel approximation results (Lemmas 4-6 from [27]-[30]) are correct under the stated conditions.
    The main error bounds are imported from the statistics literature; the paper does not prove them, and the rate depends on their exact forms.
  • domain assumption The adversary's outputs are bounded in [-M,M]^m and all adversarial servers act under full knowledge of the scheme.
    This matches the threat model in Section II and is used to bound |y_i - bar y_i| <= 2M in the proof of Theorem 2.
  • standard math The composition f composed with u_e belongs to W^(2,2) when u_e is in W^(2,2) and f is smooth.
    Quoted as Lemma 7 from [6]; used to justify the spline error bound on h = u^o_d - f composed with u_e.

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Pith. "Pith review of General Coded Computing: Adversarial Settings." pith.science (2026). https://pith.science/paper/DB3WYHEX

@misc{pith2026250208058,
  author       = {Pith},
  title        = {Pith review of: General Coded Computing: Adversarial Settings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DB3WYHEX}},
  note         = {Machine review of arXiv:2502.08058}
}
abstract

Conventional coded computing frameworks are predominantly tailored for structured computations, such as matrix multiplication and polynomial evaluation. Such tasks allow the reuse of tools and techniques from algebraic coding theory to improve the reliability of distributed systems in the presence of stragglers and adversarial servers. This paper lays the foundation for general coded computing, which extends the applicability of coded computing to handle a wide class of computations. In addition, it particularly addresses the challenging problem of managing adversarial servers. We demonstrate that, in the proposed scheme, for a system with $N$ servers, where $\mathcal{O}(N^a)$, $a \in [0,1)$, are adversarial, the supremum of the average approximation error over all adversarial strategies decays at a rate of $N^{\frac{6}{5}(a-1)}$, under minimal assumptions on the computing tasks. Furthermore, we show that within a general framework, the proposed scheme achieves optimal adversarial robustness, in terms of maximum number of adversarial servers it can tolerate. This marks a significant step toward practical and reliable general coded computing. Implementation results further validate the effectiveness of the proposed method in handling various computations, including inference in deep neural networks.

Figures

Figures reproduced from arXiv: 2502.08058 by the authors.

Figure 1
Figure 1. Log-log plot illustrating the convergence rates of approximation error for the function [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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Works this paper leans on

38 extracted references · 35 canonical work pages

  1. [6]

    Coded computing for resilient distributed computing: A learning-theoretic framework,

    P. Moradi, B. Tahmasebi, and M. A. Maddah-Ali, “Coded computing for resilient distributed computing: A learning-theoretic framework,” in The Thirty-eighth Annual Conference on Neural Information Processing Systems

    work page

  2. [1]

    Berrut approximated coded computing: Straggler resistance beyond polynomial computing,

    T. Jahani-Nezhad and M. A. Maddah-Ali, “Berrut approximated coded computing: Straggler resistance beyond polynomial computing,” IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 45, no. 1, pp. 111–122, 2022

    work page 2022

  3. [3]

    Straggler mitigation in distributed optimization through data encoding,

    C. Karakus, Y . Sun, S. Diggavi, and W. Yin, “Straggler mitigation in distributed optimization through data encoding,” Advances in Neural Information Processing Systems, vol. 30, 2017

    work page 2017

  4. [4]

    ApproxIFER: A model-agnostic approach to resilient and robust prediction serving systems,

    M. Soleymani, R. E. Ali, H. Mahdavifar, and A. S. Avestimehr, “ApproxIFER: A model-agnostic approach to resilient and robust prediction serving systems,” in Proceedings of the AAAI Conference on Artificial Intelligence , vol. 36, no. 8, 2022, pp. 8342–8350

    work page 2022

  5. [5]

    Lagrange coded computing: Optimal design for resiliency, security, and privacy,

    Q. Yu, S. Li, N. Raviv, S. M. M. Kalan, M. Soltanolkotabi, and S. A. Avestimehr, “Lagrange coded computing: Optimal design for resiliency, security, and privacy,” in The 22nd International Conference on Artificial Intelligence and Statistics . PMLR, 2019, pp. 1215–1225

    work page 2019

  6. [7]

    Efficient and Robust Distributed Matrix Computations via Convolutional Coding

    A. B. Das, A. Ramamoorthy, and N. Vaswani, “Random convolutional coding for robust and straggler resilient distributed matrix computation,” arXiv preprint arXiv:1907.08064, 2019

    work page Pith review arXiv 1907

  7. [8]

    Redundancy techniques for straggler mitigation in distributed optimization and learning,

