Hidden Neural Structure in Distance Classifiers

• The paper demonstrates that distance-based classifiers can be reformulated as neural networks with detection, pooling, and output layers, enhancing interpretability.
• It shows that probabilistic weight aggregation and manifold-based feature transforms reveal latent representations that improve robustness and scalability.
• Empirical evaluations indicate that neuralized models achieve sharper cluster separation, effective out-of-distribution detection, and efficient incremental learning.

The hidden neural network structure in distance-based classifiers refers to the perspective that, despite the absence of explicit trainable hidden layers, many classic and modern distance-based methods can be reformulated or interpreted as layered neural architectures. This perspective has both practical and theoretical ramifications, enabling the application of deep learning concepts such as network-based explainability, robust feature extraction, and scalable learning to models traditionally seen as nonparametric or Euclidean-space-based. Distance-based classifiers include models such as k-nearest neighbors (k-NN), support vector machines (SVMs) with kernels, and recent probabilistic and manifold-learning algorithms. The neuralization of these algorithms has allowed for the discovery of latent representations and the adaptation of neural network methods for interpretability, robustness, and incremental learning.

1. Neuralization of Distance-Based Models

Distance-based classifiers, notably SVMs with RBF kernels and k-NN, can be exactly rewritten as neural networks with multiple layers—a paradigm known as "neuralization" (Bley et al., 5 Aug 2025). This neural reformulation involves:

• Detection Layer: Linear units indexed by prototype pairs or pairs of support vectors from opposing classes, computing affine functions of input.
• Pooling Layers: Nonlinear aggregation via smooth maximum/minimum or rank-based pooling (e.g., softmax-like or rmax/rmin operators).
• Output Layer: Produces class probabilities or regression output by combining the pooled detector outputs.

For Gaussian kernel SVMs:

$$\text{Detection:} \quad z_{ij} = (x - m_{ij})^T w_{ij} + b_{ij}$$

$$\text{Pooling:} \quad h_j = \mathrm{smax}_{i \in C_+}^{\gamma}(z_{ij})$$

$$\text{Output:} \quad g(x) = \mathrm{smin}_{j \in C_-}^{\gamma}(h_j)$$

where $m_{ij}, w_{ij}, b_{ij}$ are constructed from support vectors $u_i, u_j$ and $C_+, C_-$ denote class partitions (Bley et al., 5 Aug 2025). For KNN, analogous layers can be defined with rmax/rmin operators.

This structure permits the direct application of neural network interpretation tools and reveals a notional "hidden" architecture in distance-based models.

2. Probabilistic Weight Aggregation and Markov Random Field Interpretations

Probabilistic distance-based classifiers using Markov random fields (MRFs) define label dependencies via distance-weighted cliques, leading to update rules structurally akin to shallow neural networks (Friel et al., 2010). The full-conditional probability for a label $y_i$ is:

$$\pi(y_i \mid y_{-i}, \beta, \sigma) \propto \exp\left\{\beta \sum_{j \ne i} w_j^i I(y_j = y_i)\right\}$$

where $w_j^i$ decays with feature distance, and $\beta, \sigma$ control aggregation strength and the spatial scale. Aggregation over distance-weighted indicator features is analogous to the computation in a single-layer NN with "connection weights" defined by a distance kernel. Parameters $\beta$ and $\sigma$ play roles similar to bias scaling and kernel width in NNs.

This perspective offers insight into how probabilistic and neural mechanisms become intertwined when label smoothing and local dependence are encoded via soft, differentiable aggregation—a key structural motif in neural networks (Friel et al., 2010).

3. Robust Feature Construction via Nonlinear and Manifold Transforms

Hidden neural network structure is also manifest when distance-based methods employ nonlinear, robust feature transformations derived from data depth, projection distances, or manifold geometry:

• Robust Distance Embeddings: Data are mapped to "distance space" via robust metrics such as bagdistance or skew-adjusted projection depth, making the transformation layer functionally analogous to a hidden layer in a neural network. The classification is then performed in this new space via simple k-NN or other rules (Hubert et al., 2015).
• Block and Nonlinear Component Groupings: Generalized distance transforms of the form

$$h_d^{\phi, \gamma}(u, v) = \phi\left(\frac{1}{d}\sum_{i=1}^d \gamma(|u_i - v_i|^2)\right)$$

act as "activation functions" across the feature dimensions before pooling/aggregation, echoing classic feedforward network operations (Roy et al., 2019).

