arXiv:2607.13251 2026 Architecture 1 ideas extracted · analyzed Aug 30, 2026
The paper gives a principled way to construct bilinear matrix-valued maps that preserve positive semidefiniteness not only for individual inputs, but also for arbitrary block-matrix lifts of those inputs. The transferable asset is the Choi/Kraus characterization: every jointly completely positive product has the explicit form \(\Phi(A,B)=\sum_r V_r^*(A\otimes B)V_r\), which is a PSD-preserving bilinear feature-fusion layer with a directly controllable low-rank parameterization. This is most useful in covariance prediction, kernel/Gram-matrix networks, uncertainty heads, and attention-like modules where an invalid indefinite output causes instability. A practical first test is to replace an unconstrained bilinear covariance head by a low-rank completely positive layer and measure PSD violations, calibration, and accuracy at matched parameter count.
Replace an unconstrained bilinear matrix fusion or covariance head with \(\Phi(A,B)=\sum_{r=1}^R V_r^*(A\otimes B)V_r\). The output is PSD by construction, and the stronger block-level property makes the layer compatible with minibatches, mixtures, and Gram-matrix inputs rather than merely preserving positivity pointwise.