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

2025-12-26 · John M. Mango, Ronald Katende arxiv

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.

📄 PDF Abstract BibTeX arXiv:2512.22084

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Orthogonalized Policy Optimization:Policy Optimization as Orthogonal Projection in Hilbert Space

2026-01-18 · Wang Zixian arxiv

We propose Orthogonalized Policy Optimization (OPO), a principled framework for large language model alignment derived from optimization in the Hilbert function space L2(pi_k). Lifting policy updates from the probability…

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Mathematical Reasoning

Differentiable Trust Region Layers for Deep Reinforcement Learning

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Trust region methods are a popular tool in reinforcement learning as they yield robust policy updates in continuous and discrete action spaces. However, enforcing such trust regions in deep reinforcement learning is diff…

Deep Reinforcement Learning reinforcement-learning Reinforcement Learning Reinforcement Learning (RL)