Zeta Ae Math Journal Volume 10 No.
May 2025, pp.
E-ISSN: 2579-5864 P-ISSN: 2459-9948
D https://doi.
org/10.
31102/zeta.
A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset Muhammad Thahiruddin1.
Siti Khotijah1.
Moh.
Fajar2.
Adib El Farras1 Prodi Matematika.
Fakultas MIPA.
Universitas Annuqayah.
Indonesia Prodi Teknologi Informasi.
Fakultas Teknik.
Universitas Annuqayah.
Indonesia * Corresponding AuthorAos Email: muhammad.
thahiruddin@ua.
ABSTRACT
This study systematically evaluates the machine learning robustness of Multi-Layer Perceptrons (MLP.
and Logistic Regression (LR) models against data perturbations using the MNIST handwritten digit dataset.
Despite their foundational roles in machine learning, the comparative resilience of MLPs and LR to diverse perturbationsAisuch as noise, geometric distortions, and adversarial attacksAiremains underexplored.
This gap is critical, as real-world applications .
, healthcare, autonomous system.
often operate with imperfect data, yet practitioners lack actionable insights into model selection under such conditions.
Existing studies predominantly focus on complex deep networks or isolated perturbation types, overlooking simpler models like LR and holistic evaluations.
To address this, we test three perturbation categories: Gaussian noise .
ua = 1 to 1.
, salt-and-pepper noise .
cy = 0.
1 to 0.
, rotational distortions .
Oo to 30Oo ), and adversarial attacks (FGSM with yun = 0.
05 to 0.
Both models were trained on 60,000 MNIST samples and tested on 10,000 perturbed images.
Results demonstrate that MLPs exhibit superior robustness under moderate noise and rotations, achieving baseline accuracies of 97.
07% .
LR's 92.
63%).
For Gaussian noise .
ua = 0.
MLP
35% accuracy compared to LR's 23.
However, adversarial attacks (FGSM, yun = 0.
reduced MLP accuracy to 0.
20%, revealing critical vulnerabilities.
Statistical analysis .
aired t-tests, .
cy < .
) confirmed significant performance differences across perturbation levels.
A linear regression .
cI2 = .
further quantified MLP's predictable accuracy decline with Gaussian noise intensity.
These findings underscore the necessity of robustness-aware model selection in noise-prone environments and highlight urgent needs for adversarial defense mechanisms in MLPs.
Practitioners are advised to prioritize MLPs for tasks with moderate distortions, while future work should integrate robustness enhancements like adversarial Keyword: Machine Learning Robustness.
Data Perturbations.
Multi-Layer Perceptrons.
Logistic Regression.
MNIST Dataset Article info:
Submitted: March 30, 2025 Accepted: May 27, 2025 How to cite this article:
Thahiruddin.
Khotijah.
Fajar.
, & Farras.
A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset.
Zeta - Math Journal, 10.
, 39-50.
https://doi.
org/10.
31102/zeta.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-ShareAlike 4.
0 International License.
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Introduction Machine learning models have become indispensable tools for classification tasks in domains ranging from healthcare to autonomous systems.
Among these models.
Multi-Layer Perceptrons (MLP.
and Logistic Regression (LR) remain foundational: MLPs excel at capturing non-linear relationships through hierarchical representations (I.
Goodfellow et al.
, 2.
, while LR provides a computationally efficient baseline for linear classification (Kuhn & Johnson, 2.
However, their robustness to data perturbations Ae such as noise, adversarial attacks, or geometric distortions - remains understudied, despite its critical importance in realworld applications where input data is rarely pristine Recent advances in adversarial machine learning have revealed vulnerabilities in neural networks, where imperceptible perturbations can drastically alter model predictions (Goldblum et al.
, 2.
For instance, adversarial attacks like the Fast Gradient Sign Method (FGSM) exploit gradient information to generate malicious inputs, undermining model reliability (Serban et al.
, 2.
Concurrently, studies on Gaussian and salt-and-pepper noise perturbations highlight how even random distortions degrade model accuracy (Ayachi.
