A comparative analysis of a neural network with calculated weights and a neural network with random generation of weights based on the training dataset size

Polad Geidarov

REVIEW 3 major objections 5 minor 7 references

A comparative analysis of a neural network with calculated weights and a neural network with random generation of weights based on the training dataset size

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A multilayer perceptron whose first-layer weights are computed analytically from 30 selected samples reaches 63% MNIST test accuracy with no training, and after three backpropagation epochs it retrains faster and more accurately than a…

desk verdict The central comparison is compromised by test-set leakage and an error-only timing protocol, but the closed-form initialization idea is legitimate and worth a rigorous re-run. read the letter →

arxiv 2505.23876 v1 pith:WK4A2D4R submitted 2025-05-29 cs.LG cs.AI

Polad Geidarov This is my paper

classification cs.LGcs.AI MSC 68T0768T10

keywords multilayerperceptronmetricrecognitionmethodscalculatedweightsMNISTnearestneighbormethodbackpropagationtrainingdatasetsizerandomweightinitialization

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

The paper claims that a multilayer perceptron built to implement the nearest-neighbor rule can have its first-layer weights calculated directly from a handful of samples rather than learned. Using 30 arbitrarily chosen MNIST digits, the construction yields 63% test accuracy in 0.5469 seconds with no training at all. The same network then undergoes three backpropagation epochs, and at every training-set size tried (60,000, 40,000, and 20,000 images) it retrains in 34–38% less time and reaches higher test accuracy than the identical architecture started from random weights. The author reads this as evidence that analytically calculated weights make an MLP faster to train and much less sensitive to shrinking training data. The payoff would be neural networks that gain a working recognition ability from a tiny set of examples, like biological learning.

What carries the argument

The carrying object is a three-layer perceptron implementing the nearest-neighbor metric recognition method. Its first layer contains $N(N-1)$ neurons, one for each ordered pair of the $N=30$ standards; each first-layer neuron holds a weight table computed analytically through the difference of squared Euclidean distances (Eq. 1), with a zero threshold. The second layer has one neuron per standard and fires only when all $N-1$ pairwise comparisons agree, which encodes 'this standard is the nearest neighbor'; the third layer sums outputs per digit class. Inputs are encoded as a 28×56 binary matrix (the image and its mirror-opposite), and the sigmoid activation is used for backpropagation. Because the first-layer weights come from formulas rather than random initialization, the network is already a functioning 30-exemplar nearest-neighbor classifier before any training, and backpropagation only polishes it.

What would settle it

Compare the two arms under a training protocol that backpropagates through every training image in every epoch, rather than only through the P misclassified images of Eq. (14), while recording total update counts. If the random-initialized network then trains in no more time than the pre-calculated network, the claimed speed advantage is an artifact of the error-selected update rule. A second check: run a plain nearest-neighbor classifier on the same 30 selected standards and the same test set; if it matches or exceeds the 63% no-training accuracy, the analytic network's initial performance is just the metric rule it encodes.

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Extended reading notes

Core claim

The central claim is that an MLP with pre-calculated weights is faster to retrain and more stable under training-set reduction than the same MLP with random initialization. On the MNIST test set, after three epochs the pre-calculated network reaches 94.49% accuracy when trained on 60,000 images, 91.94% on 40,000, and 90.78% on 20,000, while the randomly initialized network reaches 92.56%, 85.33%, and 80.25%. Total retraining time is 329 versus 499 minutes (34% less), 293.57 versus 471.8 minutes (38% less), and 196.2 versus 298.13 minutes (34% less). A 20,000-image training run with calculated weights finishes in about 196 minutes and lands close to the accuracy of a 60,000-image random-init run that takes 499 minutes. The paper's conclusion is that the earlier single comparison was not accidental and that the advantage is structural, not a quirk of one dataset size.

Load-bearing premise

The load-bearing premise is that retraining speed should be measured only on the images the network misclassifies; the pre-calculated network starts with fewer mistakes, so it automatically performs fewer updates, and that alone drives much of the reported time saving.

