Optimal Sketching for Kronecker Product Regression and Low Rank Approximation

We study the Kronecker product regression problem, in which the design matrix is a Kronecker product of two or more matrices. Formally, given $A_i \in \R^{n_i \times d_i}$ for $i=1,2,\dots,q$ where $n_i \gg d_i$ for each $i$, and $b \in \R^{n_1 n_2 \cdots n_q}$, let $\mathcal{A} = A_i \otimes A_2 \otimes \cdots \otimes A_q$. Then for $p \in [1,2]$, the goal is to find $x \in \R^{d_1 \cdots d_q}$ that approximately minimizes $\|\mathcal{A}x - b\|_p$. Recently, Diao, Song, Sun, and Woodruff (AISTATS, 2018) gave an algorithm which is faster than forming the Kronecker product $\mathcal{A} \in \R^{n_1 \cdots n_q \times d_1 \cdots d_q}$. Specifically, for $p=2$ they achieve a running time of $O(\sum_{i=1}^q \texttt{nnz}(A_i) + \texttt{nnz}(b))$, where $ \texttt{nnz}(A_i)$ is the number of non-zero entries in $A_i$. Note that $\texttt{nnz}(b)$ can be as large as $\Theta(n_1 \cdots n_q)$. For $p=1,$ $q=2$ and $n_1 = n_2$, they achieve a worse bound of $O(n_1^{3/2} \text{poly}(d_1d_2) + \texttt{nnz}(b))$. In this work, we provide significantly faster algorithms. For $p=2$, our running time is $O(\sum_{i=1}^q \texttt{nnz}(A_i) )$, which has no dependence on $\texttt{nnz}(b)$. For $p<2$, our running time is $O(\sum_{i=1}^q \texttt{nnz}(A_i) + \texttt{nnz}(b))$, which matches the prior best running time for $p=2$. We also consider the related all-pairs regression problem, where given $A \in \R^{n \times d}, b \in \R^n$, we want to solve $\min_{x \in \R^d} \|\bar{A}x - \bar{b}\|_p$, where $\bar{A} \in \R^{n^2 \times d}, \bar{b} \in \R^{n^2}$ consist of all pairwise differences of the rows of $A,b$. We give an $O(\texttt{nnz}(A))$ time algorithm for $p \in[1,2]$, improving the $\Omega(n^2)$ time required to form $\bar{A}$. Finally, we initiate the study of Kronecker product low rank and and low-trank approximation. For input $\mathcal{A}$ as above, we give $O(\sum_{i=1}^q \texttt{nnz}(A_i))$ time algorithms, which is much faster than computing $\mathcal{A}$.