We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations of tensors of any dimension, a classical result proven by Banach in 1938 shows that the optimal approximation can always be chosen to be symmetric. In contrast to these results, this article shows that the corresponding statement is no longer true for orthogonal approximations of higher rank. Specifically, for any of the four common notions of tensor orthogonality used in the literature, we show that optimal orthogonal approximations of rank greater than one cannot always be chosen to be symmetric.
Comon's Conjecture claims that for a symmetric tensor, its rank and its symmetric rank coincide. We show that this conjecture is true under an additional assumption that the rank of that tensor is not larger than its order. Moreover, if its rank is less than its order, then all rank decompositions are necessarily symmetric rank decompositions.  A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order- @math tensor is the outer product of @math nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz [J. Algebraic Geom., 4 (1995), pp. 201-222], is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most @math is not closed unless @math . We give sufficient conditions on a symmetric tensor @math to satisfy the following equality: the symmetric rank of @math , denoted as @math , is equal to the rank of @math , denoted as @math . This is done by considering the rank of the unfolded @math viewed as a matrix @math . The condition is @math . In particular, @math for @math for the cases @math . We discuss the analogues of the above results for border rank and best approximations of symmetric tensors.