    C. Karakus, Y . Sun, S. Diggavi, and W. Yin, “Redundancy techniques for straggler mitigation in distributed optimization and learning,” The Journal of Machine Learning Research , vol. 20, no. 1, pp. 2619–2665, 2019

    work page 2019

  8. [9]

    Straggler mitigation in distributed matrix multiplication: Fundamental limits and optimal coding,

    Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Straggler mitigation in distributed matrix multiplication: Fundamental limits and optimal coding,” IEEE Transactions on Information Theory , vol. 66, no. 3, pp. 1920–1933, 2020

    1920

Show all 38 references
  1. [10]

    Polynomial codes: an optimal design for high-dimensional coded matrix multiplication,

    Q. Yu, M. Maddah-Ali, and S. Avestimehr, “Polynomial codes: an optimal design for high-dimensional coded matrix multiplication,” Advances in Neural Information Processing Systems , vol. 30, 2017

    2017

  2. [11]

    Efficient and robust distributed matrix computations via convolutional coding,

    A. B. Das, A. Ramamoorthy, and N. Vaswani, “Efficient and robust distributed matrix computations via convolutional coding,” IEEE Transactions on Information Theory, vol. 67, no. 9, pp. 6266–6282, 2021

    2021

  3. [12]

    Oversketch: Approximate matrix multiplication for the cloud,

    V . Gupta, S. Wang, T. Courtade, and K. Ramchandran, “Oversketch: Approximate matrix multiplication for the cloud,” in 2018 IEEE International Conference on Big Data (Big Data) . IEEE, 2018, pp. 298–304

    2018

  4. [13]

    CodedSketch: A coding scheme for distributed computation of approximated matrix multiplication,

    T. Jahani-Nezhad and M. A. Maddah-Ali, “CodedSketch: A coding scheme for distributed computation of approximated matrix multiplication,” IEEE Transactions on Information Theory , vol. 67, no. 6, pp. 4185–4196, 2021

    2021

  5. [14]

    On the optimal recovery threshold of coded matrix multiplication,

    S. Dutta, M. Fahim, F. Haddadpour, H. Jeong, V . Cadambe, and P. Grover, “On the optimal recovery threshold of coded matrix multiplication,” IEEE Transactions on Information Theory , vol. 66, no. 1, pp. 278–301, 2020

    2020

  6. [15]

    Short-Dot: Computing large linear transforms distributedly using coded short dot products,

    S. Dutta, V . Cadambe, and P. Grover, “Short-Dot: Computing large linear transforms distributedly using coded short dot products,” vol. 65, no. 10, 2019, pp. 6171–6193

    2019

  7. [16]

    R. E. Blahut, Algebraic codes on lines, planes, and curves: an engineering approach . Cambridge University Press, 2008

    2008

  8. [17]

    Numerically stable polynomially coded computing,

    M. Fahim and V . R. Cadambe, “Numerically stable polynomially coded computing,” IEEE Transactions on Information Theory , vol. 67, no. 5, pp. 2758–2785, 2021

    2021

  9. [18]

    Analog Lagrange coded computing,

    M. Soleymani, H. Mahdavifar, and A. S. Avestimehr, “Analog Lagrange coded computing,” IEEE Journal on Selected Areas in Information Theory , vol. 2, no. 1, pp. 283–295, 2021

    2021

  10. [19]

    Numerically stable coded matrix computations via circulant and rotation matrix embeddings,

    A. Ramamoorthy and L. Tang, “Numerically stable coded matrix computations via circulant and rotation matrix embeddings,” IEEE Transactions on Information Theory, vol. 68, no. 4, pp. 2684–2703, 2021

    2021

  11. [20]

    Random Khatri-Rao-product codes for numerically-stable distributed matrix multiplication,

    A. M. Subramaniam, A. Heidarzadeh, and K. R. Narayanan, “Random Khatri-Rao-product codes for numerically-stable distributed matrix multiplication,” CoRR, vol. abs/1907.05965, 2019

    1907 arXiv

  12. [21]

    OverSketch: Approximate matrix multiplication for the cloud,

    V . Gupta, S. Wang, T. Courtade, and K. Ramchandran, “OverSketch: Approximate matrix multiplication for the cloud,” in 2018 IEEE International Conference on Big Data (Big Data) , 2018, pp. 298–304

    2018

  13. [22]

    A generalized representer theorem,

    B. Schölkopf, R. Herbrich, and A. J. Smola, “A generalized representer theorem,” in International conference on computational learning theory . Springer, 2001, pp. 416–426

