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  "body": "\n## Summary\n\nLet $A$ be a matrix with more columns than rows and let $b$ be a vector.\nSparse approximation looks for a vector $x$ with few nonzero entries such that $Ax$ is close to $b$.\nThe columns of $A$ form a redundant dictionary, so $b$ has many representations, and one wants the sparsest.\nThe exact problem is NP-hard in general.\nIn practice one uses greedy methods such as [[matching pursuit|8KJ9P9KQ5H]], or convex relaxations such as basis pursuit and the [[Lasso|KZ226116HW]].\nIt is the core of signal compression, denoising and [[compressed sensing|6V8E0AQNHA]].\n\n## Explanations\n\n### L1 human\n\nThink of the columns of a [[matrix|0VD9ZSWWJG]] $A$ as building blocks, and of $b$ as a signal to build.\nWhen there are more blocks than dimensions, the [[system of linear equations|0GT4WWGWH2]] $Ax = b$ has many solutions.\nSparse approximation asks for a solution, or a near solution, that uses as few blocks as possible.\nFor example, with $A = \\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix}$ and $b = (1, 1)$, both $x = (1, 1, 0)$ and $x = (0, 0, 1)$ solve $Ax = b$.\nThe second one uses a single column, so it is the sparsest solution.\n",
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