Optimal configuration of PRISM’s spectral shaping (single-sided vs. dual-sided, and dimension choice)

Determine whether dual-sided spectral preconditioning—applying innovation-augmented polar decomposition on both row and column correlations—is superior to single-sided preconditioning in the PRISM optimizer, and, if single-sided preconditioning is used, identify which dimension should be targeted for optimal performance, specifically whether to apply left-sided (row-correlation) or right-sided (column-correlation) preconditioning.

Background

PRISM introduces innovation-augmented polar decomposition to perform anisotropic spectral shaping with minimal overhead. The method can be implemented in single-sided form (left- or right-sided) to target either row or column correlations, and in principle could be extended to a dual-sided form akin to Kronecker-factored preconditioners.

The authors currently default to a single-sided configuration for efficiency but explicitly note that the choice of sidedness and the optimal dimension to target remain unresolved. Establishing the best configuration would clarify performance–efficiency trade-offs and guide practical deployments of PRISM.

References

While our current method defaults to a single-sided approach for efficiency—contrasting with the dual-sided preconditioners in Kronecker-factored methods—the optimal configuration remains an open question. Specifically, whether a dual-sided approach is superior , and which dimension is optimal to target in the single-sided regime, are yet to be determined.

— PRISM: Structured Optimization via Anisotropic Spectral Shaping  (2602.03096 - Yang, 3 Feb 2026) in Future Work

The cross-spectrum constant that would close this gap, and the analytic bridge from $\operatorname{tr}M$ to the anisotropy metric, are open (Section~\ref{sec:discussion}).

— Spectrum Is Not Enough: A Random-Matrix Account of Data-Free Preconditioning for Differentially Private Learning  (2610.21312 - Zhang et al., 15 Oct 2026) in Section 5, subsection “Spectral shape matching is not the operative object”