单位范数紧框架稀疏坐标补全的秩稳定性
Rank Stabilization for Sparse Coordinate Completion of Unit-Norm Tight Frames
- Hefei University of Technology（合肥工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究单位范数紧框架在每列缺失两个坐标时的补全秩稳定性，通过拟阵与子空间排列证明秩独立于长度，并给出缺陷公式与稳定化阈值下界。
AI中文摘要：
我们确定了当每个部分观测列恰好有两个缺失坐标时，单位范数紧框架的通用补全纤维维数。我们在由实框架方程定义的复代数簇上工作。缺失坐标对构成一个带标记的多重图。对于每个 $d\ge4$ 和每个框架长度 $R\ge N_d:=\binom{d+1}{2}-1$，我们证明了缺失对雅可比拟阵是某个显式子空间排列的Rado拟阵。因此，其秩由图论最小公式给出，并且与 $R$ 无关；互补的秩缺陷即为通用纤维维数。主要步骤是在长度 $N_d$ 处的实现定理。它结合了横贯-图拟阵划分、零坐标矩浸没以及一个通用残差矩阵的相容分解。基扩展和在全观测列下的稳定性随后在每个更大的长度上产生秩公式。我们还给出了四维情况下的显式缺陷公式，以及在光滑实框架轨迹上的局部实对应结果。一个公共旋转障碍将最小均匀稳定化阈值从下方界定为 $2d-1$。
英文摘要：
We determine generic completion-fiber dimensions for unit-norm tight frames when each partially observed column has exactly two missing coordinates. We work on the complex algebraic variety defined by the real frame equations. The missing coordinate pairs form a labelled multigraph. For every $d\ge4$ and every frame length $R\ge N_d:=\binom{d+1}{2}-1$, we prove that the missing-pair Jacobian matroid is the Rado matroid of an explicit subspace arrangement. Its rank is therefore given by a graph-theoretic minimum formula and is independent of $R$; the complementary rank defect is the generic fiber dimension. The main step is a realization theorem at length $N_d$. It combines a transversal--graphic matroid partition with a zero-coordinate moment submersion and a compatible decomposition of a generic residual matrix. Basis extension and stability under fully observed columns then yield the rank formula at every larger length. We also give an explicit defect formula in dimension four and a local real counterpart on the smooth real frame locus. A common-rotation obstruction bounds the smallest uniform stabilization threshold from below by $2d-1$.