Editorial extensions

If this is right

A working digit classifier can be produced with no training pass: 30 samples and 0.5469 seconds of weight calculation yield 63% accuracy on the 10,000-image MNIST test set.
Retraining a pre-calculated network is faster at every dataset size tested: 34% less time at 60k and 20k images, and 38% less at 40k images.
Accuracy degrades more gracefully when training data shrink: the pre-calculated network stays above 90% at 20k training images, while the random-init network falls to 80%.
A 20k pre-calculated run approximates the final test accuracy of a 60k random-init run while using about 60% less training time (196 versus 499 minutes).
Switching the underlying metric scheme to k-nearest neighbors should raise the no-training accuracy above 63% and improve all subsequent results, according to the paper.

Reading between the lines

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

If the comparison were rerun with every training image updated each epoch, that would tell whether the speed gap is tied to the mistake-selected update rule or to the initialization itself.
Because the no-training network is exactly a nearest-neighbor rule on 30 exemplars, an instructive baseline is plain k-nearest-neighbor on the same 30 standards with the same dataset reductions.
The reported runs are single and unseeded; repeating them over several random initializations would show the size of the run-to-run variation implied by the per-class swings in the test tables.
If the analytic-initialization idea transfers to deeper networks, the relevant comparison would broaden to other cheap initialization schemes, not only random weights.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing you should know: the main comparison is not clean. The 30 exemplars used to compute the network's weights are taken from the MNIST test set (Section 2, Fig. 5), and the same test set is then used for evaluation. The calculated-weight arm starts with labeled information from the evaluation distribution; the random-init arm starts with nothing. That alone undermines the headline claim that calculated weights are much more robust to dataset reduction.

Second: the speed advantage is largely mechanical. Eq. 14 defines training error over only the P misclassified images. The pre-calculated arm begins as a working 30-exemplar classifier, so it misclassifies fewer images each epoch and performs fewer backprop updates. Most of the 34-38% time saving follows from that protocol choice, not from a fundamentally faster learning method.

Credit where due: the architecture is transparent. The weight calculation is closed-form, the equations are concrete, and the per-class tables are unusually detailed. The direction of the results is consistent across 60k/40k/20k, and the underlying mechanism—a sensible initialization helps gradient fine-tuning—is credible. What is new relative to the author's prior paper [7] is the dataset-size sweep and the explicit robustness comparison.

Soft spots, in order of severity: (1) test-set leakage as above; (2) single unseeded run per condition, with per-class accuracies swinging tens of points between epochs—no error bars, so the magnitude of the gap is not reliably estimated; (3) no k-NN baseline, even though the architecture is exactly a neural re-implementation of nearest-neighbor; (4) speculative claims about k-NN variants that were not tested.

Also worth noting: this is a self-archived version of a 2022 journal publication, so the preprint's novelty is limited to the new dataset sizes. The citation pattern is mostly the author's own prior work, which is appropriate given the method's provenance.

Who this is for: someone thinking about analytic initialization of MLPs, or about metric-based neural architectures, might mine the equations. As experimental evidence, the current version doesn't support the strong claims.

Recommendation: send it to a serious referee, but expect major revisions. The fix is straightforward: re-run with standards drawn from the training set, report seeds and variances, count updates over all samples per epoch, and include a k-NN baseline. If the author does that, the result would be worth knowing. As is, it's a suggestive but flawed data point.

Referee Report

3 major / 5 minor

Summary. The paper compares a multilayer perceptron whose first-layer weights are computed analytically from a set of 30 reference images (Eq. 1, Fig. 5) against the same architecture with random initialization, on MNIST training sets of 60,000, 40,000, and 20,000 images. After three backpropagation epochs, it reports that the pre-calculated-weight network trains faster (34-38% less time) and reaches higher test accuracy at every training-set size (e.g., 90.78% vs. 80.25% at 20k), concluding that the calculated-weight network is much faster and more robust to training-set reduction. The paper includes detailed per-class accuracy tables and time measurements for all conditions.

