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  "body": "\n## Summary\n\n$n$-term approximation approximates a function or vector by a linear combination of at most $n$ elements.\nThese elements are chosen freely from a fixed family, called a dictionary, such as a basis.\nThe best error with $n$ terms, $\\sigma_n(f)$, measures how well $f$ can be compressed with respect to that dictionary.\nThe set of $n$-term combinations is not a linear space, so this is a basic model of [[nonlinear approximation|YS1GRQRVY7]].\nIt underlies wavelet compression, [[sparse approximation|WC8P971XGW]] and greedy algorithms such as [[matching pursuit|8KJ9P9KQ5H]].\n\n## Explanations\n\n### L1 human\n\nIn ordinary approximation one fixes in advance which basis functions to use, for example the first $n$ Fourier modes.\nIn $n$-term approximation one may pick any $n$ basis functions, and the choice depends on the target.\nA signal with a few sharp features is well described by a few well-placed [[wavelets|4PZ35C0GV3]], even if they are not the first ones.\nFor an [[orthonormal basis|0S9GGTR0EX]] of a [[Hilbert space|HM2HHE3J51]] the best choice is simple: keep the $n$ coefficients of largest absolute value.\nFor example, $f = 3e_1 + e_2 + 2e_3$ in $\\R^3$ has best $1$-term approximation $3e_1$, with error $\\sqrt{1^2 + 2^2} = \\sqrt{5}$.\n",
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