On b-bit min-wise hashing for large-scale regression and classification with sparse data

Large-scale regression problems where both the number of variables, $p$, and the number of observations, $n$, may be large and in the order of millions or more, are becoming increasingly more common. Typically the data are sparse: only a fraction of a percent of the entries in the design matrix are non-zero. Nevertheless, often the only computationally feasible approach is to perform dimension reduction to obtain a new design matrix with far fewer columns and then work with this compressed data. $b$-bit min-wise hashing is a promising dimension reduction scheme for sparse matrices which produces a set of random features such that regression on the resulting design matrix approximates a kernel regression with the resemblance kernel. In this work, we derive bounds on the prediction error of such regressions. For both linear and logistic models, we show that the average prediction error vanishes asymptotically as long as $q \|\beta^*\|_2^2 /n \rightarrow 0$, where $q$ is the average number of non-zero entries in each row of the design matrix and $\beta^*$ is the coefficient of the linear predictor. We also show that ordinary least squares or ridge regression applied to the reduced data can in fact allow us fit more flexible models. We obtain non-asymptotic prediction error bounds for interaction models and for models where an unknown row normalisation must be applied in order for the signal to be linear in the predictors.The first author was supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1 and an EPSRC programme grant

Hashing for Similarity Search: A Survey

Similarity search (nearest neighbor search) is a problem of pursuing the data items whose distances to a query item are the smallest from a large database. Various methods have been developed to address this problem, and recently a lot of efforts have been devoted to approximate search. In this paper, we present a survey on one of the main solutions, hashing, which has been widely studied since the pioneering work locality sensitive hashing. We divide the hashing algorithms two main categories: locality sensitive hashing, which designs hash functions without exploring the data distribution and learning to hash, which learns hash functions according the data distribution, and review them from various aspects, including hash function design and distance measure and search scheme in the hash coding space

Zero-Shot Hashing via Transferring Supervised Knowledge

Hashing has shown its efficiency and effectiveness in facilitating large-scale multimedia applications. Supervised knowledge e.g. semantic labels or pair-wise relationship) associated to data is capable of significantly improving the quality of hash codes and hash functions. However, confronted with the rapid growth of newly-emerging concepts and multimedia data on the Web, existing supervised hashing approaches may easily suffer from the scarcity and validity of supervised information due to the expensive cost of manual labelling. In this paper, we propose a novel hashing scheme, termed \emph{zero-shot hashing} (ZSH), which compresses images of "unseen" categories to binary codes with hash functions learned from limited training data of "seen" categories. Specifically, we project independent data labels i.e. 0/1-form label vectors) into semantic embedding space, where semantic relationships among all the labels can be precisely characterized and thus seen supervised knowledge can be transferred to unseen classes. Moreover, in order to cope with the semantic shift problem, we rotate the embedded space to more suitably align the embedded semantics with the low-level visual feature space, thereby alleviating the influence of semantic gap. In the meantime, to exert positive effects on learning high-quality hash functions, we further propose to preserve local structural property and discrete nature in binary codes. Besides, we develop an efficient alternating algorithm to solve the ZSH model. Extensive experiments conducted on various real-life datasets show the superior zero-shot image retrieval performance of ZSH as compared to several state-of-the-art hashing methods.Comment: 11 page