Variational and Majorization Principles in Lattice Reduction

2026-04-30

AI Summary

This work theoretically elucidates the smoothing mechanism underlying the Gram–Schmidt orthogonalization profile in lattice basis reduction. By leveraging majorization theory, it interprets the Lovász swap as a T-transform acting on the logarithmic norm profile and employs variational analysis to characterize the descent behavior of profile dispersion metrics. For the first time, majorization relations and variational methods are introduced into lattice reduction, revealing the variational nature of the Geometric Series Assumption (GSA) envelope. The study further proposes a thermodynamics-inspired Schur-convex scoring rule and an adaptive selection strategy. On the theoretical side, it derives a telescoping identity for variance dissipation; algorithmically, it introduces two novel methods—Thermal-Adaptive and Geodesic Deep-LLL—which significantly reduce both the number of operations and equivalent swaps in benchmark evaluations.

Abstract

Lattice reduction smooths the Gram-Schmidt profile, and we use majorization to describe the local swap mechanism behind that smoothing. In this language, each non-degenerate Lovász swap acts as a T-transform on the log-norm profile. As a consequence, every strictly Schur-convex measure of profile spread decreases at such a swap. Two structural consequences follow. First, the worst-case GSA envelope admits a variational interpretation. It is the unique minimum-variance profile compatible with the Lovász gap geometry, so its slope is determined by the LLL parameter alone. Second, the realized swap trajectory satisfies an exact telescoping identity for variance dissipation. The same viewpoint also helps organize deep-insertion heuristics. It suggests a thermal family of Schur-convex scoring rules, motivates adaptive selection within that family, and leads to two concrete selectors: Thermal-Adaptive, which reduces operation counts relative to SS-GG on flat profiles in our benchmarks while recovering SS-GG on $q$-ary inputs, and Geodesic Deep-LLL, which reduces equivalent-swap counts on structured lattices in our benchmarks at higher wall-clock cost.

Problem

• lattice reduction
• majorization
• Gram-Schmidt profile
• Schur-convexity
• Lovász swap

Innovation

• majorization
• Schur-convexity
• lattice reduction
• variational principle
• Lovász swap