论文标题
张量火车的正交分解
Orthogonal Decomposition of Tensor Trains
论文作者
论文摘要
在本文中，我们研究了将给定张量分解为张量列的问题，以使顶点的张量是正交分解的。当张量列的长度为两个时，两个顶点处的正交分解量为对称时，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和低级别但不能正交分解的，我们表明，美白过程可以将问题转化为正交情况。当张量网络具有三个或更多的长度时，顶点的张量是对称的且正交分解的，我们提供了一种算法，用于恢复它们，但会受到某些等级条件的影响。最后，在张量二维列车的情况下，顶点的张量是正交分解的，但不一定是对称的，我们表明分解问题减少了将矩阵分解为正交矩阵的新型问题，这两边都乘以对基矩阵。我们提供并比较两种解决方案，一种基于Sindhorn定理，另一种基于Procrustes的算法。我们以在我们的研究中出现的线性和多线性代数中的许多开放性问题结束。
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.
