Practical Hash Functions for Similarity Estimation and Dimensionality Reduction

Hashing is a basic tool for dimensionality reduction employed in several aspects of machine learning. However, the perfomance analysis is often carried out under the abstract assumption that a truly random unit cost hash function is used, without concern for which concrete hash function is employed. The concrete hash function may work fine on sufficiently random input. The question is if it can be trusted in the real world when faced with more structured input. In this paper we focus on two prominent applications of hashing, namely similarity estimation with the one permutation hashing (OPH) scheme of Li et al. [NIPS'12] and feature hashing (FH) of Weinberger et al. [ICML'09], both of which have found numerous applications, i.e. in approximate near-neighbour search with LSH and large-scale classification with SVM. We consider mixed tabulation hashing of Dahlgaard et al.[FOCS'15] which was proved to perform like a truly random hash function in many applications, including OPH. Here we first show improved concentration bounds for FH with truly random hashing and then argue that mixed tabulation performs similar for sparse input. Our main contribution, however, is an experimental comparison of different hashing schemes when used inside FH, OPH, and LSH. We find that mixed tabulation hashing is almost as fast as the multiply-mod-prime scheme ax+b mod p. Mutiply-mod-prime is guaranteed to work well on sufficiently random data, but we demonstrate that in the above applications, it can lead to bias and poor concentration on both real-world and synthetic data. We also compare with the popular MurmurHash3, which has no proven guarantees. Mixed tabulation and MurmurHash3 both perform similar to truly random hashing in our experiments. However, mixed tabulation is 40% faster than MurmurHash3, and it has the proven guarantee of good performance on all possible input.Comment: Preliminary version of this paper will appear at NIPS 201

Optimizing Ranking Measures for Compact Binary Code Learning

Hashing has proven a valuable tool for large-scale information retrieval. Despite much success, existing hashing methods optimize over simple objectives such as the reconstruction error or graph Laplacian related loss functions, instead of the performance evaluation criteria of interest---multivariate performance measures such as the AUC and NDCG. Here we present a general framework (termed StructHash) that allows one to directly optimize multivariate performance measures. The resulting optimization problem can involve exponentially or infinitely many variables and constraints, which is more challenging than standard structured output learning. To solve the StructHash optimization problem, we use a combination of column generation and cutting-plane techniques. We demonstrate the generality of StructHash by applying it to ranking prediction and image retrieval, and show that it outperforms a few state-of-the-art hashing methods.Comment: Appearing in Proc. European Conference on Computer Vision 201