Despite these findings, there is limited comparative research on how traditional models like the MNIST Existing literature primarily focuses on robustness evaluation of deep neural networks (DNN.
or adversarial training techniques.
For example.
Serban et al.
proposed meta-learning strategies to transfer adversarial knowledge between models (Serban et al.
, 2.
, while Kim et al.
analyzed robust generalization through large-scale empirical studies (Kim et al.
, 2.
However, these works often overlook simpler models like LR or shallow MLPs, which are still widely used in resource-constrained applications.
Furthermore, most studies evaluate robustness against isolated perturbation types .
, adversarial noise or geometric transformation.
rather than a comprehensive suite of disturbances (Madry et al.
, 2.
This creates a critical gap: practitioners lack actionable insights into the comparative stability of MLPs and LR across diverse perturbation scenarios.
This study addresses these gaps by systematically evaluating the stability of MLPs and LR under three perturbation categories:
Noise-based perturbations (Gaussian, salt-and-peppe.
Geometric distortions .
Adversarial attacks (FGSM).
The MNIST dataset is selected as the benchmark due to its simplicity and well-understood feature space, which enables controlled isolation of perturbation effects (LeCun et al.
, 1.
Its widespread adoption in robustness studies, including recent works (Hendrycks & Dietterich, 2.
Further validates its utility for systematic comparisons of model behavior under distortions.
Using the MNIST dataset as a benchmark, we aim to:
a Quantify the performance degradation of MLPs and LR under increasing perturbation magnitudes a Analyze the correlation between perturbation intensity and accuracy decline using statistical methods a Identify model-specific failure patterns through visual and quantitative diagnostics.
Our findings will provide practitioners with guidelines for selecting robust models in noise-prone environments and inform future research on perturbation-resistant architectures.
Research Method This study employs an experimental approach to evaluate the robustness of Multi-Layer Perceptron (MLP) and logistic Regression (LR) models when subjected to various perturbations on the MNIST dataset.
The methodology is divided into key stages:
Data Preparation The MNIST dataset, containing 28x28 grayscale images of handwritten digits, is used as the benchmark.
Each image is normalized to the range .
using the following formula:
ycu Oe ycuycoycnycu ycuA = ycuycoycaycu Oe ycuycoycnycu Where ycu represent the original pixel intensity ycuycoycnycu = 0 and ycuycoycaycu = 255.
The dataset is then split into a training set .
,000 image.
and a test set .
,000 image.
(LeCun et al.
, 1.
Muhammad Thahiruddin, dkk A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset Model Development Two classification models are developed:
a Multi-Layer Perceptron (MLP) The MLP architecture is adapted from prior MNIST benchmark studies (LeCun et al.
, 1.
, with two hidden layers .
and 64 neuron.
selected to balance computational efficiency and nonlinear representation capacity (I.
Goodfellow et al.
, 2.
The ReLU activation function is employed to mitigate vanishing gradients, defined as:
ReLU = .
, y.
The output layer employs the SoftMax function to generate a probability distribution over 10 classes:
yce ycycn yua.
ycn = 10 ycyc Ocyc=1 yce Where ycycn is the logit corresponding to class ycn.
The model is trained using the Adam optimizer with a learning rate of 0.
0001 (Kingma & Ba, 2.
and optimized with the sparse categorial cross entropy loss function:
ycA Ee = Oe Oc log.
ycycn ) ycA ycn=1 Where ycA is number of smaples and ycycn is the true label of ycn-th sample (I.
Goodfellow et al.
, 2.
a Logistic Regression (LR) There LR model is implemented in its multinomial form to handle the multi-class problem.
The probability that an input vector ycu belongs to class yco is given by:
ycN yce yayco ya ycayco ycE.
c = yc.
= yaycoycN ya ycayco Oc10 yc=1 yce Where ycyco and ycayco are the weight vector and bias term for class yco, respectively.
The model is optimimized using the LBFGS solver with maximum iteration count set to ensure convergence (Szegedy et al.
, 2.
