{"id":"a8c2fd9f-e5f4-40a3-baae-3731a6117524","arxiv_id":"2506.12053","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":1.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A so-called probabilistic sampling Kantorovich operator is just the classical operator applied to additively noisy input, and its main convergence theorem relies on an unstated vanishing-noise assumption.","lead":"The paper defines a probabilistic sampling Kantorovich operator by feeding noisy input into the known classical operator, and claims an L1 convergence theorem for it. The claimed theorem is not established as stated: the proof secretly assumes the noise vanishes, an assumption missing from the theorem and contradicted by the paper's own experiments.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2 is false as stated: the proof needs ε^ℶ → 0 in L1, which is neither assumed nor implied; uniform integrability and bounded L1-norm only give E||S_n(ε^ℶ)||_L1 → E||ε^ℶ||_L1, not 0.","rationale":"The paper's load-bearing result is Theorem 2, and the weakest point is exactly the step E||S_n(ε^ℶ)||_L1 → 0. The reader's weakest_assumption identifies this same gap. My independent reading confirms it: the operator P^ℶ_n is just the classical sampling Kantorovich operator applied to the noisy function f + ε^ℶ. Linearity gives P^ℶ_n(f) = S_n(f) + S_n(ε^ℶ). The first summand approximates f, and the second approximates ε^ℶ, not 0. Therefore the theorem needs ε^ℶ → 0 in L1, which is essentially the conclusion being asserted for the noise term. Since no such hypothesis is stated, the central theoretical claim fails. The proof itself contains the tell-tale phrase 'possibly additional assumptions (particularly ε^ℶ → 0 in L1)', which is an explicit recognition that the stated hypotheses are insufficient. The numerical and image experiments are presented for fixed nonzero noise and do not demonstrate the claimed limiting behavior; in fact Table 1 shows the probabilistic errors remaining at a positive distance from the classical errors at every reported n. The novelty claim also collapses with the theorem: the construction is a direct composition of SK operators with noise injection, and without a correct convergence theorem there is no new mathematical content to support acceptance. I agree with the REJECT verdict and find no reason to soften it.","tokens_in":9961,"tokens_out":2843,"duration_ms":31587,"concrete_test":"Take f = 0 and ε^ℶ = 1_{[0,1]} for every outcome ℶ. This deterministic noise satisfies sup_ℶ ∫ |ε^ℶ| dx = 1 < ∞ and is trivially uniformly integrable, so all stated hypotheses hold. Theorem 2 would predict E||P^ℶ_n(f)||_L1 → 0. But P^ℶ_n(f) = S_n(1_{[0,1]}), and the classical SK convergence theorem gives ||S_n(1_{[0,1]}) − 1_{[0,1]}||_L1 → 0, hence ||S_n(1_{[0,1]})||_L1 → 1. Compute the L1 norm of the piecewise-constant SK approximation to the indicator of [0,1] for n = 1, 2, 4, 8, ...; the values approach 1, not 0, disproving Theorem 2 as stated.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim is Theorem 2 (Section 3.1), asserting E||P^ℶ_n(f) − f||_L1 → 0 under only f ∈ L1, sup_ℶ ∫_{R^d} |ε^ℶ| dx < ∞, and uniform integrability. The proof splits the error as E||P^ℶ_n(f) − f||_L1 ≤ ||S_n(f) − f||_L1 + E||S_n(ε^ℶ)||_L1, then the first term goes to 0 by the classical SK theorem. The problem is the second term. The classical convergence theorem for SK operators says S_n(ε^ℶ) → ε^ℶ in L1 as n → ∞, not S_n(ε^ℶ) → 0. Consequently E||S_n(ε^ℶ)||_L1 → E||ε^ℶ||_L1. The stated hypotheses permit E||ε^ℶ||_L1 to be any positive number; they do not force it to 0. The proof's final sentence explicitly acknowledges this by inserting 'possibly additional assumptions (particularly ε^ℶ → 0 in L1)', but that condition is absent from the theorem's hypotheses. With fixed nonzero noise, P^ℶ_n(f) converges to f + ε^ℶ, so the claimed limit fails. This is an in-scope internal admission in the proof itself, and it is fatal to the theorem. The numerical section also does not rescue the claim: Table 1 shows probabilistic errors staying above the classical ones and not tending to 0 in a demonstrated way.