双线性观测问题中秩恢复的极限
Limits of Rank Recovery in Bilinear Observation Problems
AI总结：
研究双线性观测问题中秩恢复的极限，揭示问题修改与数值细化的区别。
摘要：
双线性观测问题出现在许多物理和信息论设定中，其中可观测变量和状态以乘积形式出现。基于秩的诊断在这些问题中常用于评估观测可访问的有效维度，通常隐含假设秩不足可以通过数值细化解决。本文通过分析双线性观测算子在系统容差变化下的秩和核性，检验这一假设。我们不聚焦于特定的重建算法，而是直接研究算子并识别出在广泛容差范围内持续存在的扩展秩平台。这些平台表明稳定的维度亏损，不会被固定问题定义内应用的细化程序所消除。为探究此行为的起源，我们将核空间分解为由变量块结构定义的代数扇区。核空间在特定扇区表现出显著但非排他的集中，揭示了内部有组织的结构，而非均匀的维度损失。通过比较细化与显式修改问题表述，进一步显示在报告的设置中，秩恢复需要改变观测问题本身的结构。此处，
英文摘要：
Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure.
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