Shah, Shaik Basheeruddin;
Pradhan, Pradyumna;
Pu, Wei;
Randhi, Ramunaidu;
Rodrigues, Miguel RD;
Eldar, Yonina C;
(2024)
Optimization Guarantees of Unfolded ISTA and ADMM Networks With Smooth Soft-Thresholding.
IEEE Transactions on Signal Processing
, 72
pp. 3272-3286.
10.1109/tsp.2024.3412981.
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Abstract
Solving linear inverse problems plays a crucial role in numerous applications. Algorithm unfolding based, model-aware data-driven approaches have gained significant attention for effectively addressing these problems. Learned iterative soft-thresholding algorithm (LISTA) and alternating direction method of multipliers compressive sensing network (ADMM-CSNet) are two widely used such approaches, based on ISTA and ADMM algorithms, respectively. In this work, we study optimization guarantees, i.e., achieving near-zero training loss with the increase in the number of learning epochs, for finite-layer unfolded networks such as LISTA and ADMM-CSNet with smooth soft-thresholding in an over-parameterized (OP) regime. We achieve this by leveraging a modified version of the Polyak-Łojasiewicz, denoted PL*, condition. Satisfying the PL* condition within a specific region of the loss landscape ensures the existence of a global minimum and exponential convergence from initialization using gradient descent based methods. Hence, we provide conditions, in terms of the network width and the number of training samples, on these unfolded networks for the PL* condition to hold, by deriving the Hessian spectral norm. Additionally, we show that the threshold on the number of training samples increases with the increase in the network width. Furthermore, we compare the threshold on training samples of unfolded networks with that of a standard fully-connected feed-forward network (FFNN) with smooth soft-thresholding non-linearity. We prove that unfolded networks have a higher threshold value than FFNN. Consequently, one can expect a better expected error for unfolded networks than FFNN.
| Type: | Article |
|---|---|
| Title: | Optimization Guarantees of Unfolded ISTA and ADMM Networks With Smooth Soft-Thresholding |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1109/tsp.2024.3412981 |
| Publisher version: | https://doi.org/10.1109/TSP.2024.3412981 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Optimization guarantees, algorithm unfolding, LISTA, ADMM-CSNet, Polyak-Łojasiewicz condition |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Electronic and Electrical Eng |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10198474 |
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