Structured Preconditioning in Affine-Invariant Geometry: Projection, Certificates, and Kronecker Separation

What the math gives to ML

The paper supplies an intrinsic alternative to ordinary Euclidean Kronecker fitting for SPD curvature matrices: project a full SPD target onto the geodesically convex family A tensor B using affine-invariant distance. The transferable asset is that the projection is unique, has logarithmic partial-trace normal equations, and can be used as a certified structured preconditioner rather than merely a covariance approximation. This is relevant for layerwise Hessian, Fisher, or gradient-covariance matrices whose inverse is too expensive to store. A second useful asset is the spectral certificate distinguishing nearest structured approximation from true condition-number minimization, enabling adaptive checks of whether a Kronecker preconditioner is actually improving optimization conditioning.