Solving Tensor Low Cycle Rank Approximation

Large language models have become ubiquitous in modern life, finding applications in various domains such as natural language processing, language translation, and speech recognition. Recently, a breakthrough work [Zhao, Panigrahi, Ge, and Arora Arxiv 2023] explains the attention model from probabilistic context-free grammar (PCFG). One of the central computation task for computing probability in PCFG is formulating a particular tensor low rank approximation problem, we can call it tensor cycle rank. Given an $n \times n \times n$ third order tensor $A$, we say that $A$ has cycle rank-$k$ if there exists three $n \times k^2$ size matrices $U , V$, and $W$ such that for each entry in each \begin{align*} A_{a,b,c} = \sum_{i=1}^k \sum_{j=1}^k \sum_{l=1}^k U_{a,i+k(j-1)} \otimes V_{b, j + k(l-1)} \otimes W_{c, l + k(i-1) } \end{align*} for all $a \in [n], b \in [n], c \in [n]$. For the tensor classical rank, tucker rank and train rank, it has been well studied in [Song, Woodruff, Zhong SODA 2019]. In this paper, we generalize the previous ``rotation and sketch'' technique in page 186 of [Song, Woodruff, Zhong SODA 2019] and show an input sparsity time algorithm for cycle rank.