Symmetric Tensor Decompositions On Varieties
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This paper discusses the problem of symmetric tensor decomposition on a given variety X: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in X. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on X. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary X. As a specific application, Vandermonde decompositions for nonsymmetric tensors can be computed by the proposed algorithm.
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