Motivated by the problem of compressing point sets into as few bits as possible while maintaining information about approximate distances between points, we construct random nonlinear maps $\varphi_\ell$ that compress point sets in the following way. For a point set $S$, the map $\varphi_\ell:\mathbb{R}^d \to N^{-1/2}\{-1,1\}^N$ has the property that storing $\varphi_\ell(S)$ (a \emph{sketch} of $S$) allows one to report pairwise squared distances between points in $S$ up to some multiplicative $(1\pm \epsilon)$ error with high probability as long as the minimum distance is not too small compared to $\epsilon$. The maps $\varphi_\ell$ are the $\ell$-fold composition of a certain type of random feature mapping. Moreover, we determine how large $N$ needs to be as a function of $\epsilon$ and other parameters of the point set. Compared to existing techniques, our maps offer several advantages. The standard method for compressing point sets by random mappings relies on the Johnson-Lindenstrauss lemma which implies that if a set of $n$ points is mapped by a Gaussian random matrix to $\mathbb{R}^k$ with $k =\Theta(\epsilon^{-2}\log n)$, then pairwise distances between points are preserved up to a multiplicative $(1\pm \epsilon)$ error with high probability. The main advantage of our maps $\varphi_\ell$ over random linear maps is that ours map point sets directly into the discrete cube $N^{-1/2}\{-1,1\}^N$ and so there is no additional step needed to convert the sketch to bits. For some range of parameters, our maps $\varphi_\ell$ produce sketches which require fewer bits of storage space.

翻译：受将点集压缩到尽可能少的比特同时保留点间近似距离信息的动机驱动，我们构建了随机非线性映射 $\varphi_\ell$，其压缩点集的方式如下：对于点集 $S$，映射 $\varphi_\ell:\mathbb{R}^d \to N^{-1/2}\{-1,1\}^N$ 具有如下性质——存储 $\varphi_\ell(S)$（即 $S$ 的\emph{草图}）可在最小距离相对于 $\epsilon$ 不太小的前提下，以高概率允许报告 $S$ 中点对间的平方距离，且误差不超过乘法因子 $(1\pm \epsilon)$。映射 $\varphi_\ell$ 是某类随机特征映射的 $\ell$ 重复合函数。此外，我们确定了 $N$ 作为 $\epsilon$ 及点集其他参数的函数所需选取的规模。与现有技术相比，我们的映射具有若干优势。通过随机映射压缩点集的标准方法依赖于约翰逊-林登斯特劳斯引理：若将 $n$ 个点通过高斯随机矩阵映射至 $\mathbb{R}^k$（其中 $k =\Theta(\epsilon^{-2}\log n)$），则点对间距离能以高概率保持乘法因子 $(1\pm \epsilon)$ 的误差。我们的映射 $\varphi_\ell$ 相对于随机线性映射的主要优势在于，其直接将点集映射至离散立方体 $N^{-1/2}\{-1,1\}^N$ 中，因此无需额外步骤将草图转换为比特。在某些参数范围内，我们的映射 $\varphi_\ell$ 生成的草图所需存储空间比特数更少。