SNM-Net Framework

• The paper introduces SNM-Net, a universal deep learning framework that leverages spherical normalization and Mahalanobis distance scoring to overcome sensor drift and unknown class interference.
• SNM-Net’s two-stage geometric decoupling via batch and L₂ normalization separates intensity from directional chemical features, enhancing robustness against environmental variations.
• Experimental results across CNN, LSTM, and Transformer backbones show significant improvements in AUROC and TPR, achieving near-theoretical detection limits and stable open-set classification.

SNM-Net is a universal deep learning framework formulated for robust open-set gas recognition, addressing two primary challenges encountered by electronic nose (E-nose) systems: (i) distributional feature shifts caused by sensor drift and (ii) performance degradation due to unknown (unseen) class interference. Distinct from prior approaches that employ Euclidean metrics and fail to capture anisotropic feature geometries or intensity variations, SNM-Net introduces a principled geometric and statistical mechanism that leverages spherical normalization and Mahalanobis distance-based scoring to achieve near-theoretical detection and stability limits in gas classification tasks (Chen et al., 28 Dec 2025).

1. Mathematical Foundations

The core innovation of SNM-Net is a two-stage geometric decoupling mechanism, achieved via cascaded batch normalization and L₂ normalization. Let $z \in \mathbb{R}^d$ denote the raw feature vector from a deep backbone:

1. Batch Normalization (BN):

$$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$

This normalizes features to zero mean and unit variance per dimension, removing first- and second-order distributional shifts.

2. L₂-Normalization (Spherical Projection):

$f = \frac{z'}{\|z'\|_2}, \quad f \in S^{d-1}$

This step projects each feature onto a $(d{-}1)$-dimensional unit hypersphere, decoupling the original vector into:

$$z = \underbrace{\|z\|_2}_{\text{intensity}} \times \underbrace{\frac{z}{\|z\|_2}}_{\text{direction (chemical properties)}}$$

The result is suppression of magnitude-induced drift (intensity) while retaining direction (gas identity), rendering the embedding robust to variations in concentration, temperature, humidity, or sensor aging.

3. Mahalanobis Distance Scoring: Each class $c$ is characterized by its empirical mean $\mu_c$ and regularized covariance $\Sigma_c^{\mathrm{reg}} = \Sigma_c + \lambda I$:

$$\mu_c = \frac{1}{N_c} \sum_{i:y_i=c} f_i,\qquad \Sigma_c = \frac{1}{N_c-1} \sum_{i:y_i=c} (f_i - \mu_c) (f_i - \mu_c)^\top$$

$$d_c(f(x)) = \sqrt{(f(x) - \mu_c)^\top (\Sigma_c^{\mathrm{reg}})^{-1} (f(x) - \mu_c)}$$

Unlike Euclidean approaches, Mahalanobis scoring yields ellipsoidal class boundaries aligned to classwise feature anisotropy.

A test sample is classified as the known class with minimum Mahalanobis distance, or is rejected as "unknown" if $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$0, with the threshold $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$1 determined (e.g., 95th percentile) from validation data.

2. Architecture-Agnostic Design

SNM-Net is intentionally constructed as a post-processing module, agnostic to the underlying feature extractor:

• CNN Backbone: Residual blocks, global average pooling, then $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$2.
• Bi-LSTM Backbone: Bidirectional encoding, final hidden state, then $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$3.
• Transformer Encoder: Two-layer multi-head self-attention (head=4), feed-forward network, global mean pooling, then $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$4.

In all architectures, the geometric decoupling (BN → L₂-norm) and statistical scoring mechanism (class mean and covariance estimation, Mahalanobis scoring) are identically applied. No backbone modification is required beyond insertion of a batch-norm and normalization layer, confirming its universal applicability for E-nose data modalities.

3. Training and Inference Protocol

The training methodology combines open-set gas recognition requirements with robust statistical estimation:

• Hyperparameters:
  • Optimizer: Adam, initial learning rate $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$5, weight decay $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$6
  • Batch size: 32, drop-out: 0.1, maximum epochs: 200, early stopping: 10
  • CAC (Class Anchor Clustering) loss weight: $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$7
  • Covariance regularization: $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$8
  • Unknown detection threshold: 95th percentile of training-known distances
• Cross-validation/Unknowns Exposure:
  • 10-fold Cross-Validation: For each fold, 6 of 10 gases randomly chosen as "known" (60% as train), remainder (40% of knowns + all 4 unknowns) as test
  • Training utilizes only "known" class identities and CAC loss
  • Inference leverages Mahalanobis scoring for detection and rejection of test samples

A sample is predicted as known-class if its minimum classwise distance is below the threshold, or otherwise is rejected as "unknown."

