Tight Bounds for the Subspace Sketch Problem with Applications

In the subspace sketch problem one is given an $n \times d$ matrix $A$ with $O(\log(nd))$ bit entries, and would like to compress it in arbitrary way build a small space data structure $Q_p$, so that for any $x \in \mathbb{R}^d$, probability at least 2/3, has $Q_p(x) = (1 \pm \varepsilon) \|Ax\|_p$, where $p \geq 0$ randomness over construction of $Q_p$. The central question is, how many bits are necessary store $Q_p$? This applications communication approximating number nonzeros product, size coresets projective clustering, memory streaming algorithms regression row-update model, embedding subspaces $L_p$ functional analysis. A major open dependence on approximation factor $\varepsilon$. We show if not positive even integer $d \Omega(\log(1/\varepsilon))$, then $\widetilde{\Omega}(\varepsilon^{-2} d)$ necessary. On other hand, $p$ integer, there upper bound $O(d^p \log(nd))$ independent Our results optimal up logarithmic factors. As corollaries our main lower bound, we obtain new bounds wide range applications, including above, which cases optimal.