Understanding Sparse JL for Feature Hashing
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Feature hashing and more general projection schemes are commonly used in machine learning to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in R^n into a lower-dimensional space 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. In practice, feature vectors often have a low ℓ_∞-to-ℓ_2 norm ratio, and for this restricted set of vectors, many sparse JL-based schemes can achieve the norm-preserving objective with smaller dimension m than is necessary for the scheme on the full space R^n. A line of work introduced by Weinberger et. al (ICML '09) analyzes the sparse JL transform with one nonzero entry per column, which is a standard feature hashing scheme. Recently, Freksen, Kamma, and Larsen (NIPS '18) closed this line of work by proving an essentially tight tradeoff between ℓ_∞-to-ℓ_2 norm ratio, distortion, failure probability, and dimension m for this feature hashing scheme. We study more general projection schemes that are constructed using sparse JL transforms permitted to have more than one (but still a small fraction of) nonzero entries per column. Our main result is an essentially tight tradeoff between ℓ_∞-to-ℓ_2 norm ratio, distortion, failure probability, and dimension m for a general sparsity s, that generalizes the result of Freksen et. al. We also connect our result to the sparse JL literature by showing that it implies lower bounds on dimension-sparsity tradeoffs that essentially match upper bounds by Cohen (SODA '16). Moreover, our proof introduces a new perspective on bounding moments of certain random variables, that could be useful in other settings in theoretical computer science.
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