{"id":"a9c0cc85-06ee-4113-b1a5-787766489ee0","arxiv_id":"2508.20290","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Neural networks approximate low-variation regions of a function first, and a preprocessing trick that subtracts an interpolant speeds up training, per the paper's 'value change' analysis.","lead":"The paper defines 'value change' (VC), a measure of how much a function oscillates inside a small neighborhood, and reports that neural networks fit low-VC regions faster, with a second claimed 'minority-tendency' where rare VC values converge first. It then proposes preprocessing that subtracts an interpolant before training, reporting faster convergence in experiments, though several stated theoretical claims are incorrect as written.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Minority-tendency evidence rests on VCDR values in the zero-density regime the paper itself declares meaningless; Tables 4.1 and 4.2 are full of 0 and 1e-7 entries, so the claimed ordering is not established.","rationale":"My read of the paper: the central claim bundles three components, the VC-tendency, the minority-tendency, and IVC-based preprocessing acceleration. The VC-tendency is supported by direct pointwise correlation plots (Figures 4.2–4.5) and is consistent with the spectral bias literature, so it is the most salvageable component. The preprocessing experiments are confounded by warm-starting: VCP-NN starts from a network already trained to approximate half the target, and VCP-obj trains on a residual, so faster early loss reduction could be ordinary warm-starting or target simplification rather than a VC-specific mechanism. The most load-bearing weakness is in the minority-tendency, because it is presented as a new universal phenomenon and is inferred solely from VCDR tables that contain the exact failure mode flagged in Definition 3.9. The tables contain exact zeros at numerous early-round cells; for near-zero target densities the ratio P_NN/P_f is not a meaningful convergence measure. The two illustrative targets give opposite orderings by absolute VC, and the 'minority' reconciliation requires knowing the target density at those VC levels, which is not reported. KDE bandwidths and seeds are also not reported anywhere, so the ordering could be noise. I therefore agree with the reader's weakest assumption and verdict. No verdict change is needed; however, a revision that replaces the VCDR evidence with density-thresholded, multi-seed diagnostics and an analytic density check could support a conditional acceptance later.","tokens_in":23140,"tokens_out":6003,"duration_ms":55383,"concrete_test":"Compute the target VC densities for sin(2x) and sin(2x)+sin(6x)+sin(10x) at L=0.2 analytically or by very large Monte Carlo sampling without KDE, and evaluate them at exactly the VC levels used in Tables 4.1 and 4.2. Then re-run the training protocol and record VCDR only at VC levels where the target density exceeds a pre-specified positive threshold (e.g., 10 times the KDE bandwidth). If the VC levels with VCDR closest to 1 at early rounds are not exactly those with lower target density, or if any ordering from §4.2 reverses when zero entries are excluded, the minority-tendency claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's strongest novel claim is the minority-tendency: during neural network approximation, VC density converges first at VC levels that are less frequent in the target function (§4.2). The only quantitative evidence is the VC density ratio VCDR = P_NN/P_f in Tables 4.1 and 4.2. Definition 3.9's remark states that when the true VC density approaches 0, slight fluctuations of the neural network's VC density make the ratio have 'no reference significance.' That is exactly the regime of these tables: Table 4.1 contains multiple exact 0 VCDR entries at early rounds, and Table 4.2 contains 0 and 5.7e-8 entries. A zero or near-zero denominator or numerator coming from KDE estimates means the ratio is controlled by bandwidth and finite-sample noise, not by approximation progress. Additionally, the two worked examples order convergence by absolute VC in opposite directions (small VC converges in Table 4.1, large VC converges in Table 4.2) and are reconciled only by post hoc relabeling of which VC values are the 'minority.' No seed variance, KDE bandwidth, or density floor is reported, so the ordering cannot be distinguished from KDE artifacts. If the minority-tendency cannot be read off the VCDR tables, the central claim of §4.2 and the 'minority-first' motivation for preprocessing is unsupported; what remains is the more modest VC-tendency correlation in Figures 4.2–4.5.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a local 'value change' (VC) metric VCL(f,x), a VC density, and an integral VC (IVC) distance, and uses these to characterize neural-network approximation. It claims a VC-tendency (regions with larger VC have larger approximation errors), a minority-tendency (neural networks first match VC density at VC levels that are less frequent in the target function), and a preprocessing framework that minimizes the IVC distance to the objective and thereby accelerates training. The supporting experiments cover linear functions, a monochrome image, flow past a cylinder, and several preprocessing variants.","tokens_in":23379,"tokens_out":7852,"duration_ms":76442,"significance":"If the empirical claims were robust, the VC metric could serve as a useful local diagnostic and preprocessing criterion for neural-network approximation, and the paper does ship code links and performs experiments on nontrivial image and PDE-related data. However, the