Significance. If the reported comparisons were valid, the paper would offer a practically interesting initialization scheme for MLPs: a closed-form, training-free construction (Eq. 1) that yields 63% MNIST test accuracy from only 30 prototypes, followed by fast fine-tuning. The architecture is clearly specified, the tables are detailed, and the no-training result is a measurable baseline. However, the experimental protocol contains a serious data-leakage flaw: the 30 standards are selected from the MNIST test set, and the same test set is used for evaluation. This confounds the initial accuracy, the speed advantage (which depends on the initial error count), and the robustness claims. The paper's central empirical contribution therefore cannot be accepted in its current form; the underlying idea may be salvageable with a corrected experimental setup, but the present evidence does not support the abstract's claims.

major comments (3)

[Section 2, Eq. (1), Fig. 5, Table 1] The 30 standards used to compute the first-layer weights via Eq. (1) are explicitly said to be 'an arbitrarily selected set of 30 digits from the MNIST test dataset' (Fig. 5). All reported evaluations, including the no-training 63% accuracy in Table 1 and the fine-tuning results in Tables 3, 5-7, and 9-11, are computed on that same 10,000-image test set. The calculated-weight arm therefore receives labeled examples from the evaluation distribution before training, while the random-init arm does not. This is transductive leakage, not merely memorization of 30 images: the standards act as prototypes, so the entire comparison is biased. The experiment must be rerun with standards drawn from the training set (or a separate validation set) and the test set used only once; without this change, the headline claims of speed and robustness advantages are not established.
[Eq. (14) and Tables 2, 4, 8] Training time is measured as the sum of times spent only on the P misclassified images (Eq. 14). Since the calculated-weight arm starts from a working classifier, it has fewer errors at epoch 1 (e.g., Table 2: 16,068 vs. 24,630 errors on 60k), so it performs fewer backpropagation updates by construction. This mechanically produces a large part of the reported 34-38% speedup. To support the claim that the pre-calculation makes training 'much faster,' the comparison should be repeated with an equivalent update protocol (e.g., all training patterns per epoch) and should include the 0.5469 s initialization cost of the calculated-weight arm. As reported, the speed comparison is not a fair apples-to-apples measure of learning efficiency.
[Tables 5-11] Each experimental condition is a single unseeded run, and the per-class test accuracies vary wildly across epochs; for example, Table 10, class 4 shows 60%, 57%, and 36% across the three epochs, and Table 5, class 5 drops from 95% to 88% between epochs 2 and 3. With no repeated runs, no seeds, and no error bars, the reported robustness gaps (e.g., 90.78% vs. 80.25% at 20k) cannot be distinguished from run-to-run variation. The conclusion that the calculated-weight network is 'much more robust' to training-set reduction (Conclusion item 3) requires repeated experiments with multiple initializations and a statistical summary.

minor comments (5)

[Eq. (13)] The sigmoid activation function is not rendered correctly in the text; please provide a clear mathematical expression.
[Tables 2, 4, 8] The column labeled 'Serr' is not defined in the captions or in Section 2; please define it explicitly and state its units.
[Section 2, thresholds] The statement that threshold values B are represented as weights w0 = -B is introduced informally; please clarify the relation between Eqs. (6), (8), (12) and the sigmoid activation.
[Conclusion, pspeedup] The 60.72% speedup reported in the conclusion compares 196 minutes (calculated weights, 20k) with 499 minutes (random init, 60k), which conflates dataset size and initialization; relabel or remove this comparison to avoid misleading readers.
[References] Several references are given only in Cyrillic without English translations; please add translated titles or at least a note for each non-English reference.

