Low Rank Approximation for General Tensor Networks

We study the problem of approximating a given tensor with $q$ modes $A \in \mathbb{R}^{n \times \ldots \times n}$ with an arbitrary tensor network of rank $k$ -- that is, a graph $G = (V, E)$, where $|V| = q$, together with a collection of tensors $\{U_v \mid v \in V\}$ which are contracted in the manner specified by $G$ to obtain a tensor $T$. For each mode of $U_v$ corresponding to an edge incident to $v$, the dimension is $k$, and we wish to find $U_v$ such that the Frobenius norm distance between $T$ and $A$ is minimized. This generalizes a number of well-known tensor network decompositions, such as the Tensor Train, Tensor Ring, Tucker, and PEPS decompositions. We approximate $A$ by a binary tree network $T'$ with $O(q)$ cores, such that the dimension on each edge of this network is at most $\widetilde{O}(k^{O(dt)} \cdot q/\varepsilon)$, where $d$ is the maximum degree of $G$ and $t$ is its treewidth, such that $\|A - T'\|_F^2 \leq (1 + \varepsilon) \|A - T\|_F^2$. The running time of our algorithm is $O(q \cdot \text{nnz}(A)) + n \cdot \text{poly}(k^{dt}q/\varepsilon)$, where $\text{nnz}(A)$ is the number of nonzero entries of $A$. Our algorithm is based on a new dimensionality reduction technique for tensor decomposition which may be of independent interest. We also develop fixed-parameter tractable $(1 + \varepsilon)$-approximation algorithms for Tensor Train and Tucker decompositions, improving the running time of Song, Woodruff and Zhong (SODA, 2019) and avoiding the use of generic polynomial system solvers. We show that our algorithms have a nearly optimal dependence on $1/\varepsilon$ assuming that there is no $O(1)$-approximation algorithm for the $2 \to 4$ norm with better running time than brute force. Finally, we give additional results for Tucker decomposition with robust loss functions, and fixed-parameter tractable CP decomposition.