Abstract An imprecise dictionary is commonly used in compressed sensing in cases where the dictionary is derived via a learning process or is based on incomplete prior knowledge. This paper investigates the issue of ambiguity in dictionary estimation and the sensitivity of a true sparse vector and signal (i.e., signals represented by true sparse vectors with respect to the true dictionary) to dictionary perturbations in compressed sensing. We first demonstrate that the inherent ambiguity in dictionary estimation cannot be resolved, even when the desired sparse vector can be obtained. By imposing conditions on the perturbation of the true dictionary, we then demonstrate that the sparse vector and signal can both be stably estimated within the bounds of the dictionary perturbation. Our analysis results are based on the condition that AD holds the restricted isometry property (RIP), where A is a precise sensing matrix and D is a dictionary mismatch. The RIP of AD cannot be efficiently determined for a general over-complete dictionary; therefore, we numerically verified our analytical results by conducting experiments under the specific but commonly encountered situation in which D (an orthonormal matrix) and A (a Gaussian sensing matrix), in which the RIP of AD holds with a high degree of probability.

中文翻译：

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压缩感知的不精确词典中的歧义和敏感性

摘要 不精确的字典通常用于压缩感知，在字典是通过学习过程导出的或基于不完整的先验知识的情况下。本文研究了字典估计中的歧义问题以及真实稀疏向量和信号（即，相对于真实字典由真实稀疏向量表示的信号）对压缩感知中字典扰动的敏感性。我们首先证明了字典估计中固有的歧义无法解决，即使可以获得所需的稀疏向量。通过对真实字典的扰动施加条件，然后我们证明稀疏向量和信号都可以在字典扰动的范围内稳定估计。我们的分析结果基于 AD 具有受限等距特性 (RIP) 的条件，其中 A 是精确的传感矩阵，D 是字典失配。对于一般的过完备字典，无法有效确定 AD 的 RIP；因此，我们通过在 D（正交矩阵）和 A（高斯传感矩阵）的特定但常见情况下进行实验，对我们的分析结果进行了数值验证，其中 AD 的 RIP 具有很高的概率。