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  "body": "\n## Definition\n\nLet $m \u003c N$ and $s \\ge 1$ be integers and let $A$ be a real $m \\times N$ [[matrix|0VD9ZSWWJG]], called the *measurement matrix*.\nA vector $x \\in \\R^N$ is *$s$-sparse* if it has at most $s$ nonzero entries.\n*Compressed sensing* is the problem of recovering an unknown $s$-sparse vector $x$ from its measurement vector $y = Ax \\in \\R^m$.\nIts standard method is *basis pursuit*: output a solution of\n$$\\min\\set{\\norm{z}_1 : z \\in \\R^N, \\ Az = y}, \\qquad \\norm{z}_1 = \\textstyle\\sum_{j=1}^N \\abs{z_j} .$$\nThe matrix $A$ *allows recovery of $s$-sparse vectors* if every $s$-sparse $x$ is the unique solution of this problem with $y = Ax$.\n\n## Remarks\n\nIt is the case $\\varepsilon = 0$ of [[sparse approximation|WC8P971XGW]], restricted to data that come from a sparse vector.\nSuppose every $2s$ columns of $A$ are linearly independent.\nThen an $s$-sparse $x$ is the only $s$-sparse solution of $Az = Ax$, since two such solutions differ by a $2s$-sparse vector in the kernel of $A$.\nA matrix with the [[restricted isometry property|B0K18PWN2A]] of order $2s$ and constant $\\delta_{2s} \u003c \\sqrt{2} - 1$ allows recovery of $s$-sparse vectors by basis pursuit.\nExample: $A = \\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix}$, $s = 1$, $x = (0, 0, 2)$, $y = (2, 2)$.\nThe solutions of $Az = y$ are $z = (2 - t, 2 - t, t)$, and $\\norm{z}_1 = 2\\abs{2 - t} + \\abs{t}$ is minimal exactly at $t = 2$, so basis pursuit returns $x$.\nNon-example: for $A = \\begin{pmatrix} 1 \u0026 1 \u0026 0 \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}$ and $s = 1$, the $1$-sparse vectors $e_1$ and $e_2$ give the same $y = (1, 0)$, so no method can recover both.\n",
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