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  "body": "\n## Summary\n\nA matrix has the restricted isometry property of order $s$ when it nearly preserves the length of every vector with at most $s$ nonzero entries.\nThe size of the distortion is measured by the restricted isometry constant $\\delta_s$.\nIf $\\delta_{2s}$ is small, $s$-sparse vectors can be recovered stably from their images by $\\ell^1$ minimization.\nThis is the basis of compressed sensing.\nComputing $\\delta_s$ exactly is hard, but random Gaussian matrices have small constants with high probability.\n\n## Explanations\n\n### L1 human\n\nA wide [[matrix|0VD9ZSWWJG]] must squash some vectors to zero, because it maps a big space into a small one.\nThe restricted isometry property says that it still keeps the lengths of all sparse vectors almost unchanged.\nIf this holds for all $2s$-sparse vectors, then two different $s$-sparse vectors cannot have the same image.\nThey even stay well separated after the map.\nFor example, the identity matrix keeps every length exactly, so $\\delta_s = 0$ for every $s$.\nA matrix with two equal columns sends $e_1 - e_2$, which has only two nonzero entries, to $0$, so it fails the property of order $2$.\n",
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