Greedy-type sparse recovery from heavy-tailed measurements

Recovering a s-sparse signal vector $x \in {\mathbb{C}^n}$ from a comparably small number of measurements $y: = (Ax) \in {\mathbb{C}^m}$ is the underlying challenge of compressed sensing. By now, a variety of efficient greedy algorithms has been established and strong recovery guarantees have been proven for random measurement matrices $A \in {\mathbb{C}^{m \times n}}$.However, they require a strong concentration of A
^∗
Ax around its mean x (in particular, the Restricted Isometry Property), which is generally not fulfilled for heavy-tailed matrices. In order to overcome this issue and even cover applications where only limited knowledge about the distribution of the measurements matrix is known, we suggest substituting A
^∗
Ax by a median-of-means estimator.In the following, we present an adapted greedy algorithm, based on median-of-means, and prove that it can recover any s-sparse unit vector $x \in {\mathbb{C}^n}$ up to a l
2
-error ${\left\| {x - \hat x} \right\|_2} < \in $ with high probability, while only requiring a bound on the fourth moment of the entries of A. The sample complexity is of the order $\mathcal{O}\left( {s\log \left( {n\log \left( {\frac{1}{ \in }} \right)} \right)\log \left( {\frac{1}{ \in }} \right)} \right)$.