4. Experimental Results and Ablation Analyses

SNM-Net was validated on the Vergara et al. public dataset: 10 gases, 18,000 samples, multi-module/sensor arrays, time-series inputs downsampled to $$z' = \mathrm{BN}(z) = \frac{z - \mu_{\mathrm{batch}}}{\sqrt{\sigma_{\mathrm{batch}}^2 + \epsilon}}$$9 maps, with 5 sensor positions (L1–L5) generating intensity drift.

• Key Metrics:
  • Known-class accuracy
  • TPR @ FPR = 5% (unknown detection rate)
  • AUROC (Area Under ROC curve)
  • Results averaged over 50 runs (positions × folds)
• Ablation Study (CNN backbone):

Configuration	Known Acc	TPR@5%	AUROC
BASE-CNN (CAC+Euclid)	0.9993±0.0024	0.8246±0.1492	0.9504±0.0474
+Mahalanobis	0.9974±0.0062	0.9045±0.0837	0.9785±0.0192
+M+BN	0.9991±0.0026	0.9347±0.0732	0.9847±0.0169
+M+L₂N	0.9986±0.0052	0.9270±0.0811	0.9829±0.0168
Full (M+BN+L₂N)	0.9988±0.0042	0.9370±0.0683	0.9846±0.0148

• Cross-Backbone Generalization (AUROC, mean ± std):
  • CNN+SNM: 0.9846±0.0148
  • LSTM+SNM: 0.9748±0.0198
  • Transformer+SNM: 0.9977±0.0028
• State-of-the-Art Comparison (Transformer backbone):

Method	Known Acc	TPR@5%	AUROC
Softmax+Trans.	1.0000±0.0002	0.8120±0.1378	0.9557±0.0406
EVM+Trans.	0.9999±0.0004	0.8227±0.1339	0.9514±0.0424
CAC+Trans.	0.9998±0.0009	0.8376±0.1529	0.9675±0.0311
SNM+Trans.	1.0000±0.0002	0.9957±0.0138	0.9977±0.0028

SNM-Net achieved a 3.0% improvement in AUROC and a 91.0% reduction in standard deviation versus Class Anchor Clustering. Across five spatial sensor locations, standard deviation of AUROC was 0.0013 (compared to baseline of 0.0163). Even at the weakest signal location (L5), TPR reached 98.34%.

5. Underlying Principles and Theoretical Insights

SNM-Net's two-stage pipeline addresses the pivotal causes of instability and inaccuracy in E-nose open-set classification:

• Spherical normalization universally suppresses signal intensity-driven drift at the feature level, ensuring downstream decisions rely on invariant geometric relationships ("directional cues") rather than context-dependent magnitude information.
• Mahalanobis distance-based class boundaries explicitly adapt to feature anisotropy via covariance modeling, enabling decision regions aligned to empirical class structure rather than imposed isotropy as in Euclidean baselines.
• This geometric-statistical combination resolves the classical trade-off between accuracy (local class specificity) and stability (robustness to sensor/environmental variation) in practical gas sensing tasks.

A plausible implication is that the framework’s statistical estimation yields classification confidence closely aligned to the true underlying data distribution, beneficial for deployment in safety-critical and industrial E-nose applications.

6. Limitations and Prospective Extensions

The empirical evaluation was limited to a single sensor array; transferability across distinct devices and manufacturing batches remains unproven. Future work includes latency and power profiling for embedded deployments. Potential methodology extensions include:

• Integration with domain adaptation to address cross-batch or cross-device variation.
• Research into adaptive threshold schemes for unknown detection under nonstationary operational conditions and environmental changes.

Continued investigation into invariance across a broader range of sensor chemistries and hardware is warranted to generalize SNM-Net’s favorable properties.

7. Summary and Impact

SNM-Net provides a mathematically and empirically validated framework for robust open-set gas recognition under drift and unknown interference. Its combination of feature-space spherical normalization and Mahalanobis statistical scoring achieves near-theoretical open-set discrimination limits (AUROC up to 0.9977, TPR at 5% FPR up to 99.57%) with stability across sensor positions (σ_AUROC ≈ 0.0028). These results substantiate its suitability as a foundational component for industrial-scale E-nose data pipelines and point to principled geometric-statistical modeling as an effective strategy for addressing real-world sensor drift and open-category challenges in chemical sensing (Chen et al., 28 Dec 2025).