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Image representation and approximate query is always a research challenge and is affected greatly by the dimension and size of images. Since hash-based methods and binary encodings in combination with other techniques, such as kernel tricks, a longer binary code and mapping vectors rotation, can maintain a linear query time and query accuracy, they have been used in this area broadly. This paper develops principal component analysis hashing (PCAH) and unequal length of binary coding to divide images into more categories, denoted as PCA-MD, to improve accuracy of the representation and lookup of images. This paper firstly proves that the eigenvector mapping is locality sensitive, which is the basis for more classes division. For the anisotropy of the eigenvectors, PCA-MD utilizes an unequal length of binary coding and fewer eigenvectors, rather than an equal code, to divide the images mapped on every eigenvector to more categories. Moreover, L1-norm distance is applied to measure the distances of images to avoid the enormous computation of Euclidean distance. Theoretical analysis and extensive experimental results demonstrate that the PCA-MD has a higher query performance and a slight longer run time than the state-of-the-art approaches based on the Hamming distance. This in turn verifies that PCAH is a locality sensitive hash and that partitioning into more categories rather than only two categories is reasonable. [ABSTRACT FROM AUTHOR]

This paper presents an extension of the Elias bound on the minimum distance of codes for discrete alphabets with general, possibly infinite valued, distances. The bound is obtained by combining a previous extension of the Elias bound, introduced by Blahut, with an extension of a bound previously introduced by the author which builds upon ideas of Gallager, Lovász, and Marton. The result can in fact be interpreted as a unification of the Elias bound and of Lovász’s bound on graph (or zero-error) capacity, both being recovered as particular cases of the one presented here. Previous extensions of the Elias bound by Berlekamp, Blahut, and Piret are shown to be included as particular cases of our bound. Applications to the reliability function are then discussed. [ABSTRACT FROM AUTHOR]

Highlights: [•] We propose to approximate the Euclidean distance using the Hamming distance. [•] The approximation relies on binary codes constructed from random projections. [•] The approximation accuracy depends mainly on the number of projections. [•] We also propose a fast retrieval method using the approximate distance. [•] This method retrieves ranked result similar to the true Euclidean distance. [Copyright &y& Elsevier]

Various kinds of distances of all negacyclic codes of length 2s over are completely determined. Using our structure theorems of negacyclic codes of length 2s over Z2a, we first calculate the Hamming distances of all such negacyclic codes, which particularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes, namely, the λ-constacyclic codes of length 2s over Z2a, where λ is any unit of Z2a with the form 4k - 1. We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes. [ABSTRACT FROM AUTHOR]

Nowadays, prototype-based binary quantization (PBQ) is a promising solution for the approximate nearest neighbor search problem, which simultaneously preserves the affinity structures of prototypes in both Euclidean space as well as those of their codes in binary space. To learn longer binary codes, space decomposition based on product quantization is usually adopted. In practice, we find that the scale between Euclidean distance and Hamming distance usually varies across these decomposed subspaces, which degenerates the performance of PBQ based methods. We make an attempt to balance the scale of these subspaces via a joint optimization problem in the classic PBQ model, and present both an iterative and alternate algorithm for optimization. We conducted experiments on 6 public databases, and demonstrated that our scale balancing based methods SKMH and SABQ outperform state-of-the-art hashing methods including popular prototype-based binary quantization methods, with up to 81.62% relative performance gains when learning 256-bit binary codes. [ABSTRACT FROM AUTHOR]