Greedy-type sparse recovery from heavy-tailed measurements

Tim Fuchs, Felix Krahmer, Richard Kueng

Abstract:

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 $\|x-\hat{x}\|_2<\epsilon$ with high probability, while only requiring a bound on the fourth moment of the entries of $A$. The sample complexity is of the order $\mathcal{O}(s\log (n\log(\frac{1}{\epsilon}))\log(\frac{1}{\epsilon}))$.

Submission Type: Full Paper