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Sketching is a powerful dimensionality reduction technique for acceleratingalgorithms for data analysis. A crucial step in sketching methods is to computea subspace embedding (SE) for a large matrix $\mathbf{A} \in \mathbb{R}^{N\times d}$. SE's are the primary tool for obtaining extremely efficientsolutions for many linear-algebraic tasks, such as least squares regression andlow rank approximation. Computing an SE often requires an explicitrepresentation of $\mathbf{A}$ and running time proportional to the size of$\mathbf{A}$. However, if $\mathbf{A}= \mathbf{T}_1 \Join \mathbf{T}_2 \Join\dots \Join \mathbf{T}_m$ is the result of a database join query on severalsmaller tables $\mathbf{T}_i \in \mathbb{R}^{n_i \times d_i}$, then thisrunning time can be prohibitive, as $\mathbf{A}$ 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 becomputed significantly faster than computing the join. For the case of a twotable join $\mathbf{A} = \mathbf{T}_1 \Join \mathbf{T}_2$ we giveinput-sparsity algorithms for computing subspace embeddings, with running timebounded by the number of non-zero entries in $\mathbf{T}_1,\mathbf{T}_2$. Thisresults in input-sparsity time algorithms for high accuracy regression,significantly improving upon the running time of prior FAQ-based methods forregression. We extend our results to arbitrary joins for the ridge regressionproblem, also considerably improving the running time of prior methods.Empirically, we apply our method to real datasets and show that it issignificantly faster than existing algorithms.

In-Database Regression in Input Sparsity Time