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  "body": "\n## Summary\n\nCompressed sensing is the recovery of a sparse vector from far fewer linear measurements than its length.\nLet $x \\in \\R^N$ have at most $s$ nonzero entries, and let $y = Ax$ for a suitable $m \\times N$ matrix $A$ with $m$ much smaller than $N$.\nThen $x$ can be found from $y$ by solving a convex problem.\nThe key sufficient condition on $A$ is the [[restricted isometry property|B0K18PWN2A]], which random matrices satisfy with high probability when $m$ is of order $s \\log(N/s)$.\nIt was developed by Candès, Romberg, Tao and Donoho in the mid-2000s and is used in medical imaging and signal processing.\n\n## Explanations\n\n### L1 human\n\nUsually, to know a vector of length $N$ one must measure $N$ numbers.\nIf one knows that only a few entries are nonzero, much fewer measurements can be enough, as long as they mix the entries well.\nThe reconstruction looks for the [[sparse|WC8P971XGW]] vector that agrees with the measurements.\nFor example, with the $2 \\times 3$ matrix $A = \\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix}$, any vector with one nonzero entry is determined by its two measurements $Ax$.\nAmong such vectors, the measurement $(2, 2)$ can only come from $(0, 0, 2)$, and the measurement $(3, 0)$ only from $(3, 0, 0)$.\n",
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