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  "body": "\n## Definition\n\nLet $A$ be an $m \\times N$ real or complex [[matrix|0VD9ZSWWJG]], usually with $m \u003c N$, let $b \\in \\mathbb{K}^m$ with $\\mathbb{K} = \\R$ or $\\C$, and let $\\norm{\\cdot}_2$ be the Euclidean norm.\nFor $x \\in \\mathbb{K}^N$, let $\\norm{x}_0$ be the number of nonzero entries of $x$; the vector $x$ is *$s$-sparse* if $\\norm{x}_0 \\le s$.\n*Sparse approximation* is either of the two following problems.\n\n1. Given an integer $s \\ge 1$, find an $s$-sparse $x$ that minimizes $\\norm{Ax - b}_2$.\n2. Given a tolerance $\\varepsilon \\ge 0$, find $x$ with $\\norm{Ax - b}_2 \\le \\varepsilon$ that minimizes $\\norm{x}_0$.\n\nFor $\\varepsilon = 0$, the second problem asks for the sparsest solution of the [[system of linear equations|0GT4WWGWH2]] $Ax = b$.\n\n## Remarks\n\nThe count $\\norm{x}_0$ is not a norm, since $\\norm{2x}_0 = \\norm{x}_0$; the notation is standard anyway.\nExample: let $A = \\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix}$, $b = (1, 1)$ and $\\varepsilon = 0$.\nThe vector $x = (0, 0, 1)$ solves $Ax = b$ with $\\norm{x}_0 = 1$, and no $x$ with $\\norm{x}_0 = 0$ works since $b \\ne 0$, so it solves problem 2.\nNon-example: $x = (1, 1, 0)$ also solves $Ax = b$, but $\\norm{x}_0 = 2$ is not minimal, so it is not a solution of problem 2.\nReplacing $\\norm{x}_0$ by the convex norm $\\norm{x}_1 = \\sum_j \\abs{x_j}$ gives basis pursuit, a linear program; a penalized form of it is the [[Lasso|KZ226116HW]].\n",
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