Understanding Sparse JL For Feature Hashing

Abstract

Feature hashing and other random projection schemes are commonly used to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in \(\mathbb\{R\}^n\) into a much lower-dimensional space \(\mathbb\{R\}^m\), while approximately preserving Euclidean norm. These schemes can be constructed using sparse random projections, for example using a sparse Johnson-Lindenstrauss (JL) transform. A line of work introduced by Weinberger et. al (ICML '09) analyzes the accuracy of sparse JL with sparsity 1 on feature vectors with small \(\ell_\infty\)-to-\(ℓ₂\) norm ratio. Recently, Freksen, Kamma, and Larsen (NeurIPS '18) closed this line of work by proving a tight tradeoff between \(\ell_\infty\)-to-\(ℓ₂\) norm ratio and accuracy for sparse JL with sparsity \(1\). In this paper, we demonstrate the benefits of using sparsity \(s\) greater than \(1\) in sparse JL on feature vectors. Our main result is a tight tradeoff between \(\ell_\in

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