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InDatabase Regression in Input Sparsity Time
Rajesh Jayaram · Alireza Samadian · David Woodruff · Peng Ye
Sketching is a powerful dimensionality reduction technique for accelerating algorithms for data analysis. A crucial step in sketching methods is to compute a subspace embedding (SE) for a large matrix $A \in \mathbb{R}^{N \times d}$. SE's are the primary tool for obtaining extremely efficient solutions for many linearalgebraic tasks, such as least squares regression and low rank approximation. Computing an SE often requires an explicit representation of $A$ and running time proportional to the size of $A$. However, if $A= T_1 \Join T_2 \Join \dots \Join T_m$ is the result of a database join query on several smaller tables $T_i \in \mathbb{R}^{n_i \times d_i}$, then this running time can be prohibitive, as $A$ itself can have as many as $O(n_1 n_2 \cdots n_m)$ rows. In this work, we design subspace embeddings for database joins which can be computed significantly faster than computing the join. For the case of a two table join $A = T_1 \Join T_2$ we give inputsparsity algorithms for computing subspace embeddings, with running time bounded by the number of nonzero entries in $T_1,T_2$. This results in inputsparsity time algorithms for high accuracy regression, significantly improving upon the running time of prior FAQbased methods for regression. We extend our results to arbitrary joins for the ridge regression problem, also considerably improving the running time of prior methods. Empirically, we apply our method to real datasets and show that it is significantly faster than existing algorithms.
Author Information
Rajesh Jayaram (Carnegie Mellon University)
Alireza Samadian (University of Pittsburgh)
David Woodruff (Carnegie Mellon University)
Peng Ye (Tsinghua University)
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