Perturbations Generation To simulate real-world distortions, three categories of perturbations are applied exclusively to the test set.
Noise Based perturbation (Gaussian and Salt-and-Peppe.
Geometric Distortions (Rotation.
(Figure .
and Adversarial Attacks (FGSM):
a Noise-Based Perturbations (Gaussian Nois.
For each test image, additive Gaussian noise is generated from a normal distribution and added to the image using following formula:
ycunoise = ycu ye.
, yua 2 ) Where yua 2 is varied .
, 0.
1, 0.
3, 0.
5, 0.
7, and 1.
to simulate different noise intensities (Fawzi et al.
, 2.
a Noise-Based Perturbations (Salt-and-Peppe.
In this case, each pixel is randomly replaced by either 0 or 1 with a probability ycy .
anging from 0.
1 to Salt-and-pepper noise parameters .
cy = 0.
1 Oe 0.
are selected to simulate realistic sensor corruption levels observed in digitized documents (Fawzi et al.
, 2.
The perturbed pixel ycuycn A is defined as:
ycy with probability ycy ycuycnA = 1, with probability with probability 1 Oe ycy { ycuycn .
This type of noise is used to mimic random pixel corruptions (Fawzi et al.
, 2.
a Geometric Distortions (Rotation.
For rotational distortions, angles yuE = 5A Oe 30A are chosen based on MNIST augmentation standards (Simard et al.
, 2.
With nearest-neighbor interpolation preserving integer pixel values.
The rotation transformation is given by:
cos yuE Oe sin yuE ycu
[ A] = [
][ ]
sin yuE cos yuE yc Where .
cu, y.
cu , yc ) represent the original and rotated pixel coordinates, respectively (Hendrycks & Dietterich, 2.
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a Adversarial Attacks (FGSM) The Fast Gradient Sign Method (FGSM) is used to generate adversarial examples.
For a give input ycu and its corresponding true label yc, the adversarial example ycuadv is computed as:
ycuadv = ycu yun UI sign.
uAyeo yc.
uE, ycu, y.
) Where yc.
uE, ycu, y.
is the loss function with respect to model parameters yuE, and yun controls the perturbation magnitude (I.
Goodfellow et al.
, 2.
This method exploits gradient information to produce small, imperceptible changes that can significantly alter the modelAos prediction.
However, due to the necessity of gradient information for FGSM Ae which is readily available in the differentiable MLP but not in the scikitlearn LR-the FGSM attacks is applied only to the MLP.
Future work may explore black-box attacks .
ZOO) (Chen et al.
, 2.
for non-differentiable models like LR.
Figure 1.
Perturbation Examples Performance Evaluation The performance of each model is evaluated by computing the classification accuracy on the perturbed test sets.
The accuracy is calculated as:
Number of Correct Predictions Accuracy = ( ) y 100% Total Number of Samples In addition.
For gaussian noise-based perturbations, the intensity of the perturbation is quantified using the L2 AnycuAn2 = oc.
cunoise,ycn Oe ycuycn ) ycn=1 Which provides a measure of the overall noise energy added to the image Muhammad Thahiruddin, dkk A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset Statistical Analysis To rigorously quantify the performance differences between MLP and LR under perturbations, two statistical approaches were employed: Paired t-test and Linear Regression Analysis (Demsar, 2.
The paired t-test isolates the effect of model architecture (MLP vs.
LR) by controlling for dataset-specific variability, ensuring observed differences are not due to random chance.
The linear regression provides actionable insights for real-world deployment: by measuring AnyaAn2 in an application, practitioners can estimate MLPAos expected accuracy without retesting.
Together, these tools offer both comparative .
hich model is better?) and predictive .
ow much will performance drop?) insights into robustness.
Paired t-test This test was conducted for each perturbation level .
Gaussian noise at .
ua = 0.
1, 0.
3, .
, 1.
) to determine whether the accuracy differences between MLP and LR were statistically significant.
The procedure is as follows:
A Data Pairing: For each perturbation intensity, the accuracy of MLP and LR was recorded on the same test set, creating paired observations .
cu = 10,000 samples per perturbation leve.
