Understanding Sparse JL for Feature Hashing
This work provides a theoretical and empirical advancement for researchers and practitioners in machine learning using dimensionality reduction, though it is incremental as it generalizes prior results.
The paper tackles the problem of improving norm-preservation in sparse Johnson-Lindenstrauss transforms for feature hashing by using sparsity greater than 1, showing that this leads to significantly better accuracy on feature vectors with small ℓ∞-to-ℓ2 norm ratios, as demonstrated through tight theoretical tradeoffs and empirical validation.
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-$\ell_2$ norm ratio. Recently, Freksen, Kamma, and Larsen (NeurIPS '18) closed this line of work by proving a tight tradeoff between $\ell_\infty$-to-$\ell_2$ 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 $\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for a general sparsity $s$, that significantly generalizes the result of Freksen et. al. Our result theoretically demonstrates that sparse JL with $s > 1$ can have significantly better norm-preservation properties on feature vectors than sparse JL with $s = 1$; we also empirically demonstrate this finding.