Fibonacci-based geometrically weighted circulant matrices: low-rank approximations and applications in data compression

This study explores the mathematical and computational characteristics of geometrically weighted circulant and symmetric geometric semicirculant matrices with the aim of identifying their potential as efficient structural tools in artificial intelligence (AI) architectures and data compressions. At the preliminary stage, a comprehensive mathematical framework was established, including the derivation of various matrix norms (such as spectral and Frobenius norms), determinants, and matrix inverses. The construction of these matrices is guided by Fibonacci numbers, whose intrinsic link to the golden ratio introduces a natural geometric decay pattern. This biologically inspired structure contributes to the balance, regularity, and interpretability of the resulting matrices, which are particularly well-suited for low-complexity modeling in AI systems. Subsequently, singular value decomposition (SVD) was employed to perform low-rank approximations, with a focus on evaluating information loss through Frobenius norm differences between original and reconstructed matrices. Techniques such as soft-thresholding and selective singular value removal were applied to assess data compression performance. Results demonstrated that symmetric geometric semicirculant matrices yielded smaller norm deviations, indicating superior data retention. Moreover, by tuning the geometric ratio parameter $r$, further improvements in matrix compactness and fidelity were achieved, especially when reducing $r$ to values like $\frac{1}{4}$. These outcomes were visually confirmed through heatmap representations, highlighting the robustness and compression potential of the proposed matrices.