Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems
This work addresses a practical problem for researchers and practitioners in signal processing and machine learning who need efficient solutions for inverse problems, though it is incremental as it builds on existing techniques in sparse recovery and deep learning.
The paper tackles the trade-off between convergence speed and reconstruction accuracy in iterative algorithms for inverse problems, showing that using a coarse estimate of a low-dimensional set can accelerate convergence while introducing an additional error related to set approximation accuracy.
Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).