An approximate sparse recovery system in $\ell_1$ norm consists of parameters $k$, $ε$, $N$, an $m$-by-$N$ measurement $Φ$, and a recovery algorithm, $\mathcal{R}$. Given a vector, $\mathbf{x}$, the system approximates $x$ by $\widehat{\mathbf{x}} = \mathcal{R}(Φ\mathbf{x})$, which must satisfy $\|\widehat{\mathbf{x}}-\mathbf{x}\|_1 \leq (1+ε)\|\mathbf{x}-\mathbf{x}_k\|_1$. We consider the 'for all' model, in which a single matrix $Φ$, possibly 'constructed' non-explicitly using the probabilistic method, is used for all signals $\mathbf{x}$. The best existing sublinear algorithm by Porat and Strauss (SODA'12) uses $O(ε^{-3} k\log(N/k))$ measurements and runs in time $O(k^{1-α}N^α)$ for any constant $α> 0$. In this paper, we improve the number of measurements to $O(ε^{-2} k \log(N/k))$, matching the best existing upper bound (attained by super-linear algorithms), and the runtime to $O(k^{1+β}\textrm{poly}(\log N,1/ε))$, with a modest restriction that $ε\leq (\log k/\log N)^γ$, for any constants $β,γ> 0$. When $k\leq \log^c N$ for some $c>0$, the runtime is reduced to $O(k\textrm{poly}(N,1/ε))$. With no restrictions on $ε$, we have an approximation recovery system with $m = O(k/ε\log(N/k)((\log N/\log k)^γ+ 1/ε))$ measurements.