The Geometry of LISTA

Hans van Gorp, Oisín Nolan, Yonina C Eldar, and Ruud J G van Sloun, “The Geometry of LISTA,” IEEE Signal Processing Magazine, August 2026. [link] [pdf] [code]

The celebrated iterative soft-thresholding algorithm (ISTA) and its accelerated variant, fast ISTA (FISTA), are classical signal processing methods used to solve the least absolute shrinkage and selection operator (LASSO) problem [1]. This problem spans various applications, including medical imaging, direction-ofarrival estimation, astronomy, and sparse coding. Within the broader trend in signal processing of transitioning from classical model-based approaches to deep learning-based methods, the learned ISTA (LISTA) algorithm was proposed as a way to solve the LASSO problem using deep learning while preserving the original ISTA structure [2]. LISTA learns a fast approximation to the LASSO problem by casting the weight matrices of the ISTA algorithm as learnable parameters and making them unique for each iteration, a technique now known as deep unfolding [3]. LISTA achieves superior reconstruction results in fewer iterations compared to ISTA. This is due to two main factors. First, by learning its weights, LISTA addresses potential modeling mismatches, such as imperfect knowledge of noise behavior or the forward model. Second, even with an accurately known model, the sparsifying basis might be too complex to implement efficiently using classical methods. Here, we consider a geometric interpretation to gain insight into why LISTA performs well with significantly fewer iterations (or folds) compared to ISTA. To that end, we use the fact that both models are continuous piecewise linear (CPWL) functions.