The weighted Euclidean norm $\|x\|_w$ of a vector $x\in \mathbb{R}^d$ with weights $w\in \mathbb{R}^d$ is the Euclidean norm where the contribution of each dimension is scaled by a given weight. Approaches to dimensionality reduction that satisfy the Johnson-Lindenstrauss (JL) lemma can be easily adapted to the weighted Euclidean distance if weights are fixed: it suffices to scale each dimension of the input vectors according to the weights, and then apply any standard approach. However, this is not the case when weights are unknown during the dimensionality reduction or might dynamically change. We address this issue by providing an approach that maps vectors into a smaller complex vector space, but still allows to satisfy a JL-like property for the weighted Euclidean distance when weights are revealed. Specifically, let $\Delta\geq 1, \epsilon \in (0,1)$ be arbitrary values, and let $S\subset \mathbb{R}^d$ be a set of $n$ vectors. We provide a weight-oblivious linear map $g:\mathbb{R}^d \rightarrow \mathbb{C}^k$, with $k=\Theta(\epsilon^{-2}\Delta^4 \ln{n})$, to reduce vectors in $S$, and an estimator $\rho: \mathbb{C}^k \times \mathbb{R}^d \rightarrow \mathbb R$ with the following property. For any $x\in S$, the value $\rho(g(x), w)$ is an unbiased estimate of $\|x\|^2_w$, and $\rho$ is computed from the reduced vector $g(x)$ and the weights $w$. Moreover, the error of the estimate $\rho((g(x), w)$ depends on the norm distortion due to weights and parameter $\Delta$: for any $x\in S$, the estimate has a multiplicative error $\epsilon$ if $\|x\|_2\|w\|_2/\|x\|_w\leq \Delta$, otherwise the estimate has an additive $\epsilon \|x\|^2_2\|w\|^2_2/\Delta^2$ error. Finally, we consider the estimation of weighted Euclidean norms in streaming settings: we show how to estimate the weighted norm when the weights are provided either after or concurrently with the input vector.

翻译：加权欧几里得范数 $\|x\|_w$ 定义为向量 $x\in \mathbb{R}^d$ 在权重 $w\in \mathbb{R}^d$ 作用下的欧几里得范数，其中每个维度的贡献按给定权重缩放。满足约翰逊-林登斯特劳斯（JL）引理的降维方法在权重固定时可轻松适配加权欧几里得距离：只需根据权重缩放输入向量的每个维度，再应用任意标准方法即可。但当权重在降维过程中未知或可能动态变化时，该策略不再适用。本文通过提出一种将向量映射至更低维复向量空间的方法解决了这一问题，该方法在权重揭示后仍能保持类JL性质以实现加权欧几里得距离。具体而言，令 $\Delta\geq 1, \epsilon \in (0,1)$ 为任意参数，$S\subset \mathbb{R}^d$ 为包含 $n$ 个向量的集合。我们构造了一个与权重无关的线性映射 $g:\mathbb{R}^d \rightarrow \mathbb{C}^k$（其中 $k=\Theta(\epsilon^{-2}\Delta^4 \ln{n})$）以压缩 $S$ 中的向量，同时设计了一个估计器 $\rho: \mathbb{C}^k \times \mathbb{R}^d \rightarrow \mathbb R$，满足以下性质：对任意 $x\in S$，$\rho(g(x), w)$ 是 $\|x\|^2_w$ 的无偏估计，且 $\rho$ 的计算仅依赖于降维后的向量 $g(x)$ 和权重 $w$。此外，该估计的误差受权重导致的范数畸变及参数 $\Delta$ 影响：当 $\|x\|_2\|w\|_2/\|x\|_w\leq \Delta$ 时，估计值与真实值的乘法误差为 $\epsilon$；否则估计值具有加性误差 $\epsilon \|x\|^2_2\|w\|^2_2/\Delta^2$。最后，我们考虑了流式场景中加权欧几里得范数的估计问题，展示了当权重在输入向量之后或同时提供时如何估计加权范数。