用算子理论揭示约束下优化的几何机制，统一梯度投影与多目标兼容性。
First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation
- 用自伴算子建模计算限制，定义可到达子空间。
- 最优方向为加权梯度，约束导致上升几何畸变。
- 揭示主导谱模式压缩，适用于多目标优化场景。
在结构约束下的优化通常通过投影或惩罚方法分析，掩盖了约束如何塑造可行动态的几何机制。本文提出一种算子理论框架，将计算或可行性限制编码为定义局部可到达子空间的自伴算子。在此框架中，最优一阶改进方向表现为伪逆加权梯度，揭示了约束如何诱导畸变的上升几何。我们进一步证明有效动态集中于主导谱模式，从而给出谱压缩的合理定义，并建立兼容性原理，刻画多个目标间共同可行方向的存在性。该框架将梯度投影、谱截断与多目标可行性统一于同一几何结构之中。
原文摘要 · Abstract (English)
Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。