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  "body": "\n## Definition\n\nLet $H$ be a real or complex [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$ and norm $\\norm{\\cdot}$.\nLet $\\mathcal{D} \\subseteq H$ be a dictionary of unit vectors, and let $f \\in H$.\n*Matching pursuit* produces residuals $r_k$, elements $g_k \\in \\mathcal{D}$ and approximants $f_k$ as follows.\n\n1. Set $r_0 = f$ and $f_0 = 0$.\n2. For $k \\ge 1$, choose $g_k \\in \\mathcal{D}$ with $\\abs{\\ip{r_{k-1}}{g_k}} = \\sup\\set{\\abs{\\ip{r_{k-1}}{g}} : g \\in \\mathcal{D}}$.\n3. Set $r_k = r_{k-1} - \\ip{r_{k-1}}{g_k} g_k$ and $f_k = f_{k-1} + \\ip{r_{k-1}}{g_k} g_k$.\n\n## Remarks\n\nBy construction $f = f_k + r_k$, and $f_k$ is an [[n-term approximant|GEK0NP5WQM]] with $n = k$.\nSince $r_k$ is orthogonal to $g_k$, $\\norm{r_k}^2 = \\norm{r_{k-1}}^2 - \\abs{\\ip{r_{k-1}}{g_k}}^2$, so the residual norms never increase.\nWhen the supremum in clause 2 is not attained, one uses a weak version that only asks for a fixed fraction $t \\in (0, 1]$ of it.\nExample: in $\\R^2$ with $\\mathcal{D} = \\set{e_1, e_2, u}$, $u = (1, 1)/\\sqrt{2}$, and $f = (2, 1)$.\nThe correlations are $2$, $1$ and $3/\\sqrt{2} \\approx 2.12$, so $g_1 = u$, $f_1 = (3/2, 3/2)$ and $r_1 = (1/2, -1/2)$.\nThen $\\ip{r_1}{u} = 0$ and $\\abs{\\ip{r_1}{e_1}} = \\abs{\\ip{r_1}{e_2}} = 1/2$; choosing $g_2 = e_1$ gives $r_2 = (0, -1/2)$, and $g_3 = e_2$ gives $r_3 = 0$.\nNon-example: choosing $g_1 = e_1$ at the first step fails clause 2, since $\\abs{\\ip{f}{e_1}} = 2 \u003c 3/\\sqrt{2}$.\n*Orthogonal matching pursuit* replaces clause 3 by taking $f_k$ to be the orthogonal projection of $f$ onto the span of $g_1, \\dots, g_k$ and $r_k = f - f_k$.\n",
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