A Hypotheses:
o Null hypothesis .
a0 ): No significant difference in mean accuracy between MLP and LR.
o Alternative hypothesis .
a1 ): MLP and LR exhibit statistically distinct mean accuracies.
o Calculation: The t-statistic was computed using:
yccI yc= ycycc AEOoycu where ( yc.
is the mean difference in accuracy (MLP Ae LR), .
cycc ) is the standard deviation of the differences, and ycu is the number of test samples.
A Interpretation: A p-value < 0.
05 rejects ya0 , confirming that MLPAos superior robustness .
r vulnerabilit.
is statistically meaningful.
For example, at yua = 0.
5 Gaussian noise, the large t-statistic .
c = 24.
19, ycy < 0.
conclusively supports MLPAos advantage over LR.
Linear Regression Analysis This analysis was applied exclusively to MLP under Gaussian noise to model the relationship between noise intensity and accuracy degradation:
A Variables:
o Independent variable .
: Noise intensity, quantified by the L2 norm of added noise (AnyaAn2 ).
o Dependent variable .
: Accuracy drop from baseline .
nperturbed accuracy Ae perturbed A Model:
Accuracy Drop = yu0 yu1 UI AnyaAn2 yun where yu0 .
and yu1 .
are regression coefficients, and yun is the error term.
A Application: The high ycI2 = 0.
98 indicates that 98% of the variance in MLPAos accuracy drop is explained by noise intensity.
The slope .
u1 = 6.
quantifies the rate of accuracy decline per unit increase in AnyaAn2 , enabling practitioners to predict performance degradation.
For instance, if AnyaAn2 = 10, the model predicts an accuracy drop of 6.
41 y 10 Oe 3.
69 = 60.
Reproducibility All experiments are repeated multiple times using different random seeds to ensure the reliability and consistency of the results.
The implementation is performed in Python using libraries such as TensorFlow, scikit-learn, and SciPy, ensuring that the methodology is reproducible.
Result and Discussion In our experiments, the baseline accuracies were 97.
07% for the MLP and 92.
63% for the LR model on the unperturbed MNIST test set.
These results provide a strong starting point for assessing model robustness under various perturbations.
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Gaussian Noise Experiment When Gaussian noise was added, both models experienced a decrease in accuracy as the noise intensity .
increased as shown in Figure 2 and Table 1.
Table 1.
Average L2 Norm under Gaussian Noise Sigma MLP Accuracy (%) LR Accuracy (%) Avg.
L2 Norm Figure 2.
Accuracy Comparison under Gaussian Noise At yua = 0.
1, the MLP accuracy was 91.
56%, while LR achieved 84.
However, at yua = 1.
0, both models exhibited near-identical failure rates, with MLP and LR accuracies collapsing to 15.
57% and 15.
This convergence suggests that extreme noise levels obliterate discriminative features, rendering both linear (LR) and non-linear (MLP) decision boundaries ineffective.
A plausible explanation is that Gaussian noise with yua Ou 1.
0 overwhelms the original signal, as pixel intensity distributions become indistinguishable from random noise (Fawzi et al.
, 2.
Under such conditions, even the MLP's hierarchical feature extraction fails to recover meaningful patterns (I.
Goodfellow et al.
, 2016.
Madry et al.
, 2.
This phenomenon underscores a critical limitation: while MLPs excel under moderate perturbations, their superiority diminishes when perturbations exceed thresholds that erase class-specific information (Fawzi et , 2.
The accuracy degradation of both models under Gaussian noise is summarized in Table 1.
As yua increases, the average L2 norm of the noise grows linearly, correlating with a predictable decline in classification For instance, at yua = 0.
5, the MLPAos accuracy drops to 35.
35%, while the LR model collapses to This trend validates the linear regression analysis .
cI2 = 0.
, where the L2 norm serves as a reliable predictor of accuracy degradation.
A linear regression analysis was conducted exclusively on the MLP under Gaussian noise to quantify the relationship between noise intensity (L2 nor.
and accuracy degradation as shown in Figure 3.
