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This work studies our recently developed algorithm, decentralized alternating projected gradient descent algorithm (Dec-AltGDmin), for recovering a low rank (LR) matrix from independent columnwise linear projections in a decentralized setting. This means that the observed data is spread across L agents and there is no central coordinating node. Since this problem is non-convex and since it involves a subspace recovery step, most existing literature from decentralized optimization is not useful. We demonstrate using extensive numerical simulations and communication, time, and sample complexity comparisons that (i) existing decentralized gradient descent (GD) approaches fail, and (ii) other common solution approaches on LR recovery literature – projected GD, alternating GD and alternating minimization (AltMin) – either have a higher communication (and time) complexity or a higher sample complexity. Communication complexity is often the most important concern in decentralized learning.
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Fast Low Rank Column-Wise Compressive Sensing for Accelerated Dynamic MRI
This work develops a novel set of algorithms, al- ternating Gradient Descent (GD) and minimization for MRI (altGDmin-MRI1 and altGDmin-MRI2), for accelerated dynamic MRI by assuming an approximate low-rank (LR) model on the matrix formed by the vectorized images of the sequence. The LR model itself is well-known in the MRI literature; our contribution is the novel GD-based algorithms which are much faster, memory-efficient, and ‘general’ compared with existing work; and careful use of a 3-level hierarchical LR model. By ‘general’, we mean that, with a single choice of parameters, our method provides accurate reconstructions for multiple acceler- ated dynamic MRI applications, multiple sampling rates and sampling schemes. We show that our methods outperform many of the popular existing approaches while also being faster than all of them, on average. This claim is based on comparisons on 8 different retrospectively undersampled multi-coil dynamic MRI applications, sampled using either 1D Cartesian or 2D pseudo- radial undersampling, at multiple sampling rates. Evaluations on some prospectively undersampled datasets are also provided. Our second contribution is a mini-batch subspace tracking extension that can process new measurements and return reconstructions within a short delay after they arrive. The recovery algorithm itself is also faster than its batch counterpart.
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- Award ID(s):
- 2115200
- PAR ID:
- 10407103
- Date Published:
- Journal Name:
- IEEE Transactions on Computational Imaging
- ISSN:
- 2573-0436
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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This work studies our recently developed decentralized algorithm, decentralized alternating projected gradient descent algorithm, called Dec-AltProjGDmin, for solving the following low-rank (LR) matrix recovery problem: recover an LR matrix from independent column-wise linear projections (LR column-wise Compressive Sensing). In recent work, we presented constructive convergence guarantees for Dec-AltProjGDmin under simple assumptions. By "constructive", we mean that the convergence time lower bound is provided for achieving any error level ε. However, our guarantee was stated for the equal neighbor consensus algorithm (at each iteration, each node computes the average of the data of all its neighbors) while most existing results do not assume the use of a specific consensus algorithm, but instead state guarantees in terms of the weights matrix eigenvalues. In order to compare with these results, we first modify our result to be in this form. Our second and main contribution is a theoretical and experimental comparison of our new result with the best existing one from the decentralized GD literature that also provides a convergence time bound for values of ε that are large enough. The existing guarantee is for a different problem setting and holds under different assumptions than ours and hence the comparison is not very clear cut. However, we are not aware of any other provably correct algorithms for decentralized LR matrix recovery in any other settings either.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TCI.2023.3263810
