Computer Science > Data Structures and Algorithms
[Submitted on 13 Dec 2022 (v1), last revised 14 Dec 2023 (this version, v2)]
Title:Dimensionality reduction on complex vector spaces for Euclidean distance with dynamic weights
View PDF HTML (experimental)Abstract: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.
Submission history
From: Paolo Pellizzoni [view email][v1] Tue, 13 Dec 2022 14:27:49 UTC (483 KB)
[v2] Thu, 14 Dec 2023 17:06:58 UTC (168 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.