central theoretical proposition is false as stated, and the quantitative evidence for the minority-tendency is taken from a regime the paper itself declares meaningless. The preprocessing experiments are suggestive but not yet isolated from generic initialization effects. The current manuscript therefore does not establish the claims it emphasizes.","major_comments":[{"comment":"Proposition 3.2 is false as written. For f(x) = -x on [-1,1], one has VCL(f,x0) = L for every x0 and every L, so the right L-derivative of VCL at L = 0 is 1, while f'(x0) = -1. The proof drops absolute values and uses inequalities that do not pass to valid limit bounds in the stated direction. Since Section 3.2 and Section 4 use VC as a tractable proxy for local slope magnitude, this proposition is load-bearing. It should be corrected, at minimum, to the statement that the L-derivative equals |f'(x0)| for differentiable f, and the proof must be rewritten accordingly.","section":"Section 3.1, Proposition 3.2"},{"comment":"The minority-tendency evidence is drawn from exactly the regime that Definition 3.9's remark declares to have 'no reference significance': when the true VC density approaches zero, small fluctuations of the neural network's VC density make the VCDR ratio unreliable. Tables 4.1 and 4.2 contain multiple exact 0 entries and values such as 5.7e-8 at early rounds, and no KDE bandwidth, density floor, sample count, or seed variance is reported. A ratio whose numerator or denominator is zero or near zero from a KDE estimate is dominated by bandwidth and finite-sample noise, not by approximation progress. Moreover, the two examples order convergence by absolute VC in opposite directions and are reconciled only by post hoc relabeling of which VC values are the minority. As presented, the claimed minority-tendency ordering is not distinguishable from KDE artifacts and therefore is not established.","section":"Section 4.2, Tables 4.1 and 4.2"},{"comment":"The comparison of methods A and B does not isolate the effect of DistIVC. Method A starts from a pretrained network that differs from the direct-initialization baseline in both its parameter values and its initial function values, so the faster convergence could be due to favorable initialization or transfer effects more generally. The C-versus-D comparison is better controlled, but the paper's main preprocessing claim would still benefit from a baseline in which the initial model is produced by an alternative criterion unrelated to IVC. Without such a control, the assertion that minimizing IVC distance specifically, rather than any reasonable preprocessing/initialization, drives the acceleration is underdetermined.","section":"Section 6.1, preprocessing experiments"}],"minor_comments":[{"comment":"The sentence 'the test loss for the f2 approximation is always smaller than the f1 approximation' contradicts the immediately preceding conclusion that the smaller-slope function f1 converges faster; this is likely a typo but should be corrected for reproducibility of the motivating example.","section":"Section 2.1"},{"comment":"The proof of property 2 concludes f1(x) = f2(x) pointwise, but the claimed zero-distance condition should only imply that f1 - f2 is constant on each VC box and hence, on a connected domain, constant globally. The theorem statement is correct, but the proof must be repaired.","section":"Section 5.2, Theorem 5.3"},{"comment":"The caption lists Layer∈{2,5}, Width∈{20,100}, while the text in Section 2.2 states Width∈{100,300}; the mismatch should be resolved.","section":"Figure 2.2 caption"},{"comment":"The experimental values of Lmin, Lmax, and the preprocessing threshold ε are not reported in the numerical sections, despite being free parameters of the proposed IVC distance and preprocessing algorithm; a sensitivity study or at least explicit default values should be provided.","section":"Definition 5.1 and Algorithm 5.1"},{"comment":"No random seed or variance information is given for the training runs displayed in Sections 4 and 6, which makes it hard to judge whether the reported orderings, especially those in the VC-density tables, are stable.","section":"General experimental reporting"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it carefully. My judgment is close to the reader's, with one adjustment: the paper has a genuinely useful empirical correlation and a sensible preprocessing trick, so it is not a zero. The VC-tendency (larger local variation → larger approximation error) is convincingly shown in Figures 4.2–4.5 for the image and cylinder data. That is consistent with the frequency-principle literature, which the paper cites but never actually connects; that missed connection hurts the novelty claim more than the result. Also, VC_L is just the classical local oscillation of f (sup of |f(y1)-f(y2)| over a box), so presenting it as a new metric is not accurate. The interpolation-subtraction preprocessing (VCP-obj) is reasonable and the experiments suggest it helps.\n\nWhat is not established is the minority-tendency. The quantitative evidence is Tables 4.1 and 4.2, VCDR values in the regime the paper itself flags as meaningless (Definition 3.9: when true density approaches 0, the ratio has no reference significance). The tables are full of exact 0 and 1e-7 entries, no seeds or KDE bandwidths are reported, and the two examples order convergence in opposite directions by absolute VC; they are reconciled only by relabeling the low-density side as 'minority' after the fact. That claim needs to be redone with density floors, variance across seeds, and a pre-registered notion of minority.