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arxiv: 2501.14857 · v2 · submitted 2025-01-24 · 🧮 math.NA · cs.NA· math.FA

Image resizing by neural network operators and their convergence rate with respect to the L^p-norm and the dissimilarity index defined through the continuous SSIM

Danilo Costarelli , Mariarosaria Natale , Michele Piconi This is my paper

Pith reviewed 2026-05-23 04:39 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.FA
keywords neural network operatorsapproximation orderL^p normcontinuous SSIMimage resizingconvergence ratesmodulus of smoothnessmultivariate operators
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The pith

Neural network operators deliver explicit approximation orders in the L^p-norm and a continuous SSIM dissimilarity index.

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

The authors develop new techniques to derive quantitative estimates for the order of approximation achieved by multivariate linear operators of the pointwise type. The estimates are obtained with respect to the L^p-norm as well as a dissimilarity index constructed from the continuous structural similarity index. Particular attention is given to neural network operators applied to C^1 functions, piecewise C^1 functions, and functions that model digital images. The analysis is further extended to general L^p spaces through the use of the multivariate averaged modulus of smoothness. Numerical tests on image resizing illustrate the practical performance of these operators relative to classical methods.

Core claim

For a family of neural network operators, sharp estimates are established in the case of C^1 and piecewise C^1 functions. Specific quantitative estimates are achieved for functions modeling digital images, including those with respect to the dissimilarity index defined through the continuous SSIM. The analysis is extended to L^p-spaces by means of a new constructive technique employing the multivariate averaged modulus of smoothness.

What carries the argument

Multivariate neural network operators, linear pointwise-type approximation operators linked to artificial neural networks.

Load-bearing premise

The target functions belong to C^1 or piecewise C^1 classes and the neural network operators satisfy the structural properties required for the modulus-of-smoothness arguments.

What would settle it

Finding a C^1 function for which the approximation error by a neural network operator in the L^p norm does not obey the stated quantitative bound, or an image whose SSIM dissimilarity deviates from the predicted estimate.

Figures

Figures reproduced from arXiv: 2501.14857 by Danilo Costarelli, Mariarosaria Natale, Michele Piconi.

Figure 2
Figure 2. Figure 2: The numerical results align with Theorem 22. In fact, as [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Discrete dissimilarity index for the NN algorithm with ramp and logistic functions, plotted with n = 5, 10, 15, 20, 25, 30 on the horizontal axis, bilinear and bicubic interpolation, and u-VPI method for the images (a) montage, (b) france [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Discrete dissimilarity index for the NN algorithm with ramp and logistic functions, plotted with n = 5, 10, 15, 20, 25, 30 on the horizontal axis, bilinear and bicubic interpolation, and u-VPI method for the images (c) mountain, (d) library. Remark 25. In some preliminary experiments, other than σℓ and σR also the hyperbolic tangent activation function σh has been considered. However, σh produced similar r… view at source ↗
Figure 3
Figure 3. Figure 3: Histogram comparison of the PSNR, S-Index, and SSIM values for different test grayscale (left) and RGB (right) images (listed along the vertical axis), with the parameter fixed at n = 30. Each bar represents the corresponding quality index for a given image, highlighting the performance of the proposed method across multiple metrics [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical dissimilarity index 1 − cSSIM obtained by NN algorithm with ramp and logistic functions is shown in (a) for montage and in (c) for france. In (b) and (d), the same numerical dissimilarity is compared with the theoretical dissimilarity cf·log n n as a function of n = 5, 10, 15, 20, 25, 30. In both (b) and (d), the numerical dissimilarity remains significantly below the theoretical curve. Moreover,… view at source ↗
read the original abstract

In literature, several algorithms for imaging based on interpolation or approximation methods are available. The implementation of theoretical processes highlighted the necessity of providing theoretical frameworks for the convergence and error estimate analysis to support the experimental setups. In this paper, we establish new techniques for deriving quantitative estimates for the order of approximation for multivariate linear operators of the pointwise-type, with respect to the $L^p$-norm and to the so-called dissimilarity index defined through the continuous SSIM. In particular, we consider a family of approximation operators known as neural network (NN) operators, that have been widely studied in the last years in view of their connection with the theory of artificial neural networks. For these operators, we first establish sharp estimates in case of $C^1$ and piecewise (everywhere defined) $C^1$-functions. Then, the case of functions modeling digital images is considered, and specific quantitative estimates are achieved, including those with respect to the mentioned dissimilarity index. Moreover, the above analysis has also been extended to $L^p$-spaces, using a new constructive technique, in which the multivariate averaged modulus of smoothness has been employed. Finally, numerical experiments of image resizing have been given to support the theoretical results. The accuracy of the proposed algorithm has been evaluated through similarity indexes such as SSIM, likelihood index (S-index) and PSNR, and compared with other rescaling methods, including bilinear, bicubic, and upscaling-de la Vall\'ee-Poussin interpolation (u-VPI). Numerical simulations show the effectiveness of the proposed method for image processing tasks, particularly in terms of the aforementioned SSIM, and are consistent with the provided theoretical analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops new techniques for deriving quantitative estimates of the approximation order for multivariate linear pointwise-type neural network operators. It first obtains sharp estimates for C^1 and piecewise C^1 functions via modulus-of-smoothness arguments, then treats functions modeling digital images (including estimates with respect to a dissimilarity index constructed from the continuous SSIM), extends the analysis to L^p spaces by means of a new averaged-modulus construction, and concludes with numerical experiments on image resizing that compare the NN operators against bilinear, bicubic, and u-VPI interpolation using SSIM, S-index, and PSNR.

Significance. If the derivations hold, the work supplies concrete new tools in multivariate approximation theory for a class of NN operators that are already studied for their links to artificial neural networks. The extension to an averaged modulus for L^p estimates and the SSIM-based dissimilarity index are technically distinctive; the numerical validation on image resizing provides direct evidence that the theoretical rates are consistent with observed performance.

minor comments (3)
  1. [Abstract] Abstract: the repeated assertion of 'sharp estimates' is not accompanied by any indication of the explicit constants or the precise modulus employed; a single sentence clarifying the key technical device would improve readability without lengthening the abstract.
  2. [Numerical experiments] Numerical section: the description of the NN operators used in the experiments omits the precise choice of activation functions, number of hidden layers, and training parameters; these details are needed for reproducibility of the reported SSIM and PSNR values.
  3. [L^p extension] Notation: the transition from the pointwise-type operator to the averaged modulus in the L^p section introduces several auxiliary functions whose dependence on the partition of unity or on the support size is not stated explicitly; a short clarifying remark or diagram would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the manuscript's contributions, and recommendation for minor revision. No specific major comments appear in the report, so there are no individual points requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation chain proceeds from structural properties of pointwise-type NN operators and standard modulus-of-smoothness estimates in C^1 and piecewise C^1 settings, through extensions to digital-image models and L^p spaces via averaged moduli, to a dissimilarity index constructed from continuous SSIM. These steps rely on operator axioms and smoothness assumptions that are stated independently of the target rates; no equation reduces a claimed prediction to a fitted parameter or self-referential normalization, and no load-bearing premise collapses to a self-citation whose content is itself defined by the present work. The numerical experiments compare against external baselines (bilinear, bicubic, u-VPI) using standard indexes, confirming the estimates remain falsifiable outside any internal fit.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only view yields no explicit free parameters, axioms, or invented entities; the work appears to rest on standard assumptions of approximation theory (e.g., properties of pointwise-type linear operators and existence of the continuous SSIM index) without introducing new fitted constants or postulated objects.

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