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  "body": "\n## Summary\n\nA tensor decomposition writes a multi-way array as a combination of simpler pieces.\nIt plays the role that a matrix factorization plays for matrices.\nThe canonical polyadic (CP) decomposition writes the tensor as a sum of simple tensors.\nThe Tucker decomposition combines a small core tensor with one factor matrix per mode.\nThe shortest CP decomposition has length equal to the [[tensor rank|W84NTF77M0]].\nTensor decompositions are used to compress data and to find hidden structure in signal processing, statistics and chemistry.\n\n## Explanations\n\n### L1 human\n\nThink of a cube of numbers, such as ratings indexed by user, product and month.\nA CP decomposition writes the cube as a sum of a few \"outer products\" of three vectors, one for each direction.\nEach term is a simple pattern: a group of users who like a group of products during certain months.\nFor example, the cube with entries $a_{ijk} = x_i y_j z_k$ is a single term, so it has a decomposition with one piece.\nThis extends the idea of the [[singular value decomposition|000NPTMX6J]] from tables to cubes.\n",
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