    2001

  14. [23]

    Smoothing noisy data with spline functions,

    G. Wahba, “Smoothing noisy data with spline functions,” Numerische mathematik, vol. 24, no. 5, pp. 383–393, 1975

    1975

  15. [24]

    SIAM, 1990

    ——, Spline models for observational data . SIAM, 1990

    1990

  16. [25]

    Flexible smoothing with b-splines and penalties,

    P. H. Eilers and B. D. Marx, “Flexible smoothing with b-splines and penalties,” Statistical science, vol. 11, no. 2, pp. 89–121, 1996

    1996

  17. [26]

    Gradient-based learning applied to document recognition,

    Y . LeCun, L. Bottou, Y . Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proceedings of the IEEE , vol. 86, no. 11, pp. 2278–2324, 1998

    1998

  18. [27]

    Spline smoothing: the equivalent variable kernel method,

    B. W. Silverman, “Spline smoothing: the equivalent variable kernel method,” The annals of Statistics , pp. 898–916, 1984

    1984

  19. [28]

    A comparison of a spline estimate to its equivalent kernel estimate,

    K. Messer, “A comparison of a spline estimate to its equivalent kernel estimate,” The Annals of Statistics , pp. 817–829, 1991

    1991

  20. [29]

    A new class of kernels for nonparametric curve estimation,

    K. Messer and L. Goldstein, “A new class of kernels for nonparametric curve estimation,” The Annals of Statistics , pp. 179–195, 1993

    1993

  21. [30]

    Splines as local smoothers,

    D. Nychka, “Splines as local smoothers,” The Annals of Statistics , pp. 1175–1197, 1995

    1995

  22. [31]

    Leoni, A first course in Sobolev spaces

    G. Leoni, A first course in Sobolev spaces . American Mathematical Society, 2024, vol. 181

    2024

  23. [32]

    R. A. Adams and J. J. Fournier, Sobolev spaces. Elsevier, 2003

    2003

  24. [33]

    Berlinet and C

    A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics . Springer Science & Business Media, 2011

    2011

  25. [34]

    Some results on tchebycheffian spline functions,

    G. Kimeldorf and G. Wahba, “Some results on tchebycheffian spline functions,” Journal of mathematical analysis and applications , vol. 33, no. 1, pp. 82–95, 1971

    1971

  26. [35]

    Splines minimizing rotation-invariant semi-norms in sobolev spaces,

    J. Duchon, “Splines minimizing rotation-invariant semi-norms in sobolev spaces,” in Constructive Theory of Functions of Several Variables: Proceedings of a Conference Held at Oberwolfach April 25–May 1, 1976 . Springer, 1977, pp. 85–100

    1976

  27. [36]

    Error bounds for derivative estimates based on spline smoothing of exact or noisy data,

    D. L. Ragozin, “Error bounds for derivative estimates based on spline smoothing of exact or noisy data,” Journal of approximation theory , vol. 37, no. 4, pp. 335–355, 1983. APPENDIX A PRELIMINARIES : S OBOLEV SPACES AND SOBOLEV NORMS Let Ω be an open interval in R and M be a ...

    1983

  28. [37]

    Assuming f ∈ fWm,2 (Ω; R), the solution to the following optimization problem is referred to as the smoothing spline: Sλ,n,m[y] := argmin g∈ eWm,2(Ω;R) 1 n nX i=1 (g (ti) − yi)2 + λ Z Ω g(m)(t) 2 dt, (28) where y = [y1, . . . , yn]. Based on Proposition 1, eHm (Ω; R) := fWm,2 ...

  29. [38]

    According to (42), as n increases, the smoothing spline behaves like a kernel smoother with a bandwidth of λ 1 4 q(t)− 1 4

    demonstrated that for t sufficiently far from the boundary, the following holds: λ 1 4 q(t)− 1 4 Gn,λ t + λ 1 4 q(t)− 1 4 x, t n→∞ − − − − →κ(x) q(t) , (42) where q(t) represents the asymptotic probability density function of the regression points. According to (42), as n incr...

  30. [39]

    proposed an alternative approximated kernel function for equidistant regression points, i.e. ti = i n, which addresses the asymptotic issues of (42): Kλ(x, t) = 2 √ 2λ 1 4 −1 e − |x−t| √ 2λ 1 4 sin |x − t|√ 2λ 1 4 + cos x − t√ 2λ 1 4 + 2 √ 2λ 1 4 −1 Φ x + t√ 2λ 1 4 , x − t√ 2λ...

    1944

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