An improved method of locality sensitive hashing for indexing large-scale and high-dimensional features

Abstract

In recent years, Locality sensitive hashing (LSH) has been popularly used as an effective and efficient index structure of multimedia signals. LSH is originally proposed for resolving the high-dimensional approximate similarity search problem. Until now, many kinds of variations of LSH have been proposed for large-scale indexing. Much of the interest is focused on improving the query accuracy for skewed data distribution and reducing the storage space. However, when using LSH, a final filtering process based on exact similarity measure is needed. When the dataset is large-scale, the number of points to be filtered becomes large. As a result, the filtering speed becomes the bottleneck of improving the query speed when the scale of data becomes larger and larger. Furthermore, we observe a “Non-Uniform” phenomenon in the most popular Euclidean LSH which can degrade the filtering speed dramatically. In this paper, a pivot-based algorithm is proposed to improve the filtering speed by using triangle inequality to prune the search process. Furthermore, a novel method to select an optimal pivot for even larger improvement is provided. The experimental results on two open large-scale datasets show that our method can significantly improve the query speed of Euclidean LSH.

Introduction

Nearest Neighbors (NNs) search takes an important role in computer vision, machine learning, data mining, and information retrieval. However, for high-dimensional data, all known techniques to solve the similarity search problem will fall prey to the curse of dimensionality [1]. In some cases, such as high-dimensional learning for video annotation, the curse of dimensionality can be solved by multimodality learning [2]. In most cases, Approximate Nearest Neighbour (ANN) algorithms have been shown to be effective approaches to drastically improve the search speed while maintaining good precision. Locality sensitive hashing [3], [4], [5], [6] is one of the most popular ANN algorithms. Until now, LSH has been successfully used for image retrieval [7], [8] and 3D object indexing [9], [10]. Euclidean LSH [4] is the most successful variation of basic LSH because it uses popular Euclidean distance as similarity metric. However, some distance metrics beyond Euclidean distance metric must be used in practical applications [11], [12], [13], [14]. Therefore, some other variations are proposed for different distance metrics. Indyk and Thaper [15] have proposed the method of embedding EMD metric into the L₂ norm, then using the original LSH scheme to find the nearest neighbor in the Euclidean space. Gorisse et al. [1] present a new LSH scheme adapted to ${\chi }^2$ distance for approximate nearest neighbors search in high-dimensional spaces. Although LSH can get perfect results in theory, there are some drawbacks when using LSH in practical.

The main limitation of LSH is that its memory consumption is too large. The reason is that many hash tables are needed to keep both high recall and high precision. In [3], [6], a large number of hash tables are used. When the scale of dataset is large, if the number of hash tables is also large, the index structure cannot be loaded into main memory to get the best performance. To reduce the storage space of LSH, some variations [16], [17], [18] based on multi-probe strategy have been brought forward. The first multi-probe strategy is proposed by entropy based LSH [16]. Through randomly generating neighbor points near the query point, additional hash buckets will be probed and all probe results are merged. As a result, more points will be returned and the recall will be improved. By using this method, less hash tables are needed. Multi-probe LSH [17] is inspired by and improves upon entropy based LSH and query adaptive method [19]. It proposes a more efficient algorithm to generate an optimal probe sequence of hash buckets that are likely to contain similar points to the query. Unlike multi-probe LSH which is based on likelihood criteria, posteriori multi-probe LSH [18] puts forward a more reliable posteriori model by taking account some prior knowledge about the query and the searched points. This prior knowledge helps to do a better quality control and accurately select the hash buckets to be probed. The multi-probe algorithms proposed in [16], [17], [18] can save much storage space for LSH while keeping the comparable query precision and recall. In recent years, some new hashing-based methods which convert each database item into a compact binary code are proposed to get faster query time with much less storage. Spectral hashing [20] learns data-dependent directions via principal component analysis to generate short binary codes. Wang et al. [21] propose a data-dependent projection learning method such that each hash function is designed to correct the errors made by the previous one sequentially. Motivated by Weiss et al. [20], Liu et al. [22] propose a graph-based hashing method which automatically discovers the neighborhood structure inherent in the data to learn appropriate compact codes in an unsupervised manner. Semi-Supervised Hashing (SSH) [23] is proposed to learn efficient hash codes which can handlesemantic similarity/dissimilarity among the data points. It is also much faster than existing supervised hashing methods and can be easily scaled to large datasets.

Another limitation of LSH is that the query accuracy strongly depends on the selection of parameters. LSH forest [24] is proposed to eliminate the different data-dependent parameters for which LSH must be constantly hand-tuned. LSH forest can guarantee LSH's performance for skewed data distributions while retaining the same storage and query overhead. This characteristic makes LSH forest suitable for large-scale dataset. Specially, LSH forest can be constructed in main memory, on disk, in parallel system, and in peer-to-peer systems.

LSH is efficient to index high-dimensional data. Its variations discussed above can make it index large-scale dataset. As a result, LSH has been a popular index structure for large-scale and high-dimensional dataset. However, whatever method is used, a final filtering process based on exact similarity measure is inevitable. When the scale of dataset becomes very large, the number of points needed to be filtered becomes large too. In this case, the cost of filtering is the main factor that influences the query speed. Specially, the most popular Euclidean LSH [4] uses the quantized projection of a data point on a randomly selected direction as the hash value, which makes the number of points in some buckets as significantly larger than others. At the same time, a query will also be mapped to these buckets with a high probability. This kind of phenomenon, which we call “Non-Uniform”, makes the cost of filtering process significantly higher. When the data scale becomes very large, the problem will be even worse. In this work, we propose a pivot-based algorithm which uses triangle inequality to accelerate the filtering process of Euclidean LSH. The selection of pivot point can significantly influence the acceleration effectiveness. Some index structures [25], [26], [27], [28] use some base points, which are similar to our pivots, to accelerate the search process. These base points are all selected randomly. However, random selection cannot guarantee to get the optimal result. In fact, there is no explicit criterion put forward for base point selection until now.

This paper extends our previous work [29] and has the following contributions: (1) provide a formal analysis of “Non-Uniform” problem of Euclidean LSH which can significantly decrease filtering efficiency; (2) propose a pivot-based algorithm using triangle inequality to accelerate the filtering process of Euclidean LSH; (3) propose an effective method to get an optimal pivot point which is superior to other methods. The difference between this paper and our previous work [29] lies in the following: (1) more complete analysis of related works; (2) more rigorous and aborative theoretical deduction; (3) analyse the factors that affect the performance of the proposed algorithm; (4) do more experiments on new dataset to validate the efficiency of the proposed method; (5) verify the feasibility of accelerating the proposed algorithm through sampling.

The rest of this paper is organised as follows. Section 2 introduces the background of our research. In Section 3, we propose our pivot-based filtering algorithm and the method to get an optimal pivot. Section 4 describes our experiments and Section 5 concludes this paper.