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'probabilistic Sampling Kantorovich operators' (PSK operators), defined by applying classical sampling Kantorovich operators to noisy measurements f^ℶ(x) = f(x) + ε^ℶ(x), where ε^ℶ is a random perturbation. The main theoretical result is Theorem 2 in Section 3.1, which asserts that, under a uniform L1-boundedness condition on the noise and an unspecified uniform integrability assumption, E‖P_n^ℶ(f) − f‖_{L1} → 0 as n → ∞. The paper also presents numerical experiments on the function e^{-x^2} and the cameraman image, comparing classical and probabilistic operators via L1 error, expected PSNR, SSIM, and MAE.","tokens_in":10299,"tokens_out":3746,"duration_ms":35630,"significance":"If Theorem 2 were correct, the paper would provide a stochastic analogue of the classical convergence theorem for sampling Kantorovich operators, potentially of interest for signal and image processing under noisy measurements. The paper also gives a comparison of classical and probabilistic operators in image quality metrics, which could be useful in practice. However, the central convergence claim is false as stated, and therefore the theoretical contribution collapses. The numerical experiments, while illustrating the behavior of the operators, cannot compensate for the flawed theorem. The paper does not provide machine-checked proofs or reproducible code, and its formal framework is too imprecise to support the claimed novelty.","major_comments":[{"comment":"Theorem 2 is false as stated. By linearity, P_n^ℶ(f) = S_n(f) + S_n(ε^ℶ). Since ε^ℶ ∈ L1, the classical convergence theorem for SK operators gives S_n(ε^ℶ) → ε^ℶ in L1, so E‖S_n(ε^ℶ)‖_{L1} → E‖ε^ℶ‖_{L1}, which is not zero unless ε^ℶ = 0 in L1. The proof explicitly acknowledges this by saying 'possibly additional assumptions (particularly ε^ℶ → 0 in L1)' are needed; that condition is absent from the theorem's hypotheses and is effectively equivalent to the desired conclusion. Thus the stated assumptions do not imply the claimed convergence.","section":"Section 3.1, Theorem 2"},{"comment":"The numerical results are consistent with the flaw in Theorem 2. With fixed Gaussian noise of standard deviation 0.02, the probabilistic L1 errors are 0.091, 0.048, 0.027, and 0.018 for n = 5, 15, 25, and 35, respectively. Rather than converging to zero, the errors appear to level off near the noise magnitude, which is exactly what one would expect when S_n(ε^ℶ) → ε^ℶ and ε^ℶ is a nonzero fixed noise field. The table therefore does not provide empirical support for the theorem.","section":"Section 4, Example 1, Table 1"},{"comment":"Lemma 2 is a boundedness result and is essentially correct, but its proof uses the assumption sup_ℶ ∫|ε^ℶ| < ∞, where ℶ denotes the sample space. This notation is ambiguous: ε^ℶ is a random field, so the supremum should be taken over realizations or over the probability space. More importantly, Lemma 2 only gives a bound in terms of E‖ε^ℶ‖_{L1}; it does not provide any mechanism for that term to vanish, and no condition in the paper ensures E‖ε^ℶ‖_{L1} → 0. Thus the lemma cannot serve as a step toward Theorem 2 as stated.","section":"Section 3.1, Lemma 2"}],"minor_comments":[{"comment":"The substitution step contains a typo: after setting w = nx − k, the displayed computation reads '1/nw ∥ξ∥L1'; it should be n^{-q}∥ξ∥_{L1}. The subsequent conclusion is correct, but the notation is garbled.","section":"Section 3.1, Lemma 1 proof"},{"comment":"The sampling cells are denoted inconsistently as D_n^a, R_n^k, and R_n^a. These notations should be unified and defined precisely in one place.","section":"Throughout"},{"comment":"The theorem statement