\n\nTwo technical fixes are needed. Proposition 3.2 is false as stated: for f(x)=-x, VC_L = L for every box, so the L-derivative is 1 while f'=-1. The proof drops the absolute value; the correct statement is |f'|. And Section 2.1's text says f2 (slope 10) has consistently smaller test loss than f1 (slope 1) at 10,000 steps, then concludes smaller slopes converge faster. As written that is backwards; likely a typo, but it should be fixed.\n\nThe preprocessing validation is partly circular: Algorithm 5.1 builds the initial model by minimizing DistIVC and then reports that smaller DistIVC correlates with faster convergence. The C-vs-D experiment compares f-g to f-g/2, which is essentially a smaller residual target versus a larger one; ordinary warm-starting could explain the speedup. The IVC distance is a fine heuristic, but it needs a baseline that separates the VC mechanism from residual-size reduction.\n\nFor people working on spectral-bias diagnostics or PINN warm-starting, there is useful material here. Bottom line: the VC-tendency correlation and the residual-subtraction preprocessing are worth a serious referee. I would not desk reject. Send to review with the expectation of major revision: fix the proposition, fix Section 2.1, redo minority-tendency properly. If that doesn't happen, reject.","headline":"Plausible VC-error correlation and a useful residual-subtraction trick, but the minority-tendency claim is unsupported and the theory as stated contains a false proposition.","tokens_in":24041,"tokens_out":5811,"would_cite":false,"duration_ms":54231,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","41A25","65D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a new local descriptor of an objective function, VC (value change), defined as the maximum difference between two values of the function inside a box of side length $L$ around a point.","keywords":["value change","VC density","VC-tendency","minority-tendency","IVC distance","neural network approximation","preprocessing","kernel density estimation"],"falsifier":"Rerun the experiments in Tables 4.1 and 4.2 replacing kernel density estimates with histograms of the same VC samples; if the order in which VC levels converge changes, the minority-tendency is an estimator artifact rather than a property of neural network approximation.","tokens_in":22771,"feed_emoji":"📈","tokens_out":8833,"duration_ms":77858,"temperature":0.7,"pith_summary":"The paper proposes a new local descriptor of an objective function, VC (value change), defined as the maximum difference between two values of the function inside a box of side length $L$ around a point. It claims this descriptor organises neural-network approximation behavior: regions with larger VC retain higher prediction errors throughout training (the VC-tendency), and the distribution of VC values is learned first where its target density is smallest (the minority-tendency). On top of this, the paper defines an integral VC (IVC) distance that measures how much two functions differ in local variation, and uses it to build preprocessing schemes that initialize a network with the target's variation shape before training. The resulting preprocessing accelerates training and improves accuracy on linear functions, monochrome image approximation, and flow past a cylinder. If the tendencies are robust, VC gives a derivative-free, data-only way to predict where a neural approximation will be unreliable and to fix it in advance.","feed_headline":"Function's local jumps predict where neural nets fail","feed_subtitle":"New VC metric ties steep local variation to training errors and speeds up learning via shape-matching preprocessing.","key_machinery":"The central object is the value change $VC_L(f,x)$, the maximum amplitude of $f$ inside a box of side $L$ around $x$; its $L$-dependent derivative recovers $|f'(x)|$ for $C^1$ functions, which is why the paper reads it as a multi-scale, derivative-free generalization of slope. Around this object sit its distribution (VC density), the ratio of two densities (VCDR), and the integrated version $IVC(f,x)$ averaged over $L$, whose spatial integral defines the IVC distance. These objects do the work of translating difficulty into a measurable geometric quantity: VC-tendency connects pointwise error to VC magnitude, the VC density ratio tracks how the approximation progresses in distribution, and the IVC distance provides the objective that the preprocessing algorithm tries to minimize before ordinary training begins.","core_discovery":"The central claim is that the local variation of the objective function $f$, measured by $VC_L(f,x) = \\sup_{y_1,y_2} |f(y_1)-f(y_2)|$ over a box of side $L$ centered at $x$, controls how difficult that neighborhood is to approximate. Experiments on a monochrome image and on flow past a cylinder show that, once pixels are sorted by VC, average, maximum, and median prediction errors increase with VC, and this ordering strengthens as training proceeds. In distributional terms, the paper claims a minority-tendency: comparing the VC density of the network with that of the target, the levels of VC that are rare in the target are the ones whose densities converge first, regardless of whether rare means small VC or large VC. From VC it builds a norm, the IVC distance $Dist_{IVC}(f_1,f_2)=\\int_\\Omega IVC(f_1-f_2,x)\\,dx$, which vanishes exactly when the two functions differ by a constant, and shows that preprocessing that reduces this distance speeds up the later training, with numerical support on linear, image, and cylinder-flow tasks.","pith_inferences":["VC could be used as a sampling or weighting rule: train with