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Despite the success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-$k$ tensor $A_k$ achieving the above infinum. 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Woodruff, Peilin Zhong, Zhao Song","submitted_at":"2017-04-26T17:59:11Z","abstract_excerpt":"We consider relative error low rank approximation of $tensors$ with respect to the Frobenius norm: given an order-$q$ tensor $A \\in \\mathbb{R}^{\\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\\|A-B\\|_F^2 \\leq (1+\\epsilon)$OPT, where OPT $= \\inf_{\\textrm{rank-}k~A'} \\|A-A'\\|_F^2$. Despite the success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-$k$ tensor $A_k$ achieving the above infinum. 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