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arxiv: 2604.08117 · v1 · submitted 2026-04-09 · 💻 cs.NE

Internal noise in deep neural networks: interplay of depth, neuron number, and noise injection step

D.A. Maksimov , V.M. Moskvitin , N. Semenova This is my paper

Pith reviewed 2026-05-10 17:53 UTC · model grok-4.3

classification 💻 cs.NE
keywords deep neural networksinternal noiseGaussian noiseactivation functionnoise injectionfeedforward networksanalog neural networksnoise filtering
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The pith

Activation functions filter internal Gaussian noise more effectively when it is introduced before them rather than after.

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

The paper studies how the placement of internal Gaussian noise relative to the activation function influences accuracy in deep feedforward neural networks. It shows that the activation function nonlinearly filters noise, yielding consistently higher accuracy for pre-activation injection than post-activation, especially for additive noise. Post-activation multiplicative noise degrades performance less than additive noise, while noise in early layers accumulates more harmfully through later weight matrices. Pooling reduces noise effects in both configurations. These results matter for building networks tolerant to internal perturbations, as occur in analog hardware.

Core claim

The activation function acts as an effective nonlinear filter of noise. Networks with noise introduced before the activation function consistently achieve higher accuracy than those with noise applied after it, with additive noise being more effectively suppressed in this case. For noise introduced after the activation function, multiplicative noise is less detrimental than additive noise, and earlier hidden layers contribute more significantly to performance degradation due to cumulative noise amplification governed by the statistical properties of subsequent weight matrices. Pooling-based noise reduction improves performance in both cases.

What carries the argument

The noise injection step relative to the activation function, where the activation serves as a nonlinear filter that suppresses perturbations in the neuron's input channel.

If this is right

  • Accuracy improves when noise is injected before rather than after the activation function.
  • Additive noise is suppressed more effectively by pre-activation filtering than multiplicative noise.
  • Post-activation multiplicative noise causes less accuracy loss than additive noise.
  • Noise introduced in earlier hidden layers degrades final performance more than noise in later layers.
  • Pooling operations consistently mitigate noise effects whether injection occurs before or after activation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hardware implementations of neural networks could prioritize low-noise linear operations before nonlinear activations to improve robustness.
  • The filtering effect may vary with different activation shapes, suggesting targeted tests for ReLU versus sigmoid or other functions.
  • Training routines that inject noise at the pre-activation stage could enhance generalization in noisy real-world deployments.
  • The layer-wise accumulation result points to possible benefits from depth-dependent noise scaling or regularization.

Load-bearing premise

The performance differences stem primarily from the position of noise injection relative to the activation function, independent of the specific network depth, width, activation type, training procedure, or dataset chosen.

What would settle it

Running the same networks with identical hyperparameters but finding no accuracy advantage or even an advantage for post-activation noise injection across multiple depths and widths would falsify the central filtering claim.

Figures

Figures reproduced from arXiv: 2604.08117 by D.A. Maksimov, N. Semenova, V.M. Moskvitin.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of considered deep neural networks [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Impact of additive (solid curves) and multiplicative (dashed [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Impact of additive (panel (a)) and multiplicative (panel (b)) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Impact of additive (left panels) and multiplicative (right pan [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Noise reduction pooling technique with [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Impact of additive (a) and multiplicative (b) noise of different [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Noise reduction pooling technique with [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
read the original abstract

This paper examines the influence of internal Gaussian noise on the performance of deep feedforward neural networks, focusing on the role of the noise injection stage relative to the activation function. Two scenarios are analyzed: noise introduced before and after the activation function, for both additive and multiplicative noise influence. The case of noise before activation function is similar to perturbations in the input channel of neuron, while the noise introduced after activation function is analogous to noise occurring either within the neuron itself or in its output channel. The types of noise and the method of their introduction were inspired by analog neural networks. The results show that the activation function acts as an effective nonlinear filter of noise. Networks with noise introduced before the activation function consistently achieve higher accuracy than those with noise applied after it, with additive noise being more effectively suppressed in this case. For noise introduced after the activation function, multiplicative noise is less detrimental than additive noise, and earlier hidden layers contribute more significantly to performance degradation due to cumulative noise amplification governed by the statistical properties of subsequent weight matrices. The study also demonstrates that pooling-based noise reduction is effective in both cases when noise is introduced before and after the activation function, consistently improving network performance.

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

1 major / 1 minor

Summary. The paper examines the effects of internal Gaussian noise on deep feedforward neural networks, comparing noise injection before versus after the activation function for both additive and multiplicative cases. It claims that the activation function functions as a nonlinear noise filter, with pre-activation injection yielding consistently higher accuracy (additive noise suppressed more effectively in this position). Post-activation, multiplicative noise is less harmful than additive; earlier layers degrade performance more due to cumulative amplification by subsequent weight matrices; and pooling reduces noise effectively in both regimes. The work draws analogies to analog hardware and explores interactions with network depth and neuron count.

