$\ell_p$不经意子空间嵌入是在$r \times n$矩阵$Π$上的分布，使得对于任何固定的$n \times d$矩阵$A$，$$\Pr_Π[\textrm{for all }x, \ \|Ax\|_p \leq \|ΠAx\|_p \leq κ\|Ax\|_p] \geq 9/10,$$，其中$r$是嵌入的维数，$κ$是嵌入的失真，并且对于$n$维向量$y$，$\|y\|_p$是$\ell_p$范数。另一个重要的属性是$Π$的稀疏性，即每列非零条目的最大数量，因为这决定了计算$Π\cdot A$的运行时间。虽然对于$p = 2$，在维度、失真和稀疏性方面存在接近最优的折衷，但是对于$1 \leq p < 2$的重要情况，所知要少得多。在本文中，我们获得了几乎最佳的权衡$\ell_p$不经意子空间嵌入的每一个$1 \leq p < 2$。我们证明了对于每个$1 \leq p < 2$，任何具有维度$r$的不经意子空间嵌入都有失真$κ= Ω\left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \cdot \log^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).$当$r = \mathrm{poly}(d)$在应用中时，这给出了$κ= Ω(d^{1/p}\log^{-2/p} d)$下界，并表明Sohler和Woodruff（STOC，2011）对$p = 1$的不经意子空间嵌入以及Meng和Mahoney（STOC，2013）对$1 < p < 2$的不经意子空间嵌入在$\mathrm{poly}(\log(d))$因子上是最优的。我们还给出了每个$1 \leq p < 2$的稀疏不经意子空间嵌入，它们在维数和失真上都是最优的，直到$\mathrm{poly}(\log d)$因子。不经意的子空间嵌入对于分布式和流媒体环境至关重要，以及逐项$\ell_p$低秩近似。我们的研究结果为这些应用程序提供了改进的算法。
英文摘要：
An $\ell_p$ oblivious subspace embedding is a distribution over $r \times n$ matrices $Π$ such that for any fixed $n \times d$ matrix $A$, $$\Pr_Π[\textrm{for all }x, \ \|Ax\|_p \leq \|ΠAx\|_p \leq κ\|Ax\|_p] \geq 9/10,$$ where $r$ is the dimension of the embedding, $κ$ is the distortion of the embedding, and for an $n$-dimensional vector $y$, $\|y\|_p$ is the $\ell_p$-norm. Another important property is the sparsity of $Π$, that is, the maximum number of non-zero entries per column, as this determines the running time of computing $Π\cdot A$. While for $p = 2$ there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparisty, for the important case of $1 \leq p < 2$, much less was known. In this paper we obtain nearly optimal tradeoffs for $\ell_p$ oblivious subspace embeddings for every $1 \leq p < 2$.
We show for every $1 \leq p < 2$, any oblivious subspace embedding with dimension $r$ has distortion $κ= Ω\left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \cdot \log^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).$ When $r = \mathrm{poly}(d)$ in applications, this gives a $κ= Ω(d^{1/p}\log^{-2/p} d)$ lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for $p = 1$ and the oblivious subspace embedding of Meng and Mahoney (STOC, 2013) for $1 < p < 2$ are optimal up to $\mathrm{poly}(\log(d))$ factors. We also give sparse oblivious subspace embeddings for every $1 \leq p < 2$ which are optimal in dimension and distortion, up to $\mathrm{poly}(\log d)$ factors.
Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise $\ell_p$ low rank approximation. Our results give improved algorithms for these applications.
Deterministic oblivious distribution (and tight compaction) in linear time
在N个元素的数组中，M个位置和M个元素被“标记”。我们展示了如何排列数组中的元素，使所有标记的元素结束在标记的位置，在时间O（N）（在标准的字RAM模型），确定性，和obliviously -即与一个序列的内存访问，只依赖于N，而不是在哪些元素或位置被标记。作为一个推论，我们肯定地回答了一个开放的问题，即存在一个O（N）运行时间的确定性遗忘算法，用于紧密压缩（将M个标记元素移动到数组的前M个位置），这是几个加密结构的构建块。我们的O（N）结果提高了确定性紧压缩，随机紧压缩和随机松散压缩的更简单问题（将M个标记元素移动到前O（M）位置）的运行时上限-到目前为止分别为O（N lg N），O（N lg N）和O（N lg*N）。
Tight Bounds on $\ell_1$ Approximation and Learning of Self-Bounding Functions
Google Brain（谷歌大脑）；Carnegie Mellon University（卡内基梅隆大学）；Stanford University（斯坦福大学）
We study the complexity of learning and approximation of self-bounding functions over the uniform distribution on the Boolean hypercube ${0,1}^n$ . Informally, a function $f:{0,1}^n \rightarrow \mathbb{R}$ is self-bounding if for every $x \in {0,1}^n$ , $f(x)$ upper bounds the sum of all the $n$ marginal decreases in the value of the function at $x$ . Self-bounding functions include such well-known classes of functions as submodular and fractionally-subadditive (XOS) functions. They were introduced by Boucheron et al. (2000) in the context of concentration of measure inequalities. Our main res…
