2018/05/22 by Casper Benjamin Freksen, Freksen, Casper Benjamin, Lior Kamma +3 · 1 citation
Computer Science · #Advanced Image and Video Retrieval Techniques #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Face and Expression Recognition #Functional Analysis (math.FA) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Machine Learning and Algorithms
paper · pdf · doi:10.48550/arxiv.1805.08539
openalex publication_date 2018/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Feature hashing, also known as \em the hashing trick, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms. Loosely speaking, feature hashing uses a random sparse projection matrix A : ℝn → ℝm (where m ≪ n) in order to reduce the dimension of the data from n to m while approximately preserving the Euclidean norm. Every column of A contains exactly one non-zero entry, equals to either -1 or 1. Weinberger et al. showed tail bounds on ‖Ax‖22. Specifically they showed that for every ε, δ, if ‖x‖∞ / ‖x‖2 is sufficiently small, and m is sufficiently large, then Pr[ | ‖Ax‖22 - ‖x‖22 | lt; ε ‖x‖22 ] ≥ 1 - δ . These bounds were later extended by Dasgupta \etal (2010) and most recently refined by Dahlgaard et al. (2017), however, the true nature of the performance of this key technique, and specifically the correct tradeoff between the pivotal parameters ‖x‖∞ / ‖x‖2, m, ε, δ remained an open question. We settle this question by giving tight asymptotic bounds on the exact tradeoff between the central parameters, thus providing a complete understanding of the performance of feature hashing. We complement the asymptotic bound with empirical data, which shows that the constants "hiding" in the asymptotic notation are, in fact, very close to 1, thus further illustrating the tightness of the presented bounds in practice.