uses R^d in the supremum condition while the rest of the paper works on R^q; these should be consistent.","section":"Section 3.1, Theorem 2"},{"comment":"The noise level is not specified consistently: Algorithm 1 uses a Gaussian with std = 0.02, while Example 1 writes ε_k ∼ N(0, τ²) without assigning a value to τ. This makes the numerical experiments hard to reproduce.","section":"Algorithm 1 and Example 1"},{"comment":"The last column header 'Var(|Sn(f) − f|)' in the table for PSK operators is misleading; it should refer to the variance of the error for the probabilistic operator P_n^ℶ(f), not for the classical Sn(f).","section":"Table 3"},{"comment":"There are numerous typographical and grammatical errors, including 'cordiality' for 'cardinality', 'Preciously' for 'Precisely', and 'receptively' for 'respectively'. A thorough editing pass is needed.","section":"Overall presentation"}],"recommendation":"reject","confidential_remarks":"The central theorem of the paper is not merely missing a technical hypothesis; it is false as stated, and the internal 'possibly additional assumptions' remark in the proof confirms that the authors were aware of the gap. The numerical experiments actually reinforce the problem by showing nonzero limiting errors under fixed noise. Given that the main result is the raison d'être of the paper, rejection seems appropriate. The paper also has substantial presentation issues and limited novelty: the 'probabilistic' operator is just the classical operator applied to a noisy input, and no meaningful stochastic convergence theory is developed beyond the flawed Theorem 2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper's central result, Theorem 2, is false as stated, and its own proof admits it. The PSK operator is just the classical SK operator applied to noise-corrupted data: P^ℶ_n(f) = S_n(f+ε^ℶ) = S_n(f)+S_n(ε^ℶ) by linearity. So the convergence theorem reduces to the known L1 convergence of S_n(f) plus a vanishing condition on S_n(ε^ℶ). That condition is E||S_n(ε^ℶ)||→0, which by the classical theorem is equivalent to ε^ℶ→0 in L1. The hypotheses only give sup_ℶ ||ε^ℶ||_1 < ∞ and uniform integrability; they allow E||ε^ℶ||_1 to be any positive number. In the proof's last line the authors write 'possibly additional assumptions (particularly ε^ℶ→0 in L1)' — but that assumption is absent from the theorem statement and is effectively the conclusion. With fixed nonzero noise the operator converges to f+ε, not f. This is a load-bearing flaw.\n\nWhat is done well: Lemma 1 and Lemma 2 are standard but correct; the linearity observation is explicit; the paper engages with the correct classical references. The numerical experiments cover several metrics, but they are not reproducible (no code or data), and the data availability statement says no data were generated, which conflicts with the image experiments. The numerics also do not rescue the theory: Table 1 shows probabilistic errors stuck above classical ones and not tending to zero, and Table 3's E[PSNR] values are roughly flat.\n\nThe novelty claim is not substantiated. There is no genuinely new operator or theorem here. The paper reads as a teaching example of a known result with noise added. That could be useful for a tutorial, but not as a research contribution. I would not send this to a serious referee; it needs either a corrected theorem with the noise-decay condition, or a reframing as an expository note. As it stands, reject.","headline":"The main theorem is false as stated: the proof needs ε→0 in L1, which is neither assumed nor implied, and the 'novel' operator is just the classical SK operator applied to noisy inputs.","tokens_in":10826,"tokens_out":3597,"would_cite":false,"duration_ms":34936,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A25","41A35","46E30","47A58","47B38","94A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Noisy samples can still