Minimum-correction second-moment matching: theory, algorithms and applications

We address the problem of finding closest matrix $$\tilde{\varvec{U}}$$ to a given $$\varvec{U}$$ under constraint that prescribed second-moment $$\tilde{\varvec{P}}$$ must be matched, i.e. $$\tilde{\varvec{U}}^{\mathrm {T}}\tilde{\varvec{U}}=\tilde{\varvec{P}}$$ . obtain closed-form formula for unique global optimizer full-rank case, is related by an SPD (symmetric positive definite) linear transform. This result generalized rank-deficient cases as well infinite dimensions. highlight geometric intuition behind theory and study problem’s rich connections minimum congruence transform, polar decomposition, optimal transport, data assimilation. In special case $$\tilde{\varvec{P}}=\varvec{I}$$ , minimum-correction matching reduces well-studied orthonormalization problem. investigate general strategies numerically computing analyze existing decomposition square root algorithms. modify stabilize two Newton iterations previously deemed unstable root, such they can now used efficiently compute both orthogonal factor root. then verify higher performance various new algorithms using benchmark with randomly generated matrices. Lastly, we complete applications stochastic Lorenz-96 dynamical system in chaotic regime. reduced subspace tracking dynamically equations, maintain numerical orthonormality continuity time-varying base vectors. ensemble filtering assimilation, prior samples are transformed into posterior ones covariance Kalman update while also minimizing corrections samples.