A Fast Algorithm for Low Rank + Sparse column-wise Compressive Sensing

This paper focuses studies the following low rank + sparse (LR+S) column-wise compressive sensing problem. We aim to recover an $n \times q$ matrix, $\X^* =[ \x_1^*, \x_2^*, \cdots , \x_q^*]$ from $m$ independent linear projections of each of its $q$ columns, given by $\y_k :=\A_k\x_k^*$, $k \in [q]$. Here, $\y_k$ is an $m$-length vector with $m < n$. We assume that the matrix $\X^*$ can be decomposed as $\X^*=\L^*+\S^*$, where $\L^*$ is a low rank matrix of rank $r << \min(n,q)$ and $\S^*$ is a sparse matrix. Each column of $\S$ contains $\rho$ non-zero entries. The matrices $\A_k$ are known and mutually independent for different $k$. To address this recovery problem, we propose a novel fast GD-based solution called AltGDmin-LR+S, which is memory and communication efficient. We numerically evaluate its performance by conducting a detailed simulation-based study.

Fast Low Rank column-wise Compressive Sensing for Accelerated Dynamic MRI

This work develops a novel set of algorithms, alternating 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 accelerated 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 under sampled multi-coil dynamic MRI applications, sampled using either 1D Cartesian or 2D pseudo radial under sampling, at multiple sampling rates. Evaluations on some prospectively under sampled 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.