John M. Mango | Search arXiv

arXiv2025

A Frobenius-Optimal Projection for Enforcing Linear Conservation in Learned Dynamical Models

We consider the problem of restoring linear conservation laws in data-driven linear dynamical models. Given a learned operator $\widehat{A}$ and a full-rank constraint matrix $C$ encoding one or more invariants, we show that the matrix closest to $\widehat{A}$ in the Frobenius norm and satisfying $C^\top A = 0$ is the orthogonal projection $A^\star = \widehat{A} - C(C^\top C)^{-1}C^\top \widehat{A}$. This correction is uniquely defined, low rank and fully determined by the violation $C^\top \widehat{A}$. In the single-invariant case it reduces to a rank-one update. We prove that $A^\star$ enforces exact conservation while minimally perturbing the dynamics, and we verify these properties numerically on a Markov-type example. The projection provides an elementary and general mechanism for embedding exact invariants into any learned linear model.

math.DS↗

arXiv2025

Preserving Extreme Singular Values with One Oblivious Sketch

We study when a single linear sketch can control the largest and smallest nonzero singular values of every rank-$r$ matrix. Classical oblivious embeddings require $s=Θ(r/\varepsilon^{2})$ for $(1\pm\varepsilon)$ distortion, but this does not yield constant-factor control of extreme singular values or condition numbers. We formalize a conjecture that $s=O(r\log r)$ suffices for such preservation. On the constructive side, we show that combining a sparse oblivious sketch with a deterministic geometric balancing map produces a sketch whose nonzero singular values collapse to a common scale under bounded condition number and coherence. On the negative side, we prove that any oblivious sketch achieving relative $\varepsilon$-accurate singular values for all rank-$r$ matrices must satisfy $s=Ω((r+\log(1/δ))/\varepsilon^{2})$. Numerical experiments on structured matrix families confirm that balancing improves conditioning and accelerates iterative solvers, while coherent or nearly rank-deficient inputs manifest the predicted failure modes.

math.NA↗