arXiv:2608.15125 2026 Architecture 2 ideas extracted · analyzed Sep 1, 2026
The paper identifies a highly structured family of positive-semidefinite matrices whose validity is controlled by scalar sequence differences. A Min matrix is a sum of rank-one prefix-indicator outer products, giving an explicit O(n)-parameter PSD representation and an O(n d) matrix-vector product. The paper also proves that entrywise maps preserving Loewner order on these cones are exactly the nondecreasing convex functions, turning a matrix-order constraint into a scalar architectural constraint. These results can be transferred into structured attention, graph message passing, or covariance modules where guaranteed PSD/order behavior is useful.
Replace or augment a dense attention similarity matrix with a Min-cone matrix generated by a monotone scalar sequence. The resulting matrix is positive semidefinite by construction, has only O(n) learned scalar parameters, and can be multiplied by values in O(n d) time using cumulative sums rather than forming an n-by-n matrix.
Apply a trainable scalar gate entrywise to a Min/Max structured affinity or covariance matrix while enforcing that the gate is nonnegative, nondecreasing, and convex. This preserves Loewner ordering on the structured cone and avoids unconstrained elementwise nonlinearities that can destroy PSD or order relations.