Minimization of the largest normalized higher-order determinant

Determine which singular-vector geometries minimize the largest attainable normalized determinant for higher exterior powers, thereby resolving whether highly incoherent singular vectors are the minimizing configurations.

Background

The normalized pivot determinant measures the conditioning and mutual alignment of selected row and column frames in the singular-value-weighted geometry. After establishing upper and lower determinant bounds, the paper asks which singular-vector configurations make the largest attainable normalized determinant as small as possible. Rank-one analysis in Appendix A suggests that highly incoherent configurations may be natural candidates, but the analogous optimization problem for higher exterior powers is left unresolved.

References

A complementary question is which singular-vector geometries minimize the largest attainable normalized determinant. The rank-one analysis in Appendix \ref{apx:cosine_vs_coherence} suggests highly incoherent configurations as natural candidates; the corresponding problem for higher exterior powers remains open, but we believe the answer will similarly be highly incoherent singular vectors.

— A Geometric View of Adaptive Cross Approximation via Exterior Algebra  (2609.17947 - Loe et al., 16 Sep 2026) in Section 2, subsection “The normalized pivot determinant”