Muhammad Thahiruddin, dkk A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset Figure 3.
Gaussian Noise Impact on MLP Accuracy The analysis yielded a slope of 6.
4069, indicating that each unit increase in L2 norm reduces MLP accuracy by 6.
This steep decline arises from the MLPAos hierarchical architecture: additive Gaussian noise propagates cumulatively through its layers, progressively distorting feature representations .
, edges, shape.
learned by early layers and destabilizing the non-linear decision boundaries in later layers.
The nearperfect ycI2 = 0.
9812 reflects the MLPAos predictable response to Gaussian noiseAia consequence of its continuous, structured perturbation that uniformly disrupts pixel intensities, unlike discrete or spatially localized distortions .
, salt-and-pepper nois.
The statistically significant p-value .
confirms that this relationship is not random but inherent to the MLPAos sensitivity to Gaussian noise.
While the intercept .
suggests an implausible accuracy increase at zero noise, this artifact stems from extrapolating the regression model beyond the tested L2 norm range .
ua = 0.
1Ae 1.
, where baseline accuracy is already saturated .
07%).
The continuous and additive nature of Gaussian noise uniquely enables such linear modeling, as its effects scale systematically with intensity.
In contrast, simpler models like LR lack hierarchical feature extraction, rendering their performance degradation less linearly correlated with noise magnitude.
These findings underscore how the MLPAos architectural complexityAiwhile enabling robustness to moderate noiseAi also introduces predictable vulnerabilities under structured, high-intensity perturbations.
The paired t-test results further confirm that the performance differences between the MLP and LR are statistically significant for most noise levels .
< 0.
, except at extremely high noise levels .
ua = 1.
where both models perform similarly poorly as shown in Table 2.
Table 2.
Paired t-test Result for Gaussian Noise Sigma t-stat p-value The paired t-test results (Table .
confirm that the performance differences between MLP and LR are statistically significant .
cy < 0.
under low to moderate noise levels .
ua = 0.
1Ae 0.
, with the highest tstatistic observed at yua = 0.
c = 40.
, marking the optimal point where MLPAos hierarchical architecture most effectively compensates for noise without losing critical discriminative features.
However, under extreme noise .
ua = 1.
, both models collapse to near-identical performance .
cy = 0.
1283, yc = 1.
, with accuracies plummeting to 15%Aijust marginally above random guessing .
% for MNIST).
The declining t-statistics from yua = 0.
3 to 0.
08 Ie 10.
reflect a diminishing MLP advantage as noise intensifies, though the differences remain statistically robust.
These findings validate MLPAos consistent superiority in practical scenarios with moderate noise .
ua O 0.
, while highlighting its inability to withstand perturbations that Zeta Ae Math Journal.
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obliterate fundamental data structures .
ua Ou 1.
This underscores that model selection must account for estimated noise intensity in target environments, with MLP remaining the preferred choice provided perturbations do not exceed critical thresholds.
Salt-and-Pepper Noise Experiment Both models show declining performance with increased salt-and-pepper noise as shown in Figure 4.
Figure 4.
Accuracy Comparison under Salt-and-Pepper Noise As illustrated in Figure 4, both models exhibit rapid accuracy degradation as salt-and-pepper noise intensity .
increases, with the MLP maintaining a consistent advantage over LR across all perturbation levels.
At lower noise intensities .
cy = 0.
, the MLP achieves 80.
27% accuracy compared to LRAos 65.
demonstrating its ability to mitigate localized pixel corruption through hierarchical feature learning.
However, as noise escalates .
cy = 0.
, the MLPAos accuracy drops to 23.
57%, while LR collapses to 17.
77%, reflecting the destructive impact of extreme pixel-level disruptions on both models.
The statistical significance of MLPAos superiority is confirmed in Table 3, where paired t-tests yield ycy < 0001 for all noise levels as follows Table 3.
