CEMK: A Categorical Energy Meta-Kernel for Universal Approximation with O(n log n) Complexity

Abstract

This paper introduces CEMK (Categorical Energy Meta-Kernel), a novel learning paradigm that departs from traditional deep learning by replacing matrix multiplication with frequency-domain morphism composition. Built upon category theory, the Fourier transform, and the least-action principle from physics, CEMK achieves universal approximation with a theoretical complexity of O(n log n). The framework integrates four key innovations: (1) A learnable position encoding that preserves sign information, enabling the model to learn odd functions. (2) A random Fourier feature kernel that performs similarity measurement independent of input dimensionality. (3) A frequency-domain morphism composition mechanism where weights are transformed by FFT, multiplied element-wise in the spectral domain, and inverse-transformed, reducing the core computational bottleneck from O(n^3) to O(n log n). (4) An energy functional that enforces spectral sparsity by penalizing high-frequency components, aligning with the least-action principle. We present a complete verification framework with four rigorous tests. On universal approximation, a single CEMK model achieves R-squared scores of 1.0000 (linear), 1.0000 (quadratic), 1.0000 (trigonometric), 0.9999 (step), and 0.9737 (multivariate) all exceeding the 0.92 threshold. The empirical complexity exponent from log-log fitting is 0.1092, empirically confirming near-O(n log n) scaling. Resource consumption is extremely lightweight: 100 training epochs on CPU take 1.04 seconds with 314 MB peak memory. For robustness, when 10 percent of test samples are corrupted with NaN or 1e10 outliers, the model does not crash and R-squared drops only from 1.0000 to 0.8821, a decrease of 0.1179 well within the 0.30 tolerance. This work demonstrates that a mathematically grounded framework combining category theory, frequency-domain computation, and physical constraints can achieve universal approximation, high efficiency, interpretability, and resilience to missing data, offering a viable path toward general intelligence that is both explainable and computationally sustainable.

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