SIAM Journal on Computing, Volume 50, Issue 4, Page 1287-1335, January 2021.
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 an arbitrary way to build a small space data structure $Q_p$, so that for any given $x \in \mathbb{R}^d$, with probability at least 2/3, one has $Q_p(x) = (1 \pm \varepsilon) \|Ax\|_p$, where $p \geq 0$ and the randomness is over the construction of $Q_p$. The central question is, how many bits are necessary to store $Q_p$? This problem has applications to the communication of approximating the number of nonzeros in a matrix product, the size of coresets in projective clustering, the memory of streaming algorithms for regression in the row-update model, and embedding subspaces of $L_p$ in functional analysis. A major open question is the dependence on the approximation factor $\varepsilon$. We show if $p \geq 0$ is not a positive even integer and $d = \Omega(\log(1/\varepsilon))$, then $\widetilde{\Omega}(\varepsilon^{-2} d)$ bits are necessary. On the other hand, if $p$ is a positive even integer, then there is an upper bound of $O(d^p \log(nd))$ bits independent of $\varepsilon$. Our results are optimal up to logarithmic factors. As corollaries of our main lower bound, we obtain new lower bounds for a wide range of applications, including the above, which in many cases are optimal.

中文翻译：

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应用程序子空间草图问题的紧边界

SIAM Journal on Computing，第 50 卷，第 4 期，第 1287-1335 页，2021 年 1 月。
在子空间草图问题中，给定一个 $n \times d$ 矩阵 $A$ 和 $O(\log(nd))$ 位条目，并希望以任意方式对其进行压缩以构建小空间数据结构$Q_p$，所以对于任何给定的 $x \in \mathbb{R}^d$，至少有 2/3 的概率，有 $Q_p(x) = (1 \pm \varepsilon) \|Ax\| _p$，其中 $p \geq 0$ 和随机性超过了 $Q_p$ 的构造。核心问题是，存储 $Q_p$ 需要多少位？此问题适用于矩阵乘积中非零值的近似数、投影聚类中的核心集大小、行更新模型中回归流算法的内存以及函数分析中 $L_p$ 的嵌入子空间的通信. 一个主要的悬而未决的问题是对近似因子 $\varepsilon$ 的依赖。我们证明如果 $p \geq 0$ 不是正偶数且 $d = \Omega(\log(1/\varepsilon))$，那么 $\widetilde{\Omega}(\varepsilon^{-2} d )$ 位是必要的。另一方面，如果 $p$ 是正偶数，则存在 $O(d^p \log(nd))$ 位独立于 $\varepsilon$ 的上限。我们的结果在对数因子下是最佳的。作为我们主要下界的推论，我们为包括上述在内的广泛应用获得了新的下界，这在许多情况下是最佳的。