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Relations Encoded in Multiway Arrays
Unlike matrix rank, hypermatrix rank is not lower semi-continuous. As a result, optimal low rank approximations of hypermatrices may not exist. Characterizing hypermatrices without optimal low rank approximations is an important step in implementing algorithms with hypermatrices. The main result of this thesis is an original coordinate-free proof that real 2 by 2 by 2 tensors that are rank three do not have optimal rank two approximations with respect to the Frobenius norm. This result was previously only proved in coordinates. Our coordinate-free proof expands on prior results by developing a proof method that can be generalized more readily to higher dimensional tensor spaces. Our proof has the corollary that the nearest point of a rank three tensor to the second secant set of the Segre variety is a rank three tensor in the tangent space of the Segre variety. The relationship between the contraction maps of a tensor generalizes, in a coordinate-free way, the fundamental relationship between the rows and columns of a matrix to hypermatrices. Our proof method demonstrates geometrically the fundamental relationship between the contraction maps of a tensor. For example, we show that a regular real or complex tensor is tangent to the 2 by 2 by 2 Segre variety if and only if the image of any of its contraction maps is tangent to the 2 by 2 Segre variety.