Semantic-guided RBF Methods

• Semantic-guided RBF is a method that augments classical radial basis function models by integrating semantic metrics, enhancing representation fidelity and interpretability.
• It employs adaptive fusion of cosine and Euclidean kernels via gradient descent to tailor similarity measures to domain-specific semantic features.
• Applications span time-varying signal classification, deep CNN architectures, quantum feature mapping, and statistically guided shape modeling in medical imaging.

Semantic-guided Radial Basis Function (RBF) approaches augment classical RBF models with mechanisms, metrics, and representations designed to infuse semantic structure or meaning into model learning and operation. This can involve tailoring the RBF kernel or the distance metric to reflect semantic features, aligning the learned representations with interpretable patterns or domain knowledge, or integrating RBFs into architectures that facilitate interpretability and semantic retrieval. Semantic guidance in RBFs arises in varied contexts, including similarity metric learning, process neural network optimization, quantum kernel architectures, and shape modeling from images.

1. Foundations of Semantic Guidance in RBFs

Classical RBF neural networks operate by computing activations based on the distance between input vectors and fixed centers, typically with the Euclidean metric. Semantic guidance modifies this paradigm by replacing or augmenting the raw geometric distance with metrics or kernels that capture temporal, contextual, or application-dependent relationships. For instance, generalized Fréchet distance can compare time-varying signals, reflecting more nuanced, semantically relevant similarity than Euclidean measures (Wang et al., 2014). In convolutional contexts, learning a similarity distance metric as part of the RBF operation enables the classifier to separate semantically similar and dissimilar instances (&&&1&&&). Semantic kernels, such as those based on cosine similarity or domain-specific embedding distances, further extend this principle (Khan et al., 2019).

2. Adaptive and Hybrid Distance Metrics

A key mechanism for semantic guidance in RBFs is the adaptive or learned fusion of multiple distance metrics. The adaptive kernel RBF framework fuses Euclidean and cosine distances:

$$\varphi_i(x, x_i) = \alpha_1 \varphi_{i1}(x \cdot x_i) + \alpha_2 \varphi_{i2}\left(\|x - x_i\|\right)$$

where $\varphi_{i1}$ is a cosine kernel and $\varphi_{i2}$ is a Gaussian (Euclidean) kernel. Mixing weights $\alpha_1$ and $\alpha_2$ are normalized and dynamically updated using gradient descent to optimize the network’s objective function:

$$\alpha_1(n+1) = \alpha_1(n) + \eta e(n) \sum_{i=1}^{m_1} w_i(n) \frac{|\alpha_1(n)||\alpha_2(n)|}{\alpha_1(n)(|\alpha_1(n)| + |\alpha_2(n)|)^2} [\varphi_{i1} - \varphi_{i2}]$$

alongside analogous updates for $\alpha_2$ (Khan et al., 2019). This dynamic adaptation allows RBF networks to emphasize metrics that most effectively capture the semantic or structural properties of the task, such as semantic category boundaries in classification tasks.

3. Semantic Similarity Metrics for Temporal and Structured Data

In process neural networks, semantic guidance can be achieved by deploying similarity measures sensitive to domain structure. The generalized Fréchet distance:

$$d(X(t), X'(t)) = \sqrt{\sum_{i=1}^n [d_F(x_i(t), x'_i(t))]^2}$$

where $d_F$ is the discrete Fréchet distance, enables the comparison of time-varying function samples along their actual temporal trajectories, substantially improving recognition and classification of dynamic signals (Wang et al., 2014). This metric provides a semantic-like measure, closely aligning with interpretable patterns such as temporal spike trains in EEG signal recognition or phoneme sequence similarity in speech processing.

4. Semantic-guided RBFs in Deep and Interpretable Architectures

In contemporary vision architectures, semantic guidance is achieved through RBF layers attached atop deep convolutional embeddings. CNN embeddings are mapped to RBF hidden layers where each activation is a function of the trainable similarity metric:

$r^2 = (x - c_j)^T R_j (x - c_j)$

where $R_j$ is a positive-definite (potentially full-covariance) matrix. This architecture supports cluster-based interpretability. Similarity of samples can be semantically ranked by distances in the learned metric space, enabling direct retrieval of similar and dissimilar training instances and facilitating human-in-the-loop model analysis (Amirian et al., 2022).

A quadratic activation function:

$h(r) = 1 - \frac{r^2}{\sigma^2}$

streamlines the computational graph and allows direct gradient flow during training, supporting end-to-end adaptation of semantic similarity metrics alongside classification objectives. Unsupervised loss terms (e.g., k-means objectives on clusters) further guide the network to organize embeddings around semantically meaningful centers.

5. Quantum RBF Networks and Semantic Feature Maps

Quantum RBF networks instantiate the classical kernel mapping and center weighting using quantum state preparation and tensor products of single-qubit states (Shao, 2019). The input is encoded as a superposition

$$|f(x)\rangle \propto \sum_t \exp\left(-\frac{\|x - x^{(t)}\|^2}{2\sigma^2}\right) |t\rangle$$

allowing high-dimensional, kernelized feature maps. Center weights are efficiently encoded as tensor products:

$$w(\theta) = \bigotimes_{j=1}^m [\cos \theta_j, \sin \theta_j]^T$$

Semantic guidance in quantum RBFs could be achieved by tailoring the feature map $f(x)$ to reflect semantic (rather than purely geometric) similarities, such as leveraging word embeddings or medical concept vectors. Quantum architectures support efficient manipulation of these high-dimensional maps, suggesting that quantum RBFs are natural candidates for large-scale semantic modeling, conditional on the structure of the semantic feature map.

6. RBF-based Statistical Shape Modeling and Semantic Priors

In geometric modeling, RBFs support semantic-guided continuous representations via implicit surface models. Image2SSM introduces an RBF-shape formulation:

$$f(x) = \sum_{j=1}^J w_j \phi(\|x - t_j\|_2) + c^T x + c_0$$

where control points $t_j$, weights $w_j$, and an affine component define the implicit surface as the zero-level set $f(x) = 0$ (Xu et al., 2023). Self-supervised losses enforce correspondence (via entropy-based metrics) and geometric alignment (via cosine similarity of normals) across populations of anatomical surfaces. The continuous, differentiable, and compact RBF-based model facilitates the incorporation of semantic priors—particularly anatomical landmarks and tissue classes—into shape analysis pipelines. The approach’s scalability and accuracy make it suitable for large-cohort studies demanding semantic interpretability.

7. Prospects and Future Directions in Semantic-guided RBFs

Across multiple domains, semantic-guided RBFs have demonstrated improvements in interpretability, representation fidelity, and classification performance. Adaptive fusion of semantic and geometric kernels suggests robust multi-modal learning frameworks. The use of learned similarity metrics aligns RBF behavior with domain-specific notions of similarity, supporting both human and automated semantic reasoning. In vision and medical imaging, integrating cluster-based RBF layers yields interpretable decision processes. Quantum RBF architectures provide promising efficiency for scaling semantic kernels, conditional on expressive feature maps.

Open research directions include:

• Designing domain-specific semantic kernels for text, image, and signal data.
• Extending adaptive fusion frameworks to include semantic relevance weighting and compositional kernels.
• Embedding semantic priors into quantum feature maps for enhanced quantum-classical models.
• Developing continuous semantic representations in shape and geometry processing.
• Empirically validating semantic guidance mechanisms on large, annotated datasets across domains.

These approaches collectively advance the capabilities of RBF models from purely geometric similarity approximators toward interpretable, semantically-grounded machine learning systems.