Circularity Check

2 steps flagged · score 6.0 of 10

The calculated-weight advantage is inflated by test-set-derived standards and error-proportional timing: the 63% no-training figure is in-sample and the 34-38% speedup is arithmetically tied to the error counts being compared.

fitted input called prediction [Section 2, Eq. (1), Fig. 5, Table 1]
"An arbitrarily selected set of 30 digits from the MNIST test dataset is used as standards (Fig. 5). ... Formula (1) is used as an expression of the proximity measure, and the values of all weights are calculated by this formula. ... Table 1 shows the overall results and the individual results for each recognized digit on the MNIST test dataset (10,000 images) after the analytical calculation of the weights. The final result without training the neural network was already 63%."

The first-layer weights are constructed from 30 labeled exemplars drawn from the same MNIST test set on which Table 1 reports 63% accuracy. A classifier that stores test-set prototypes as its weights is evaluated on those very images, so the no-training accuracy is an in-sample measure, not an out-of-sample prediction. The random-initialization arm receives no such test-set information. Because the calculated arm's initial accuracy advantage is what reduces P in Eq. (14), the later speed and robustness comparisons inherit this leakage. The result is not equivalent to Eq. (1) alone, but the headline 'already 63% on the test set' is a direct consequence of constructing the classifier from a subset of that test set.
self definitional [Section 2, Eq. (14); Tables 2, 4, 8; Fig. 9]
"The learning error ... was calculated for each epoch using the formula: ... where ... P is the number of incorrectly identified images from the MNIST training set for which the weights were corrected by the backpropagation algorithm. ... the process of retraining an analytically calculated neural network takes 34% less time, pspeedup = (499-329)*100/499 ≈ 34%."

Per-epoch training time is defined as the time spent correcting the P misclassified images, and P is exactly the complement of the recognition accuracy reported in the same tables. A better-initialized network therefore has fewer P by construction, so its measured training time is smaller for that reason alone. In the 60k comparison, the random arm has 1935 errors vs 1199 for the calculated arm, matching the 256 vs 159 minute times; the speedup is an arithmetic consequence of Eq. (14), not an independent training-speed discovery. The same definition applies to all three dataset-size comparisons and to the 60.72% cross-size speedup, so the central 'much faster' claim reduces to the initial accuracy gap.

full rationale

The paper is experimental rather than a formal derivation, and the analytic weight formula Eq. (1) is itself a legitimate closed-form nearest-neighbor construction: given any fixed set of standards, the weights are deterministic and no parameter is fitted to the test labels. In that narrow sense the method is not circular. However, the two load-bearing empirical claims are not clean. First, the standards used in Eq. (1) are taken from the MNIST test set, so the 63% no-training result and all subsequent comparisons are contaminated by transductive use of the evaluation set; the calculated arm begins with information the random arm does not have. Second, the training-time metric in Eq. (14) counts only P incorrectly identified images, so the 34-38% time advantage is mechanically determined by the accuracy advantage, not by faster gradient steps or cheaper updates. The self-citations to the author's prior papers [1,2,3,5,6,7] are frequent but not uniquely load-bearing: the formulas and experiments are reproduced in this paper, and no uniqueness theorem is invoked to forbid alternatives. The kNN equivalence is openly acknowledged, so that is not a hidden circular renaming. Overall, the headline speed and robustness advantages are partial artifacts of the evaluation protocol, warranting a score of 6 rather than a clean 0-2.

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

The central claim adds no free parameters fitted to test labels, which is the paper's genuine strength; the weights are closed-form functions of 30 exemplars. The hand-chosen quantities (threshold 150, N=30, learning rates, init range) and the error-only update policy (Eq. 14) are the parameters the experimental outcome actually depends on, and the last of these explains most of the reported speedup mechanically. The axioms are domain assumptions about the equivalence between the constructed network and the nearest-neighbor rule, plus a fair-timing assumption that the paper does not defend. No invented entities.

free parameters (5)

