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The cosparse analysis model for signals assumes that the signal of interest can be multiplied by an analysis dictionary , leading to a sparse outcome. This model stands as an interesting alternative to the more classical synthesis-based sparse representation model. In this paper, we propose a theoretical study of the performance guarantee of the thresholding algorithm for the pursuit problem in the presence of noise. Our analysis reveals two significant properties of &#x03A9;, which govern the pursuit performance: the first is the degree of linear dependencies between sets of rows in &#x03A9;, depicted by the cosparsity level. The second property, termed the restricted orthogonal projection property, is the level of independence between such dependent sets and other rows in &#x03A9;. We show how these dictionary properties are meaningful and useful, both in the theoretical bounds derived and in a series of experiments that are shown to align well with the theoretical prediction.

Performance Guarantees of the Thresholding Algorithm for the Cosparse Analysis Model