converge to the true function","keywords":["probabilistic sampling Kantorovich operators","stochastic uniform integrability","sampling Kantorovich operators","L1 approximation","peak signal-to-noise ratio","structural similarity index","image reconstruction","pointwise convergence"],"falsifier":"Take $f=0$ and let the noise be a fixed, non-random function, say $\\varepsilon^\\beth(x)=1$ on $[0,1]$ and $0$ elsewhere. Then the boundedness and uniform-integrability hypotheses hold, yet the classical convergence theorem gives $S_n(\\varepsilon^\\beth)\\to\\varepsilon^\\beth$ in $L^1$, so the expected error tends to $\\|\\varepsilon^\\beth\\|_{L^1}=1$ rather than $0$.","tokens_in":9730,"feed_emoji":"🎲","tokens_out":11668,"duration_ms":102760,"temperature":0.7,"pith_summary":"This paper introduces a probabilistic sampling Kantorovich (PSK) operator, built by feeding noisy measurements $f^\\beth(x)=f(x)+\\varepsilon^\\beth(x)$ into the classical sampling operator $S_n$. The central aim is to show that the expected total error $\\mathbb{E}[\\|P_n^\\beth(f)-f\\|_{L^1}]$ drops to zero as the sampling density $n$ grows, assuming the noise is bounded in total size and uniformly integrable. If that holds, then averaging corrupted local samples still recovers the original function on average, which is exactly what one wants for reconstruction problems where data arrives with random errors. The paper also compares classical and probabilistic operators on a test function and on a standard test image, reporting PSNR, SSIM, and MAE, and finds the probabilistic version more stable under randomness.","feed_headline":"Sampling with noise can still recover the true function","feed_subtitle":"A new probabilistic operator drives expected reconstruction error to zero as sampling density grows.","key_machinery":"The central object is the PSK operator $P_n^\\beth(f)=S_n(f+\\varepsilon^\\beth)$, where $S_n$ is the classical sampling Kantorovich operator that averages $f$ over small cells $D_n^a=\\prod_j[a_j/n,(a_j+1)/n]$ and convolves with a kernel $\\xi$ satisfying unit-sum, boundedness, and decay conditions. The identity that carries the proof is linearity, $P_n^\\beth(f)=S_n(f)+S_n(\\varepsilon^\\beth)$, which separates deterministic approximation from the transported noise. The boundedness lemma $\\|S_n(g)\\|_{L^1}\\le \\|\\xi\\|_{L^1}\\|g\\|_{L^1}$, valid for all $n$, is what lets the expectation of the noise term be controlled and the final convergence be asserted.","core_discovery":"Stated in the paper's own terms, the core claim is Theorem 2: for $f\\in L^1(\\mathbb{R}^q)$ and a noise process $\\varepsilon^\\beth$ with $\\sup_\\beth\\int_{\\mathbb{R}^q}|\\varepsilon^\\beth(x)|\\,dx<\\infty$, the PSK operator satisfies $\\mathbb{E}[\\|P_n^\\beth(f)-f\\|_{L^1}]\\to0$ as $n\\to\\infty$ under an assumption of uniform integrability. The proof decomposes the operator as $P_n^\\beth(f)=S_n(f)+S_n(\\varepsilon^\\beth)$, applies the classical convergence theorem to $S_n(f)$, and uses the boundedness lemma $\\|S_n(g)\\|_{L^1}\\le L\\|g\\|_{L^1}$ to handle the transported noise. The theorem's stated hypotheses are boundedness and uniform integrability; the proof's final paragraph notes that concluding the noise term vanishes may need the additional condition $\\varepsilon^\\beth\\to0$ in $L^1$.","pith_inferences":["The paper's Theorem 2 as written is missing an explicit noise-decay assumption: if $\\varepsilon^\\beth$ is fixed and nonzero, the expected error cannot converge to zero, so a corrected statement should either require $\\varepsilon^\\beth\\to0$ in $L^1$ or aim for the noisy signal $f+\\varepsilon^\\beth$ as the target.","Because the bound is linear, one could lift the qualitative theorem to a rate estimate: if the classical operator converges at a known rate and $\\mathbb{E}[\\|\\varepsilon^\\beth\\|_{L^1}]$ decays at a known rate, the same proof gives a rate for $\\mathbb{E}[\\|P_n^\\beth(f)-f\\|_{L^1}]$.","The numerical section compares expected values of nonlinear image metrics; an untested extension would be to check whether optimizing the kernel or window size under expected SSIM, rather than classical SSIM, changes the reconstructed image."],"forward_implications":["When the hypotheses of Theorem 2 hold, noisy measurements can be used for approximation: increasing the sampling density drives the expected total reconstruction error to zero.","The expected error is controlled by the classical approximation error plus the expected noise level, so the rate at which noise decays in $L^1$ determines how fast the probabilistic reconstruction becomes accurate.","The operator is uniformly stable under sampling: $\\mathbb{E}[\\|P_n^\\beth(f)\\|_{L^1}]$ stays bounded by a constant times the sum of the signal size and the expected noise size, independently of $n$.","In image experiments, the paper's reported metrics indicate that PSK operators keep expected PSNR and SSIM relatively stable across window sizes, while classical operators have better metrics only in ideal noiseless conditions."],"supporting_citations":[{"why":"Supplies the classical sampling Kantorovich convergence theorem $\\|S_n(f)-f\\|_{L^1}\\to0$ that the proof invokes for the deterministic part of the operator.","marker":"[17]"},{"why":"Provides the uniform-integrability criterion used to pass from convergence of random variables to convergence of their expectations.","marker":"[14]"},{"why":"Supplies the SSIM image-quality measure used in the numerical comparison between classical and probabilistic operators.","marker":"[28]"}],"fun_headline_variants":["Probabilistic sampling operator drives error to zero","Expected reconstruction error vanishes under noise","New operator converges to true function despite noise","Sampling Kantorovich operator handles noisy data","Uniform integrability ensures noise-free approximation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on the transported noise term shrinking to zero on average, which only follows if the original noise itself is shrinking in size on average, a condition the theorem does not state.","fun_headline_variants_meta":{"raw":{"variants":["Probabilistic sampling operator drives error to zero","Expected reconstruction error vanishes under noise","New operator converges to true function despite noise","Sampling Kantorovich operator handles noisy data","Uniform integrability ensures noise-free approximation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000416,"raw_usage":{"total_tokens":2110,"prompt_tokens":872,"completion_tokens":1238,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":1173}},"tokens_in":488,"tokens_out":1238,"duration_ms":9829,"temperature":1.0,"reasoning_tokens":1173,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:37:51.737005+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $f=0$ and let the noise be a fixed, non-random function, say $\\varepsilon^\\beth(x)=1$ on $[0,1]$ and $0$ elsewhere. Then the boundedness and uniform-integrability hypotheses hold, yet the classical convergence theorem gives $S_n(\\varepsilon^\\beth)\\to\\varepsilon^\\beth$ in $L^1$, so the expected error tends to $\\|\\varepsilon^\\beth\\|_{L^1}=1$ rather than $0$.","supporting_citations":[{"cited_title":"Bardaro, G","cited_arxiv_id":null,"evidence_quote":"Supplies the classical sampling Kantorovich convergence theorem $\\|S_n(f)-f\\|_{L^1}\\to0$ that the proof invokes for the deterministic part of the operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the uniform-integrability criterion used to pass from convergence of random variables to convergence of their expectations."},{"cited_title":"Wang, A.C","cited_arxiv_id":null,"evidence_quote":"Supplies the SSIM image-quality measure used in the numerical comparison between classical and probabilistic operators."}],"review_version":1}