arXiv:2608.15444 2026 Regularization 1 ideas extracted · analyzed Sep 1, 2026
The paper studies the convex cone of symmetric matrices whose every d-by-d principal submatrix is positive semidefinite, a hierarchy that is cheaper to enforce than global positive semidefiniteness. The transferable asset is the explicit local constraint: for d=2, every pair of coordinates satisfies a determinant inequality without requiring a full eigendecomposition. A practical neural-network use is to impose this as a soft or exact parameterization on learned similarity, covariance, metric, or preconditioner matrices, obtaining pairwise-valid geometry at quadratic rather than cubic cost. Because local positive semidefiniteness does not imply global positive semidefiniteness, experiments should explicitly measure whether the cheaper constraint improves stability or conditioning without excessive loss of expressivity.
Replace a costly global PSD constraint on a learned symmetric similarity or covariance matrix with the paper's 2-local PSD constraint. Every 2-by-2 principal submatrix is guaranteed valid, preventing excessively large pairwise correlations while avoiding eigendecomposition or Cholesky factorization of the full matrix.