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An approximate sparse recovery system in $\ell_1$ norm consists of parameters$k$, $\epsilon$, $N$, an $m$-by-$N$ measurement $\Phi$, and a recoveryalgorithm, $\mathcal{R}$. Given a vector, $\mathbf{x}$, the system approximates$x$ by $\widehat{\mathbf{x}} = \mathcal{R}(\Phi\mathbf{x})$, which must satisfy$\|\widehat{\mathbf{x}}-\mathbf{x}\|_1 \leq(1+\epsilon)\|\mathbf{x}-\mathbf{x}_k\|_1$. We consider the 'for all' model, inwhich a single matrix $\Phi$, possibly 'constructed' non-explicitly using theprobabilistic method, is used for all signals $\mathbf{x}$. The best existingsublinear algorithm by Porat and Strauss (SODA'12) uses $O(\epsilon^{-3}k\log(N/k))$ measurements and runs in time $O(k^{1-\alpha}N^\alpha)$ for anyconstant $\alpha > 0$.  In this paper, we improve the number of measurements to $O(\epsilon^{-2} k\log(N/k))$, matching the best existing upper bound (attained by super-linearalgorithms), and the runtime to $O(k^{1+\beta}\textrm{poly}(\logN,1/\epsilon))$, with a modest restriction that $\epsilon \leq (\log k/\logN)^{\gamma}$, for any constants $\beta,\gamma > 0$. When $k\leq \log^c N$ forsome $c>0$, the runtime is reduced to $O(k\textrm{poly}(N,1/\epsilon))$. Withno restrictions on $\epsilon$, we have an approximation recovery system with $m= O(k/\epsilon \log(N/k)((\log N/\log k)^\gamma + 1/\epsilon))$ measurements.

For-all Sparse Recovery in Near-Optimal Time