Pixel binarization threshold = 150
Pixels with values above 150 are active in the first half of the 28x56 input and below 150 in the mirrored half (Section 2); this hand-chosen threshold determines all inputs and therefore all results.
Number and identity of reference samples N = 30
The 30 standards are 'arbitrarily selected' from the MNIST test set (Fig. 5); the untrained 63% baseline and the advantage of the calculated-weight arm depend on this hand-picked set.
Backprop learning rates = 0.1 and 0.02
Two epochs at 0.1 then one at 0.02, reused from [7] (Section 2); applied identically to both arms, so it does not by itself bias the comparison.
Random initialization range = [-0.5, 0.5]
The baseline network's weights are drawn uniformly in this range (Tables 2, 4, 8); the range directly sets how far the random arm starts from a working classifier and shapes the early-epoch gaps.
Error-only update policy P = misclassified images only
Eq. (14) limits backprop corrections to the P wrongly recognized images; since training time scales with P, the pre-calculated arm's lower error count mechanically produces most of the reported 34-38% speedup.

assumptions (5)

standard math Formula (1) correctly computes the signed squared-Euclidean-distance comparison between an input cell and two reference cells.
Invoked in Section 1, Eq. (1) as the proximity measure for all first-layer weights; the equation is asserted and is garbled in the preprint's rendering.
domain assumption The threshold activations (3), (6), (8) with thresholds 0, N-1, and the third-layer rule implement exactly the nearest-neighbor decision rule.
Section 1, Eqs. (2)-(8); the equivalence is stated, not proven, and it is the basis for claiming the network is a functional classifier immediately after weight calculation.
domain assumption Replacing threshold activations with the sigmoid (13) during backprop preserves the nearest-neighbor behavior well enough for the retrained network to remain a metric recognizer.
Section 2, Eq. (13); the paper does not analyze the deviation between the two activation functions during training.
ad hoc to paper Training time spent on the P misclassified images only (Eq. 14) is a fair basis for comparing training speed between the two arms.
Section 2, Eq. (14); the pre-calculated arm starts with fewer errors, so this protocol bakes most of the speedup into the measurement.
domain assumption MNIST is a valid benchmark and drawing the 30 embedded standards from the test set does not bias the comparison.
Section 2, Fig. 5; the standards are taken from the test set rather than the training set, and the evaluation set includes those exact 30 images.

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Cite this review

Pith. "Pith review of A comparative analysis of a neural network with calculated weights and a neural network with random generation of weights based on the training dataset size." pith.science (2026). https://pith.science/paper/WK4A2D4R

@misc{pith2026250523876,
  author       = {Pith},
  title        = {Pith review of: A comparative analysis of a neural network with calculated weights and a neural network with random generation of weights based on the training dataset size},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WK4A2D4R}},
  note         = {Machine review of arXiv:2505.23876}
}

read the original abstract

The paper discusses the capabilities of multilayer perceptron neural networks implementing metric recognition methods, for which the values of the weights are calculated analytically by formulas. Comparative experiments in training a neural network with pre-calculated weights and with random initialization of weights on different sizes of the MNIST training dataset are carried out. The results of the experiments show that a multilayer perceptron with pre-calculated weights can be trained much faster and is much more robust to the reduction of the training dataset.

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Works this paper leans on

7 extracted references · 7 canonical work pages

[7]

Geidarov P. Sh. Comparative Analys is of the Results of Training the neural network with Calculated Weights and with Random Generation of the Weights // Automation and Remote Control, 2020, V. 81, №. 7, C. 1211-1229

work page 2020
[1]

Neural networks on the basis of the sample method // Automatic Control And Calculater Sciences

Geidarov, P.Sh. Neural networks on the basis of the sample method // Automatic Control And Calculater Sciences. 2009. V. 43. № 4. pp. 203 -210

work page 2009
[2]

Clearly Defined Neural Networks Architectur e // Optical Memory & Neural Networks, 2015

Geidarov, P.Sh. Clearly Defined Neural Networks Architectur e // Optical Memory & Neural Networks, 2015. V. 24. I. 3. pp. 209 -219

work page 2015
[3]

Алгоритм реализации метода ближайшего соседа в много cлойном персептроне // Труды СПИИРАН

