Computing Data-Aware Diagonal Frame Decompositions

We present a method for computing a specific type of matrix decomposition that combines a structured framework with data-driven adaptation. This framework allows for an efficient and stable representation of signals or images using frames by identifying sets of vectors that naturally capture the essential features of the given data. The resulting decomposition enables the development of regularization techniques that enhance the quality of reconstruction by preserving important signal characteristics while suppressing noise. We develop an optimization-based algorithm to compute the decomposition and establish convergence properties of the resulting iterates. Numerical experiments on synthetic data and tomographic image reconstruction illustrate the behavior of the proposed method and its performance compared to conventional spectral regularization methods.