Dimensionality reduction on complex vector spaces for dynamic weighted Euclidean distance

The weighted Euclidean distance between two vectors is a Euclidean distance where the contribution of each dimension is scaled by a given non-negative weight. The Johnson-Lindenstrauss (JL) lemma can be easily adapted to the weighted Euclidean distance if weights are known at construction time. Given a set of $n$ vectors with dimension $d$, it suffices to scale each dimension of the input vectors according to the weights, and then apply any standard JL reduction: the weighted Euclidean distance between pairs of vectors is preserved within a multiplicative factor $\epsilon$ with high probability. However, this is not the case when weights are provided after the dimensionality reduction. In this paper, we show that by applying a linear map from real vectors to a complex vector space, it is possible to update the compressed vectors so that the weighted Euclidean distances between pairs of points can be computed within a multiplicative factor $\epsilon$, even when weights are provided after the dimensionality reduction. 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.