代数信号模型中可迁移性与等变性的基本极限 I：有限维
Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions
- Department of Electrical Engineering, University of Colorado (Denver)（科罗拉多大学丹佛分校电气工程系）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文通过代数信号模型间的同态研究可迁移性的基本极限，引入谱迁移效率量化信息保持，并证明其在采样、压缩感知和机器学习中的具体应用。
中文摘要 AI 辅助
我们通过代数信号模型之间的同态来研究代数信号处理中可迁移性的基本极限。同态是两个模型的信号空间之间的线性映射，且与滤波操作可交换，因此对信号进行滤波和跨域传输可以按任意顺序进行。此类映射的存在性完全由两个模型移位算子的滤波特征值之间的重合决定，但仅存在性是不够的：同态空间总是包含破坏所有信息的平凡元素。我们引入谱迁移效率 $\eta(\theta)\in[0,1]$ 来量化信息保持质量，证明每个同态可分解为重合类上的无约束块，推导出同态空间的维度，并精确刻画无损迁移何时可实现。在正规移位算子之外，我们量化了偏离正规性的惩罚，并展示滤波器导数如何修复谱缺陷。该理论在三种情形中产生具体结果：对于采样，特征值交错将子采样下的可迁移性转化为显式的滤波器设计约束；对于压缩感知，$\eta(\theta)$ 控制受限等距常数和所得测量的相干性，且恢复在重合类上解耦；对于机器学习，谱混叠作为受控的对称性破缺出现，使得不匹配域之间的迁移成为可能。
英文摘要
We study the fundamental limits of transferability in algebraic signal processing through homomorphisms between algebraic signal models. Homomorphisms are linear maps between the signal spaces of two models that commute with filtering, so filtering a signal and transferring it across domains can be done in either order. The existence of such maps is governed entirely by coincidences among the filtered eigenvalues of the two models' shift operators, but existence alone is insufficient: the space of homomorphisms always contains trivial elements that destroy all information. We introduce the spectral transfer efficiency $η(θ)\in[0,1]$ to quantify information-preserving quality, prove that every homomorphism decomposes into unconstrained blocks over coincidence classes, derive the dimension of the homomorphism space, and characterize exactly when lossless transfer is achievable. Beyond normal shift operators, we quantify a departure-from-normality penalty and show how filter derivatives can repair spectral defectiveness. The theory yields concrete consequences in three settings: for sampling, eigenvalue interlacing converts transferability under subsampling into an explicit filter design constraint; for compressed sensing, $η(θ)$ controls the restricted isometry constant and the coherence of the resulting measurements, and recovery decouples across coincidence classes; and for machine learning, spectral aliasing emerges as the controlled symmetry breaking that makes transfer between mismatched domains possible at all.