• Manifold-Aware Architectures: In Distance Learner, shared MLP trunk layers feed class-specific output heads that estimate distance-to-manifold for each class. This branching structure parallels multi-output neural architectures and ensures that hidden layers encode geometric proximity to class manifolds rather than just labels (Chetan et al., 2022).

4. Hidden Structures Enhance Explainability and Robustness

Uncovering latent neural architectures in distance-based models enables the direct application of sophisticated explainability techniques:

• Layer-Wise Relevance Propagation (LRP): The neuralized view makes LRP applicable to SVMs and k-NN by decomposing decisions into relevance assigned to input features via the detection and pooling layers. Analytical relevance propagation is enabled by the differentiable aggregation, in contrast to black-box perturbation methods (Bley et al., 5 Aug 2025).
• Noise-Canceling Generative Architectures: Generative distance classifiers refine feature distances by subtracting distances to a common reference, canceling instance-specific noise and mapping semantics directly ("what-they-are") in high-dimensional representation—a conceptual parallel to the semantic disentanglement in hidden layers of deep models (Lin et al., 2022).

A plausible implication is that by mapping classic models into neuralized frameworks, debate over their interpretability relative to deep learning classifiers can be reframed—making transparency and post-hoc interpretation more practical.

5. Implications for Model Design, Adaptivity, and Scalability

The neural perspective on distance-based classifiers has practical effects on design and scalability:

• Adaptive Model Capacity: Compact-sized probabilistic neural networks (CS-PNN) demonstrate one-pass, data-driven growing methods that adapt the number and spread of hidden RBF units as the data evolve, mirroring the adaptive structuring seen in dynamic neural networks (Hoya et al., 1 Jan 2025). Such RBF-based networks are inherently distance-based yet gain flexibility and robustness by dynamic adaptation of centers and radii, making the hidden structure both explicit and plastic.
• Incremental and Lifelong Learning: Distance-based models expressed via neural architectures allow for efficient incremental learning and unlearning—from updating or pruning hidden units to recalibrating the hidden representations as new classes or instances are observed (Hoya et al., 1 Jan 2025, Lin et al., 2022). This capability is less practical for fixed-architecture discriminative neural networks and supports efficient adaptation in streaming or nonstationary environments.
• Cluster Separation Optimization in Hidden Layers: Auxiliary loss functions targeting hidden-layer linear separability (e.g., weighted cross-entropy over hidden classifications) induce representations more amenable to distance-based discrimination, thus directly benefiting hybrid methods that rely on Euclidean geometry in the latent space (Apicella et al., 2023).

6. Theoretical and Empirical Evaluations

Quantitative and theoretical results reinforce the significance of hidden neural structures in distance-based models:

• Cluster Quality and Decision Boundaries: Training rules that penalize overconfident net activations (e.g., LCNN’s VC-bound-regularized loss) encourage crisper, sparser, and more separable clusters in hidden representations, improving the efficacy of distance-based post-processing (Jayadeva et al., 2017).
• Robustness to Distributional Shift: Hidden distance-based structure supports out-of-distribution and adversarial detection, as embeddings can be directly penalized or trained under perturbations to improve class separability and detection reliability (Mandelbaum et al., 2017, Chetan et al., 2022).
• High-Dimensional Consistency: Theoretical analysis under high-dimension, low-sample-size (HDLSS) regimes demonstrates that appropriately generalized or block-grouped distance transformations yield vanishing misclassification probability as dimension grows, so long as the hidden transformation preserves class-separating structure (Roy et al., 2019).

Empirical evaluations show that neuralized or robustly constructed distance-based classifiers can match or outperform both classic k-NN and state-of-the-art neural networks in a broad range of synthetic and real-world tasks, while retaining interpretability and scalability.

7. Conclusion

Uncovering hidden neural network structures in distance-based classifiers bridges the conceptual gap between nonparametric methods and modern deep learning. Neuralization makes hidden layered architectures explicit, enabling the application of interpretability techniques, dynamic adaptivity, and robust geometric learning. This synthesis clarifies classical results and opens avenues for hybrid models that leverage the statistical robustness and direct feature-space geometry of distance-based approaches with the representational power and analytic tractability of neural networks. The evolution of distance-based methods into the neural paradigm contributes both to practical advances in explainability, scalability, and adaptivity, and to a more unified understanding of pattern recognition architectures in contemporary machine learning.