points weighted by VC density or by VC magnitude to test whether the observed ordering is a cause or merely a correlation.","The minority-tendency, if real, suggests a curriculum: fit rare VC levels first and common levels later, paralleling but not identical to frequency-based curriculum learning; a direct experiment would be to train on a function whose VC density is deliberately made bimodal and inspect the convergence order of the two modes.","The VCDR tables' zero and $10^{-7}$ entries indicate that the ratio estimator needs regularization, such as additive smoothing or log-density comparison, before the minority-tendency can be tested rigorously; this extends the paper's own caveat into a concrete recommendation.","The IVC distance could double as an online diagnostic: monitoring $Dist_{IVC}(\\psi_{NN}, f)$ during training would reveal when the network has stopped improving in shape even while the $L^2$ loss is still decreasing."],"forward_implications":["Training behavior of a fixed network can be predicted, up to ordering, from a single static computation of VC over the sampling domain, without simulating training.","High-VC regions, such as edges, discontinuities, and steep gradients, should be expected to hold the largest residual errors, so sampling or capacity should be concentrated there.","Initializing or splitting a training problem so that the network's VC shape matches the target, meaning a small IVC distance, yields faster convergence at equal accuracy, as demonstrated on the three test problems.","The minority-tendency gives a distribution-level description of approximation order that is distinct from low-frequency-first: rare VC levels can converge before common ones.","IVC distance is a genuine metric on functions modulo additive constants, so it can be used as a shape-only similarity measure between a model and a target."],"supporting_citations":[{"why":"Establishes the universal approximation result that justifies the paper's setting: a neural network can in principle approximate the objective functions considered.","marker":"[7]"},{"why":"Provides the neural tangent kernel description of gradient-descent training that this paper positions its local-variation account against.","marker":"[12]"},{"why":"Supplies the direct numerical simulation data for the flow-past-a-cylinder experiments that test both VC-tendency and the preprocessing method.","marker":"[27]"},{"why":"Prior frequency-principle result that neural networks learn low-frequency components first; the minority-tendency is presented as a distribution-level pattern in the same spirit.","marker":"[40]"},{"why":"Overview of the frequency principle that frames why a VC-based ordering of approximation difficulty is worth investigating.","marker":"[39]"}],"fun_headline_variants":["Local variation metric predicts neural net error hotspots","VC metric ties function jumps to approximation errors","Preprocessing by IVC distance speeds up neural net training","Rare VC levels converge first: minority-tendency in NN fitting","Objective value change now quantifies approximation difficulty"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The minority-tendency rests on trusting the estimated ratio of two VC density curves even where the target curve is nearly zero; the paper states that the ratio has no reference significance in that regime, and the supporting tables contain many zero and near-zero entries.","fun_headline_variants_meta":{"raw":{"variants":["Local variation metric predicts neural net error hotspots","VC metric ties function jumps to approximation errors","Preprocessing by IVC distance speeds up neural net training","Rare VC levels converge first: minority-tendency in NN fitting","Objective value change now quantifies approximation difficulty"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1458,"prompt_tokens":965,"completion_tokens":493,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":418}},"tokens_in":581,"tokens_out":493,"duration_ms":4880,"temperature":1.0,"reasoning_tokens":418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:49:22.002948+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun the experiments in Tables 4.1 and 4.2 replacing kernel density estimates with histograms of the same VC samples; if the order in which VC levels converge changes, the minority-tendency is an estimator artifact rather than a property of neural network approximation.","supporting_citations":[{"cited_title":"C YBENKO , Approximation by superpositions of a sigmoidal function , Mathematics of Control, Signals and Sys- tems, 2 (1989), pp","cited_arxiv_id":null,"evidence_quote":"Establishes the universal approximation result that justifies the paper's setting: a neural network can in principle approximate the objective functions considered."},{"cited_title":"J ACOT, F","cited_arxiv_id":null,"evidence_quote":"Provides the neural tangent kernel description of gradient-descent training that this paper positions its local-variation account against."},{"cited_title":"T REBOTICH AND D","cited_arxiv_id":null,"evidence_quote":"Supplies the direct numerical simulation data for the flow-past-a-cylinder experiments that test both VC-tendency and the preprocessing method."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior frequency-principle result that neural networks learn low-frequency components first; the minority-tendency is presented as a distribution-level pattern in the same spirit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Overview of the frequency principle that frames why a VC-based ordering of approximation difficulty is worth investigating."}],"review_version":2}