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  "body": "\n## Problem\n\nLet $F$ be a field and let $m, n \\ge 1$ be integers.\nLet $\\Omega$ be a set of positions $(i, j)$ with $1 \\le i \\le m$ and $1 \\le j \\le n$.\nFor each $(i, j) \\in \\Omega$, let a value $M_{ij} \\in F$ be given.\nFind an $m \\times n$ [[matrix|0VD9ZSWWJG]] $X$ over $F$ such that $X_{ij} = M_{ij}$ for all $(i, j) \\in \\Omega$ and the [[rank|0ZC9YTXZRY]] of $X$ is as small as possible.\n\n## Remarks\n\nWithout more structure the problem has no unique answer: the entries outside $\\Omega$ can be anything, so one asks for the completion of lowest rank.\nLow-rank completion is NP-hard in general.\nOver $\\R$, a standard relaxation replaces the rank by the nuclear norm, the sum of the [[singular values|09VMDSRXWJ]] of $X$, which gives a convex problem.\nCandès and Recht proved in 2009 that nuclear norm minimization recovers the unknown matrix exactly with high probability, under some conditions.\nThe matrix must have low rank and satisfy an incoherence condition, and enough entries must be sampled at random.\nOther versions ask for a completion that is positive semidefinite, or that has another prescribed property.\n\n## Explanations\n\n### L1 human\n\nImagine a table of ratings, one row per user and one column per film, where most cells are empty.\nMatrix completion asks to fill the empty cells in the simplest way, meaning with a matrix of small rank.\nFor example, if the known entries are $X_{11} = 1$, $X_{12} = 2$ and $X_{21} = 3$, the completion of rank $1$ is forced: $X_{22} = 6$.\nThe problem became famous through recommender systems such as the Netflix Prize.\n",
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      "first_stated": "2009",
      "posed_by": "Emmanuel Candès and Benjamin Recht"
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    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
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        "what": "a source with a locator",
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