Gradient Based Method for the Fusion of Lattice Quantizers

In practical applications, lattice quantizers leverage discrete lattice points to approximate arbitrary points in the lattice. An effective lattice quantizer significantly enhances both the accuracy and efficiency of these approximations. In the context of high-dimensional lattice quantization, previous work proposed utilizing low-dimensional optimal lattice quantizers and addressed the challenge of determining the optimal length ratio in orthogonal splicing. Notably, it was demonstrated that fixed length ratios and orthogonality yield suboptimal results when combining low-dimensional lattices. Building on this foundation, another approach employed gradient descent to identify optimal lattices, which inspired us to explore the use of neural networks to discover matrices that outperform those obtained from orthogonal splicing methods. We propose two novel approaches to tackle this problem: the Household Algorithm and the Matrix Exp Algorithm. Our results indicate that both the Household Algorithm and the Matrix Exp Algorithm achieve improvements in lattice quantizers across dimensions 13, 15, 17 to 19, 21, and 22. Moreover, the Matrix Exp Algorithm demonstrates superior efficacy in high-dimensional settings.