Exact max-range optimization over the full invertible family

Determine an exact optimization method for the finite-set max-range objective over the full invertible transformation family GL(K), including practical condition-number constraints.

Background

The paper analyzes restricted diagonal, orthogonal, partial-rotation, and hierarchical gauge families, but does not solve the broader optimization problem over all invertible contraction gauges. The unresolved problem is to optimize the finite-set maximum-range objective over GL(K) while incorporating practically relevant condition-number restrictions.

References

In these regimes, the full product-error objective ranks candidate transform chains. Prior work explores gradient-based optimization over affine and structured families \citep{ma2024affinequant,sun2025flatquant}; exact optimization of the finite-set max-range objective over the full invertible family $\mathrm{GL}(K)$, including practical condition-number constraints, remains open.

— Contraction-Gauge Preconditioning for Quantized Matrix Multiplication  (2607.18745 - Sao et al., 21 Jul 2026) in Section 9.2, subsection “Rotation--scaling composition”