Nearest neighbor search is known as a challenging issue that has been studied for several decades. Recently, this issue becomes more and more imminent in viewing that the big data problem arises from various fields. In this paper, a scalable solution based on hill-climbing strategy with the support of k-nearest neighbor graph (kNN) is presented. Two major issues have been considered in the paper. Firstly, an efficient kNN graph construction method based on two means tree is presented. For the nearest neighbor search, an enhanced hill-climbing procedure is proposed, which sees considerable performance boost over original procedure. Furthermore, with the support of inverted indexing derived from residue vector quantization, our method achieves close to 100 recall with high speed efficiency in two state-of-the-art evaluation benchmarks. In addition, a comparative study on both the compressional and traditional nearest neighbor search methods is presented. We show that our method achieves the best trade-off between search quality, efficiency and memory complexity.
Hill-climbing strategy is adopted in our search scheme due to its simplicity and efficiency. Different from @cite_6 @cite_27 , the search procedure is directed by residue vector quantization (RVQ) @cite_18 . Firstly, the candidate vectors are quantized by RVQ. However, different from compressional methods, the quantized codes are used only to build inverted indexing. Candidate vectors sharing the same RVQ codes (indexing key) are organized into one inverted list. In comparison to @cite_27 , only 4 bytes are required to keep the index for one vector. When a query comes, the distance between the query and the RVQ codes are calculated, the candidates reside in the lists of top-ranked codes are taken as the seeds for hill-climbing procedure. The inverted indexing built from RVQ codes plays the similar role as k-d trees in @cite_27 , while requiring very small amount of memory.
@cite_27 @cite_18 @cite_6
2766596763 2171572695 2100540720 2132234208
The recent years have witnessed the emerging of vector quantization (VQ) techniques for efficient similarity search. VQ partitions the feature space into a set of codewords and encodes data points as integer indices using the codewords. Then the distance between data points can be efficiently approximated by simple memory lookup operations. By the compact quantization, the storage cost, and searching complexity are significantly reduced, thereby facilitating efficient large-scale similarity search. However, the performance of several celebrated VQ approaches degrades significantly when dealing with noisy data. In addition, it can barely facilitate a wide range of applications as the distortion measurement only limits to @math norm . To address the shortcomings of the squared Euclidean ( @math norm) loss function employed by the VQ approaches, in this paper, we propose a novel robust and general VQ framework, named RGVQ, to enhance both robustness and generalization of VQ approaches. Specifically, a @math -norm loss function is proposed to conduct the @math -norm similarity search, rather than the @math norm search, and the @math -th order loss is used to enhance the robustness. Despite the fact that changing the loss function to @math norm makes VQ approaches more robust and generic, it brings us a challenge that a non-smooth and non-convex orthogonality constrained @math - norm function has to be minimized. To solve this problem, we propose a novel and efficient optimization scheme and specify it to VQ approaches and theoretically prove its convergence. Extensive experiments on benchmark data sets demonstrate that the proposed RGVQ is better than the original VQ for several approaches, especially when searching similarity in noisy data. A new data structure for efficient similarity search in very large datasets of high-dimensional vectors is introduced. This structure called the inverted multi-index generalizes the inverted index idea by replacing the standard quantization within inverted indices with product quantization. For very similar retrieval complexity and pre-processing time, inverted multi-indices achieve a much denser subdivision of the search space compared to inverted indices, while retaining their memory efficiency. Our experiments with large datasets of SIFT and GIST vectors demonstrate that because of the denser subdivision, inverted multi-indices are able to return much shorter candidate lists with higher recall. Augmented with a suitable reranking procedure, multi-indices were able to significantly improve the speed of approximate nearest neighbor search on the dataset of 1 billion SIFT vectors compared to the best previously published systems, while achieving better recall and incurring only few percent of memory overhead. The fixed slope lossy algorithm derived from the kth-order adaptive arithmetic codeword length function is extended to finite-state decoders or trellis-structured decoders. When this algorithm is used to encode a stationary, ergodic source with a continuous alphabet, the Lagrangian performance converges with probability one to a quantity computable as the infimum of an information-theoretic functional over a set of auxiliary random variables and reproduction levels, where spl lambda >0 and - spl lambda are designated to be the slope of the rate distortion function R(D) of the source at some D; the quantity is close to R(D)+ spl lambda D when the order k used in the arithmetic coding or the number of states in the decoders is large enough, An alternating minimization algorithm for computing the quantity is presented; this algorithm is based on a training sequence and in turn gives rise to a design algorithm for variable-rate trellis source codes. The resulting variable-rate trellis source codes are very efficient in low-rate regions. With k=8, the mean-squared error encoding performance at the rate 1 2 bits sample for memoryless Gaussian sources is comparable to that afforded by trellis-coded quantizers; with k=8 and the number of states in the decoder=32, the mean-squared error encoding performance at the rate 1 2 bits sample for memoryless Laplacian sources is about 1 dB better than that afforded by the trellis-coded quantizers with 256 states, with k=8 and the number of states in the decoder=256, the mean-squared error encoding performance at the rates of a fraction of 1 bit sample for highly dependent Gauss-Markov sources with correlation coefficient 0.9 is within about 0.6 dB of the distortion rate function. Recent indexing techniques inspired by source coding have been shown successful to index billions of high-dimensional vectors in memory. In this paper, we propose an approach that re-ranks the neighbor hypotheses obtained by these compressed-domain indexing methods. In contrast to the usual post-verification scheme, which performs exact distance calculation on the short-list of hypotheses, the estimated distances are refined based on short quantization codes, to avoid reading the full vectors from disk. We have released a new public dataset of one billion 128-dimensional vectors and proposed an experimental setup to evaluate high dimensional indexing algorithms on a realistic scale. Experiments show that our method accurately and efficiently re-ranks the neighbor hypotheses using little memory compared to the full vectors representation.