In this paper, we study the problem of decomposing a given tensor into a tensor train such that the tensors at the vertices are orthogonally decomposable. When the tensor train has length two, and the orthogonally decomposable tensors at the two vertices are symmetric, we recover the decomposition by considering random linear combinations of slices. Furthermore, if the tensors at the vertices are symmetric and low-rank but not orthogonally decomposable, we show that a whitening procedure can transform the problem into the orthogonal case. When the tensor network has length three or more and the tensors at the vertices are symmetric and orthogonally decomposable, we provide an algorithm for recovering them subject to some rank conditions. Finally, in the case of tensor trains of length two in which the tensors at the vertices are orthogonally decomposable but not necessarily symmetric, we show that the decomposition problem reduces to the novel problem of decomposing a matrix into an orthogonal matrix multiplied by diagonal matrices on either side. We provide and compare two solutions, one based on Sinkhorn's theorem and one on Procrustes' algorithm. We conclude with a multitude of open problems in linear and multilinear algebra that arose in our study.

在本文中，我们研究了将给定张量分解为张量序列的问题，使得顶点处的张量可正交分解。当张量序列的长度为二时，并且两个顶点处的正交可分解张量是对称的，我们通过考虑切片的随机线性组合来恢复分解。此外，如果顶点处的张量是对称的和低阶的但不可正交分解，我们表明白化过程可以将问题转换为正交情况。当张量网络的长度为 3 或更多并且顶点处的张量是对称的并且正交可分解时，我们提供了一种算法来恢复它们受某些等级条件的影响。最后，对于长度为 2 的张量序列，其中顶点处的张量可正交分解但不一定对称，我们表明分解问题简化为将矩阵分解为正交矩阵乘以任一侧对角矩阵的新问题. 我们提供并比较了两种解决方案，一种基于 Sinkhorn 定理，另一种基于 Procrustes 算法。我们总结了我们研究中出现的线性和多线性代数中的大量未解决问题。