🤖 AI Summary
This paper addresses the problem of efficient dimensionality reduction for weighted Euclidean distances when weights are either unknown or dynamically changing. We propose the first weight-agnostic linear dimensionality reduction framework: it maps real vectors into a low-dimensional complex space, where a carefully designed combination of complex-valued random projections and a weight-adaptive estimator yields an unbiased, high-accuracy estimate of the squared weighted norm. The method supports both post-hoc weight revelation and streaming inputs. In a $k = Theta(varepsilon^{-2}Delta^4 log n)$-dimensional complex space, it satisfies a Johnson–Lindenstrauss-type guarantee with simultaneous multiplicative and additive error bounds. Our key contribution is the first linear, unbiased, and streaming-scalable dimensionality reduction scheme for dynamic-weight settings—significantly extending the applicability of the classical JL lemma to weighted metrics.
📝 Abstract
The weighted Euclidean norm $|x|_w$ of a vector $xin mathbb{R}^d$ with weights $win mathbb{R}^d$ 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 $Deltageq 1, epsilon in (0,1)$ be arbitrary values, and let $Ssubset mathbb{R}^d$ be a set of $n$ vectors. We provide a weight-oblivious linear map $g:mathbb{R}^d
ightarrow mathbb{C}^k$, with $k=Theta(epsilon^{-2}Delta^4 ln{n})$, to reduce vectors in $S$, and an estimator $
ho: mathbb{C}^k imes mathbb{R}^d
ightarrow mathbb R$ with the following property. For any $xin S$, the value $
ho(g(x), w)$ is an unbiased estimate of $|x|^2_w$, and $
ho$ is computed from the reduced vector $g(x)$ and the weights $w$. Moreover, the error of the estimate $
ho((g(x), w)$ depends on the norm distortion due to weights and parameter $Delta$: for any $xin S$, the estimate has a multiplicative error $epsilon$ if $|x|_2|w|_2/|x|_wleq Delta$, 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.
