Recovery of Low Rank and Jointly Sparse Matrices with Two Sampling Matrices

Subspace Aware Recovery of Low Rank and Jointly Sparse Signals

“…The matrix is uniquely identified, provided spark( D ) ≥ 2 k − r + 1 and the matrix Q 1 is full rank. We observe that this is the same condition we obtained in our earlier work [22]. As described earlier, this approach results in an asymmetric sampling pattern, where some frames are measured at a high rate.…”

“…As X has N columns, the total number of observations thus equals $${\mathcal{N}}_{\text{MMV}}\ge false(2k-r+1false)N.$$We will show that the combined number of measurements required to estimate P and Q and hence X is considerably smaller, particularly when the rank of X is small. We also observe that our earlier work in [22] employs two measurement matrices and also requires fewer measurements than (19). We contrast the approach in this paper with that in [22] at the end of Section IV.…”

“…We also observe that our earlier work in [22] employs two measurement matrices and also requires fewer measurements than (19). We contrast the approach in this paper with that in [22] at the end of Section IV.…”

“…This work generalizes our earlier work on the recovery of jointly sparse and low-rank matrix, where we first recover the column subspace, followed by subspace aware recovery of the matrix [22]. In [22], we require some of the columns to be sampled at a high rate (require around $\mathrm{scriptO}\left(k\right)$ measurements) for column subspace estimation, followed by very low sampling rate $\mathrm{scriptO}\left(r\right)$ for the remaining frames [22].…”

“…In [22], we require some of the columns to be sampled at a high rate (require around $\mathrm{scriptO}\left(k\right)$ measurements) for column subspace estimation, followed by very low sampling rate $\mathrm{scriptO}\left(r\right)$ for the remaining frames [22]. This approach may not be ideally suited for dynamic imaging applications and has not been used in the MRI context before, where the time available to measure every frame is limited.…”