Paired t-test Result under Salt-and-Pepper Noise Amount t-stat p-value Table 3 presents the paired t-test results comparing the performance of MLP and Logistic Regression (LR) across increasing salt-and-pepper noise intensities .
cy = 0.
10 to 0.
All p-values .
cy < 0.
confirm statistically significant differences in accuracy between the two models at every noise level, with t-statistics ranging from yc = 30.
87 at ycy = 0.
10 to yc = 12.
28 at ycy = 0.
The declining t-statistic trend reflects a narrowing performance gap as noise intensifies, suggesting that while MLP consistently outperforms LR, its relative advantage diminishes under extreme corruption.
For instance, at ycy = 0.
10, the large t-statistic .
c = .
underscores MLPAos strong robustness, leveraging hierarchical feature extraction to mitigate localized pixel corruption.
Even at ycy = 0.
50, where both models struggle (MLP: 23.
LR: 17.
77%), the significant t-statistic .
c = 12.
confirms MLPAos persistent superiority, albeit reduced.
These results validate MLPAos reliability in noise-prone environments, particularly at moderate intensities .
cy O 0.
, where its architectural complexity provides critical resilience.
The findings emphasize MLPAos suitability for applications like lowquality image processing, while highlighting the need for supplementary noise-reduction strategies in highcorruption scenarios.
A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset Muhammad Thahiruddin, dkk Rotation Experiment The effects of rotational perturbations are shown in Figure 5.
Figure 5.
Accuracy Comparison under Rotation At 5A rotation, the MLP maintained an accuracy of 96.
86% while LR was at 91.
However, with increasing rotation up to 30A, the MLP's accuracy declined to 70.
16% and LR to 55.
Rotational distortions gradually degrade model performance, with the MLP maintaining superior accuracy across all angles.
For instance, at 30Oo rotation, the MLP achieves 70.
16% accuracy, significantly higher than LRAos 55.
highlighting its ability to tolerate geometric variations.
The statistical significance of the MLPAos robustness is reinforced in Table 4.
Table 4.
Paired t-test Result for Rotation Angle (A) t-stat p-value Table 4 details the statistical significance of performance differences between MLP and Logistic Regression (LR) under rotational distortions, with angles ranging from 5Oo to 25Oo .
All p-values .
cy < 0.
confirm that MLPAos superior robustness is statistically significant at every rotation level, while the increasing t-statisticsAifrom yc = 20.
12 at 5Oo to yc = 31.
65 at 25Oo Aireveal a critical trend: MLPAos advantage over LR grows more pronounced as geometric distortions intensify.
At mild rotations .
Oo ).
MLP retains nearbaseline accuracy .
86% vs.
LRAos 91.
71%), reflected in the relatively lower t-statistic .
c = 20.
However, at extreme angles .
Oo ), where MLP achieves 70.
16% accuracy compared to LRAos 55.
65%, the t-statistic peaks .
c = 31.
, emphasizing MLPAos ability to hierarchically adapt to reoriented features .
, edges, curve.
that LRAos linear decision boundaries fail to generalize.
This pattern underscores how MLPAos architectural complexityAienabling spatial invariance through layered feature extractionAibecomes increasingly vital under severe geometric stress.
The results validate MLPAos reliability in applications prone to orientation variations .
, document scanning, robotic.
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FGSM Adversarial Attacks FGSM attacks were implemented solely on the MLP due to the differentiability requirements, which are not supported in the scikit-learn LR framework as shown in Figure 6.
Figure 6.
MLP Accuracy under FGSM Attack Figure 6 illustrates the catastrophic vulnerability of the MLP model to adversarial attacks generated via the Fast Gradient Sign Method (FGSM), where even minimal perturbation magnitudes .
drastically degrade classification accuracy.
At yun = 0.
05, the MLPAos accuracy plummets to 35.
70% .
rom a baseline of 97.
07%),
and by yun = 0.
30, performance collapses to near-zero .
20%), effectively rendering the model non-functional.
This exponential decline underscores how adversarial perturbationsAismall, strategically crafted noiseAi exploit the MLPAos gradient-dependent architecture to mislead its hierarchical feature extraction.
