We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions, as well as approximations with tree tensor networks and more general tensor networks. For tensor train decomposition, we give a bicriteria $(1 + \eps)$-approximation algorithm with a small bicriteria rank and $O(q \cdot \nnz(A))$ running time, up to lower order terms, which improves over the additive error algorithm of \cite{huber2017randomized}. We also show how to convert the algorithm of \cite{huber2017randomized} into a relative error algorithm, but their algorithm necessarily has a running time of $O(qr^2 \cdot \nnz(A)) + n \cdot \poly(qk/\eps)$ when converted to a $(1 + \eps)$-approximation algorithm with bicriteria rank $r$. To the best of our knowledge, our work is the first to achieve polynomial time relative error approximation for tensor train decomposition. Our key technique is a method for obtaining subspace embeddings with a number of rows polynomial in $q$ for a matrix which is the flattening of a tensor train of $q$ tensors. We extend our algorithm to tree tensor networks. In addition, we extend our algorithm to tensor networks with arbitrary graphs (which we refer to as general tensor networks), by using a result of \cite{ms08_simulating_quantum_tensor_contraction} and showing that a general tensor network of rank $k$ can be contracted to a binary tree network of rank $k^{O(\deg(G)\tw(G))}$, allowing us to reduce to the case of tree tensor networks. Finally, we give new fixed-parameter tractable algorithms for the tensor train, Tucker, and CP decompositions, which are simpler than those of \cite{swz19_tensor_low_rank} since they do not make use of polynomial system solvers. Our technique of Gaussian subspace embeddings with exactly $k$ rows (and thus exponentially small success probability) may be of independent interest.

翻译：我们研究张量的低秩近似，重点关注张量列分解、Tucker分解，以及树型张量网络与更一般张量网络的近似方法。针对张量列分解，我们提出一种双准则$(1 + \eps)$-近似算法，其双准则秩较小且运行时间（忽略低阶项）为$O(q \cdot \nnz(A))$，优于\cite{huber2017randomized}的加性误差算法。同时，我们展示了如何将\cite{huber2017randomized}的算法转化为相对误差算法，但转化后的$(1 + \eps)$-近似算法（双准则秩为$r$）的运行时间必然为$O(qr^2 \cdot \nnz(A)) + n \cdot \poly(qk/\eps)$。据我们所知，本文首次实现了张量列分解的多项式时间相对误差近似。我们的关键技术是一种子空间嵌入方法，其行数为$q$的多项式量级，适用于由$q$个张量组成的张量列展平矩阵。我们将算法扩展至树型张量网络。此外，通过利用\cite{ms08_simulating_quantum_tensor_contraction}的结果，我们进一步将算法推广至任意图结构的张量网络（称为一般张量网络），并证明秩为$k$的一般张量网络可约化为秩为$k^{O(\deg(G)\tw(G))}$的二叉树网络，从而归约至树型张量网络情形。最后，我们提出了张量列分解、Tucker分解和CP分解的新型固定参数可解算法，这些算法比\cite{swz19_tensor_low_rank}的算法更简洁，因其无需使用多项式系统求解器。此外，我们提出的精确$k$行高斯子空间嵌入技术（尽管成功概率呈指数级小）可能具有独立研究价值。