Title: Compressive Recovery Defense: A Defense Framework for $\ell_0, \ell_2$ and $\ell_\infty$ norm attacks.. Abstract: We provide recovery guarantees for compressible signals that have been corrupted with noise and extend the framework introduced in \cite{bafna2018thwarting} to defend neural networks against $\ell_0$, $\ell_2$, and $\ell_{\infty}$-norm attacks. In the case of $\ell_0$-norm noise, we provide recovery guarantees for Iterative Hard Thresholding (IHT) and Basis Pursuit (BP). For $\ell_2$-norm bounded noise, we provide recovery guarantees for BP, and for the case of $\ell_\infty$-norm bounded noise, we provide recovery guarantees for Dantzig Selector (DS). These guarantees theoretically bolster the defense framework introduced in \cite{bafna2018thwarting} for defending neural networks against adversarial inputs. Finally, we experimentally demonstrate the effectiveness of this defense framework against an array of $\ell_0$, $\ell_2$ and $\ell_\infty$-norm attacks.   