Significance. If the empirical ordering holds under controlled conditions, the findings could inform robust network design for noisy environments and analog implementations. The consistent pre- versus post-activation performance gap and the pooling benefit are potentially useful observations for practitioners. As a purely simulation-based study without derivations or parameter-free predictions, its impact depends on the breadth of architectures, datasets, and statistical rigor in the full experiments.

major comments (1)
  1. [Abstract] The central claim that performance differences arise primarily from noise-injection position (rather than from specific choices of depth, width, activation type, or training procedure) requires explicit controls; the abstract does not indicate whether neuron numbers and layer widths were held fixed across the before/after comparisons or whether statistical tests (e.g., error bars, significance levels) confirm the reported ordering.
minor comments (1)
  1. [Abstract] The description of 'cumulative noise amplification governed by the statistical properties of subsequent weight matrices' would benefit from a brief quantitative illustration (e.g., variance propagation formula or reference to a specific figure) to clarify the mechanism.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for highlighting the need for greater clarity in the abstract. We address the major comment point by point below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Abstract] The central claim that performance differences arise primarily from noise-injection position (rather than from specific choices of depth, width, activation type, or training procedure) requires explicit controls; the abstract does not indicate whether neuron numbers and layer widths were held fixed across the before/after comparisons or whether statistical tests (e.g., error bars, significance levels) confirm the reported ordering.

    Authors: We agree that the abstract should explicitly state the experimental controls. In all before/after comparisons, network depth, layer widths, neuron counts per layer, activation functions, and training procedures (including optimizer, learning rate, and epochs) were held identical; only the noise injection position (pre- versus post-activation) and noise type (additive versus multiplicative) were varied. This isolates the effect of injection stage. Results are based on multiple independent runs with different random seeds; mean accuracies and standard deviations are reported in all figures and tables, with error bars shown to indicate variability. We will revise the abstract to include a concise statement confirming these fixed controls and the presence of statistical measures supporting the observed ordering. This change will be reflected in the next manuscript version. revision: yes

Circularity Check

0 steps flagged

Empirical simulation study with no derivation chain

full rationale

The manuscript reports direct numerical experiments on feedforward networks with Gaussian noise injected before versus after the activation function, for additive and multiplicative cases. All performance comparisons, accuracy orderings, and observations about noise filtering and pooling are presented as outcomes of those simulations across varying depths, widths, and layers. No equations, ansatzes, fitted parameters renamed as predictions, uniqueness theorems, or self-citations appear as load-bearing steps in the reported chain; the central claim follows immediately from the experimental design without reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The work implicitly assumes standard feedforward architectures, Gaussian noise statistics, and common activation functions.

pith-pipeline@v0.9.0 · 5520 in / 1015 out tokens · 22627 ms · 2026-05-10T17:53:21.452248+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Internal noise in deep neural networks: interplay of depth, neuron number, and noise injection step

    In optical implementations of neural networks, inter-neuronal connections rely on various physi- cal mechanisms, including holography 9, diffraction 10,11, in- tegrated Mach–Zehnder modulator networks 12, wavelength- division multiplexing13, and optical interconnects fabricated using 3D printing technologies 14–16. In addition, particular attention should...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    Impact of additive (solid curves) and multiplicative (dashed curves) noise of varying intensities on the accuracy of trained deep neural networks with one hidden layer (a) and two hidden layers (b). A. Network’s depth To gain a more comprehensive understanding of the ro- bustness of deep neural networks to noise, we examine the effects of noise introduced...

    work page 1947

  3. [3]

    The noise influences were intro- duced separately into the 2nd (blue curves), 3rd (orange curves) and 4th layer (green curves)

    Impact of additive (panel (a)) and multiplicative (panel (b)) noise of different intensities on the accuracy of trained deep neural networks with 5 layers (3 hidden). The noise influences were intro- duced separately into the 2nd (blue curves), 3rd (orange curves) and 4th layer (green curves). TABLE II. Statistics of connection matrices of ANN with 5 laye...

    work page 1917

  4. [4]

    Finding a roadmap to achieve large neuromorphic hardware systems

    Noise reduction pooling technique withm=3 for networks with additive (a) and multiplicative (b) noise of different intensities beforeactivation function. The noise influences were introduced sep- arately into the 2nd (blue curves), 3rd (orange curves) and 4th layer (green curves) of trained networks. Solid lines were obtained for net- works without noise ...

    work page doi:10.1140/epjs/s11734-025-01697-7 2013