(1 + eps)-Approximate Sparse Recovery

The problem central to sparse recovery and compressive sensing is that of \emph{stable sparse recovery}: we want a distribution $\math cal{A}$ of matrices $A \in \R^{m \times n}$ such that, for any $x \in \R^n$ and with probability $1 - \delta >, 2/3$ over $A \in \math cal{A}$, there is an algorithm to recover $\hat{x}$ from $Ax$ with\begin{align} \norm{p}{\hat{x} - x} \leq C \min_{k\text{-sparse } x'} \norm{p}{x - x'}\end{align}for some constant $C >, 1$ and norm $p$. The measurement complexity of this problem is well understood for constant $C >, 1$. However, in a variety of applications it is important to obtain $C = 1+\eps$ for a small $\eps >, 0$, and this complexity is not well understood. We resolve the dependence on $\eps$ in the number of measurements required of a $k$-sparse recovery algorithm, up to polylogarithmic factors for the central cases of $p=1$ and $p=2$. Namely, we give new algorithms and lower bounds that show the number of measurements required is $k/\eps^{p/2} \textrm{polylog}(n)$. For $p=2$, our bound of $\frac{1}{\eps}k\log (n/k)$ is tight up to \emph{constant} factors. We also give matching bounds when the output is required to be $k$-sparse, in which case we achieve $k/\eps^p \textrm{polylog}(n)$. This shows the distinction between the complexity of sparse and non-sparse outputs is fundamental.