Unlike random noise .
Gaussian or salt-and-peppe.
, adversarial attacks target the modelAos decision boundaries, causing disproportionate harm despite imperceptible changes to human observers.
The results starkly contrast with MLPAos robustness to other perturbations, highlighting a critical security flaw: while MLPs handle moderate natural noise well, their lack of adversarial training leaves them defenseless against malicious inputs.
This necessitates urgent integration of defense mechanisms .
, adversarial training, input preprocessin.
to safeguard real-world deployments, particularly in high-stakes domains like autonomous systems or cybersecurity where adversarial threats are prevalent.
Analysis The experimental results demonstrate that while both models suffer under increased perturbations, the MLP generally exhibits higher robustness compared to the LR model.
Several factors contribute to the observed robustness of the MLP:
Non-linear Feature Extraction.
The MLP's layered architecture enables it to learn complex, non-linear representations of the input data.
This allows the network to extract features that are more invariant to noise and small distortions.
contrast, the LR model, being inherently linear, lacks this capability and is less able to capture intricate patterns in the data.
Adaptive Learning through Multiple Layers:
The hierarchical structure of the MLP provides multiple levels of abstraction.
Early layers can capture basic patterns .
dges, simple shape.
, while later layers combine these to form more robust This multi-level processing helps the MLP to better tolerate perturbations, as even if some features are degraded, others can still contribute to correct classification.
Statistical Validation.
The paired t-test results across various perturbations consistently indicate that the differences in model performance are statistically significant.
For most noise levels, the MLP significantly outperforms LR.
Only at extremely high perturbation levels .
, yua = 1.
0 in Gaussian nois.
do both models converge to similarly low accuracy levels, likely because the perturbations overwhelm any useful signal.
Muhammad Thahiruddin, dkk A Robustness Study of Multi-Layer Perceptrons and Logistic Regression to Data Perturbation: MNIST Dataset Implications of Regression Analysis.
The strong linear regression .
ith an R-squared of 0.
observed for the MLP under Gaussian noise con rms that its performance degradation can be predicted reliably based on the noise intensity.
This insight is valuable for practical applications where it may be possible to estimate the level of noise and anticipate performance drops.
The absence of similar analysis for LR is partly due to its linear simplicity and the lack of a suitable continuous noise intensity metric in its context.
Conclusions This study directly addressed the gaps identified in the background by systematically evaluating the robustness of MLPs and LR under diverse perturbations.
First, the comparative analysis revealed that MLPs consistently outperform LR in moderate noise .
ua O 0.
and rotational distortions .
uE O 25Oo ), achieving accuracy margins of up to 14.
6% .
alt-and-pepper nois.
51% .
This resolves the ambiguity in model selection for practitioners working with imperfect data, such as low-quality imaging or sensor-driven Second, we quantified critical failure thresholds: both models collapse to near-random performance at extreme noise .
ua = 1.
or rotations .
Oo ), demonstrating that perturbations exceeding L2 norm = 06 or yuE = 25Oo erase class-discriminative features.
Third, the strong linear relationship .
cI2 = 0.
between Gaussian noise intensity and MLPAos accuracy degradation established a predictive robustness metric, enabling practitioners to estimate performance declines without costly retestingAia novel contribution for classical models.
However, the MLPAos vulnerability to adversarial attacks .
20% accuracy at yun = 0.
exposes a critical limitation unaddressed in prior works: hierarchical architectures, while robust to random noise, remain highly susceptible to gradient-based manipulations.
This underscores the need to integrate adversarial trainingAia defense mechanism previously reserved for DNNsAiinto MLP workflows.
These findings redefine robustness benchmarks for foundational models.
For tasks involving moderate noise or geometric variations.
MLPs are unequivocally preferable to LR.
Yet, in adversarial-prone environments, neither model suffices without targeted hardening.
Future work should expand predictive modelling .
, linking salt-and-pepper noise to accuracy via pixel corruption rate.
and explore hybrid architectures that marry MLPAos feature invariance with adversarial resilience.
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