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      Semialgebraic Geometry of Nonnegative Tensor Rank

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          Abstract

          We study the semialgebraic structure of \(D_r\), the set of nonnegative tensors of nonnegative rank not more than \(r\), and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. Under some mild condition (non-defectivity), we show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-\(r\) decompositions if the real tensor space is \(r\)-identifiable. We determine conditions under which a best nonnegative rank-\(r\) approximation has a unique nonnegative rank-\(r\) decomposition: for \(r \le 3\), this is always the case; for general \(r\), this is the case when the best nonnegative rank-\(r\) approximation does not lie on the boundary of \(D_r\).

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          Journal
          2015-12-11
          2016-03-01
          Article
          1601.05351
          cfeee455-09ab-49c4-acea-5a7169c44dc5

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          19 pages
          math.RA

          Algebra
          Algebra

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