Гейдаров, П.Ш. Алгоритм реализации метода ближайшего соседа в много cлойном персептроне // Труды СПИИРАН. 2017. Т. 51. C. 123 -151. Gejdarov, P.Sh. Algoritm realizacii metoda blizhajsh ego soseda v mnogoclojnom perseptrone [Algorithm for implementing the nearest neighbor method in a multilayer perceptron ]. Trudy SPIIRAN [ SPIIRAS Proceedings] 2017. V. 51...

work page 2017
[4]

Техническая диагностика, М.:Машиностроение

Биргер И.А. Техническая диагностика, М.:Машиностроение. 1978. Birger I.A. Tehnicheskaja diagnostika. [Technical diagnostics]. M.:Mashinostroenie. 1978

work page 1978
[5]

Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания

Гейдаров П.Ш. Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания. Часть 1 // Информационно-Управляющие Системы, Санкт-Петербург. 2020. №2. C. 20-30. Gejdarov P.Sh. Algoritm vychislenija znachenij vesov sinapsov pervogo sloja nejronnoj seti na osnove metricheskih metodov raspoznavanija. Chast...

work page 2020
[6]

Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания

Гейдаров П.Ш. Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания. Часть 2 // Информационно-Управляющие Системы, Санкт-Петербург. 2020. №3. C. 25-38. Gejdarov P.Sh. Algoritm vychislenija znachenij vesov sinapsov pervogo sloja nejronnoj seti na osnove metricheskih metodov raspoznavanija. Chast...

work page 2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

Geidarov P. Sh. Comparative Analys is of the Results of Training the neural network with Calculated Weights and with Random Generation of the Weights // Automation and Remote Control, 2020, V. 81, №. 7, C. 1211-1229

work page 2020

[2] [1]

Neural networks on the basis of the sample method // Automatic Control And Calculater Sciences

Geidarov, P.Sh. Neural networks on the basis of the sample method // Automatic Control And Calculater Sciences. 2009. V. 43. № 4. pp. 203 -210

work page 2009

[3] [2]

Clearly Defined Neural Networks Architectur e // Optical Memory & Neural Networks, 2015

Geidarov, P.Sh. Clearly Defined Neural Networks Architectur e // Optical Memory & Neural Networks, 2015. V. 24. I. 3. pp. 209 -219

work page 2015

[4] [3]

Алгоритм реализации метода ближайшего соседа в много cлойном персептроне // Труды СПИИРАН

Гейдаров, П.Ш. Алгоритм реализации метода ближайшего соседа в много cлойном персептроне // Труды СПИИРАН. 2017. Т. 51. C. 123 -151. Gejdarov, P.Sh. Algoritm realizacii metoda blizhajsh ego soseda v mnogoclojnom perseptrone [Algorithm for implementing the nearest neighbor method in a multilayer perceptron ]. Trudy SPIIRAN [ SPIIRAS Proceedings] 2017. V. 51...

work page 2017

[5] [4]

Техническая диагностика, М.:Машиностроение

Биргер И.А. Техническая диагностика, М.:Машиностроение. 1978. Birger I.A. Tehnicheskaja diagnostika. [Technical diagnostics]. M.:Mashinostroenie. 1978

work page 1978

[6] [5]

Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания

Гейдаров П.Ш. Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания. Часть 1 // Информационно-Управляющие Системы, Санкт-Петербург. 2020. №2. C. 20-30. Gejdarov P.Sh. Algoritm vychislenija znachenij vesov sinapsov pervogo sloja nejronnoj seti na osnove metricheskih metodov raspoznavanija. Chast...

work page 2020

[7] [6]

Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания

Гейдаров П.Ш. Алгоритм вычисления значений весов синапсов первого слоя нейронной сети на основе метрических методов распознавания. Часть 2 // Информационно-Управляющие Системы, Санкт-Петербург. 2020. №3. C. 25-38. Gejdarov P.Sh. Algoritm vychislenija znachenij vesov sinapsov pervogo sloja nejronnoj seti na osnove metricheskih metodov raspoznavanija. Chast...

work page 2020