Dynamic Kernel Projection Layer (DKPL) for Hilbert Space Mapping

This repository introduces a novel, high-performance architectural component designed to project linearly inseparable feature spaces into an infinite-dimensional Hilbert space approximation. Unlike traditional static kernel methods, the Dynamic Kernel Projection Layer (DKPL) features fully learnable spatial scales and bandwidths that dynamically optimize structural curvature during standard backpropagation.

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🔬 Core Mathematical Architecture

Standard dense neural layers rely heavily on Euclidean linear transformations ($\mathbf{W} \cdot \mathbf{x} + \mathbf{b}$), requiring excessive depth to approximate non-linear boundaries. DKPL resolves this by mapping inputs into a highly expressive kernel space in a single operational step.

1. Parametric Spatial Scaling

Incoming feature matrices $\mathbf{X}$ are scaled along an empirical learnable spatial tensor $\mathbf{W}{scale}$ to automatically adjust the sensitivity of individual feature components: $$\mathbf{X}{scaled} = \mathbf{X} \odot \mathbf{W}_{scale}$$

2. Pairwise Non-Euclidean Distance Calculation

Using the fundamental algebraic expansion property ($|\mathbf{a} - \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2\mathbf{a}\mathbf{b}^T$), the layer tracks localized distance variances cleanly inside eager execution: $$D_{i,j} = |\mathbf{x}i|{scaled}^2 + |\mathbf{x}j|{scaled}^2 - 2\mathbf{x}_i \cdot \mathbf{x}_j^T$$

3. Learnable RBF Kernel Matrix & Hilbert Projection

The continuous dynamic similarity matrix is driven by a learnable bandwidth factor $\sigma$: $$\mathbf{K}(i, j) = \exp\left( -\frac{D_{i,j}}{2\sigma^2} \right)$$

The final representation is fetched by projecting the non-linear self-similarity matrices into the target architectural dimensions: $$\mathbf{O} = \mathbf{K}(\mathbf{X}, \mathbf{X}) \cdot (\mathbf{X} \cdot \mathbf{W}_{projection})$$

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🚀 Key Advantages

• Instant Non-Linear Separation: Eradicates the necessity for deep network structures in highly complex classification domains (e.g., cybersecurity patterns, medical segmentations).
• Learnable Manifold Variance: Both the spatial curvature ($\mathbf{W}_{scale}$) and bandwidth ($\sigma$) dynamically absorb gradient updates.
• Pure PyTorch Design: Operates inside standard autograd loops without custom hardware-level dependency constraints.

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📜 License

This project is open-sourced under the terms of the MIT License.