Fusion Hashing: A General Framework for Selfimprovement of Hashing
Abstract
Hashing has been widely used for efficient similarity search based on its query and storage efficiency. To obtain better precision, most studies focus on designing different objective functions with different constraints or penalty terms that consider neighborhood information. In this paper, in contrast to existing hashing methods, we propose a novel generalized framework called fusion hashing (FH) to improve the precision of existing hashing methods without adding new constraints or penalty terms. In the proposed FH, given an existing hashing method, we first execute it several times to get several different hash codes for a set of training samples. We then propose two novel fusion strategies that combine these different hash codes into one set of final hash codes. Based on the final hash codes, we learn a simple linear hash function for the samples that can significantly improve model precision. In general, the proposed FH can be adopted in existing hashing method and achieve more precise and stable performance compared to the original hashing method with little extra expenditure in terms of time and space. Extensive experiments were performed based on three benchmark datasets and the results demonstrate the superior performance of the proposed framework.
I Introduction
The amount of big data has grown explosively in recent years and the approximate nearest neighbor (ANN) search, which takes a query point and finds its ANNs within a large database, has been shown to be useful for many practical applications, such as computer vision, information retrieval, data mining, and machine learning. Hashing is a primary technique in ANN and has become one of the most popular candidates for performing ANN searches because it outperforms many other methods in most real applications [1] [2].
Hashing attempts to convert documents, images, videos, and other types of data into a set of short binary codes that preserve the similarity relationships in the original data. By utilizing these binary codes, ANN searches can be performed more easily on largescale datasets because of the high efficiency of pairwise comparisons based on Hamming distance [3]. Learningbased hashing is one of the most accurate hashing methods because it can achieve better retrieval performance by analyzing the underlying characteristics of data. Therefore, learningbased hashing has become popular because the learned compact hash codes can index and organize massive amounts of data effectively and efficiently.
Learningbased hashing is the task of learning a (compound) hash function that maps an input item to a compact code . The hash function can have a form based on a linear projection, kernel, spherical function, neural network, nonparametric function, etc. Hash functions are an important factor influencing search accuracy when utilizing hash codes. The time cost of computing hash codes is also important. A linear function can be efficiently evaluated, but kernel functions and nearestvectorassignmentbased functions provide better search accuracy because they are more flexible. Nearly all methods utilizing a linear hash function can be extended to kernelized hash functions. The most commonly used hash functions take the form of a generalized linear projection:
(1) 
where if and (or equivalently ). Otherwise, is the projection vector and is the bias variable. Here, is a prespecified general linear function. Different choices of yield different properties for hash functions, leading to a wide range of hashing approaches. For example, locality sensitive hashing (LSH) keeps as an identity function, whereas shiftinvariant kernelbased hashing and spectral hashing set to be a shifted cosine or sinusoidal function [4] [5].
Various algorithms have been developed and exploited to optimize hash function parameters. Randomized hashing approaches [6] [7] often utilize random projections or permutations. Learningbased hashing frameworks exploits data distributions and various levels of supervised information to determine the optimal parameters for hash functions. Supervised information includes pointwise labels, pairwise relationships, and ranking orders [8] [9] [10] [11] [12].
In general, most existing hashing methods attempt to design a loss function (objective function) that can preserve the similarity order in the target data (i.e., minimize the gap between the ANN search results computed from the hash codes and true search results obtained from the input data by adding constraints or penalty terms).
In contrast to exiting hashing methods, in this study, we explored a novel strategy that can facilitate the selfimprovement of existing hashing methods without adding or changing any terms in their objective functions. The proposed strategy is a twostep method. We first learn several different hash codes by utilizing a given hashing method, then fuse the codes according to various rules. Finally, a simple linear hash function is learned for outofsample extension. We call this novel framework fusion hashing (FH). FH can be utilized to provide selfimprovement to existing hashing method. The main contributions of this study are summarized as follows:

A general framework for hashing selfimprovement is proposed. The proposed FH method can be applied to existing hashing methods without changing the objective function of the original hashing method and results in better precision compared to the original hashing method.

Two hash code fusion strategies are proposed. In the proposed framework, two hash code fusion strategies are proposed and we perform theoretical analysis to guide the fusion process. Through the fusion of hash codes, we can learn new hash functions for outofsample extension.

Experiments based on three largescale datasets demonstrate that the proposed framework can improve different types of hashing methods in terms of precision.
The remainder of this paper is organized as follows. In Section 2, we describe the proposed FH method in detail. Evaluations based on experiments are presented in Section 3. We discuss the conclusions of our study in Section 4.
Ii Proposed Method
The proposed FH is a twostep framework that first optimizes binary codes utilizing hash code fusion, then estimates hash function parameters based on the optimized hash codes. Given an existing hashing method, the proposed FH provides selfimprovement capabilities. A flowchart for the proposed FH framework is presented in Fig. 1. FH consists of hash code fusion and hash learning steps. In the hash code fusion step, we first run a given hashing method times to obtain hash matrices for all samples. We then fuse the hash matrices utilizing two different fusion strategies. In the hash learning step, we learn a simple linear hash function based on the fused hash codes for outofsample extension.
In the following subsections, we first present some notations for FH and then describe the hash fusion and learning steps.
Iia Problem Statement and Notation
Generally, a hash function can have a form based on a linear projection, kernel, spherical function, neural network, etc. However, the linear function (or its variations, such as kernel and bilinear functions) is one of the most popular hash function forms because it is very efficient and easily optimized. Additionally, nearly all methods utilizing a linear hash function can be extended to kernelized hash functions [13]. Therefore, the theoretical analysis and hash learning methods proposed in this paper are largely based on linear hash functions.
In this paper, boldface lowercase letters, such as , denote vectors and boldface uppercase letters, such as , denote matrices. Furthermore, and are utilized to denote the norm and transpose of a matrix , respectively. Boldface denotes a vector where all elements are one. A few additional notations utilized in the proposed FH method are listed in Table I.
Notation  Description 

number of samples  
length of hash code  
run times for a given hashing method  
a given hashing method  
the row of the hash matrix  
hash matrix obtained from the run of hashing method  
final hash matrix  
original feature matrix of size  
projection matrix between the hash matrix and feature matrix 
IiB Hash Code Fusion
As discussed above, given a hashing method, we execute times to get hash codes for use as a training set. Next, we fuse these hash codes into a final hash code for use as a training sample. The motivation for hash code fusion is twofold. First, more accurate and stable codes can be obtained through hash code fusion. Second, synergy and relationships between different hash codes can be exploited through hash fusion. In this paper, we propose two fusion strategies. To describe these strategies, we first present some definitions and theorems, then outline the specific processes for the two fusion strategies.
It is known that learningbased hashing attempts to preserve the similarity relationships between samples in the original space based on Hamming distance. Therefore, different objective functions have been designed based on similarity preservation. In such optimization problems, there is a trivial solution in which all the hash codes of the samples are same (i.e., ). To avoid this solution, the code balance condition was introduced in [13]. It states that the number of data items mapped to each hash code must be the same. Bit balance and bit uncorrelation are utilized to approximate the code balance condition. Bit balance means that each bit has an approximately 50% chance of being or . Bit uncorrelation means that different bits are uncorrelated. These two conditions are formulated as
(2) 
where is an dimensional allones vector and is an identity matrix of size .
The property of code balance has proved to be very significant for hashing [14] [15]. In this study, to evaluate balance, we propose a definition called balance degree.
Balance degree: Given a hash matrix , the balance degree of the bit for the samples is defined as the absolute value of the sum of the row in the hash matrix. For example, if the vector is the row of the hash matrix, then the balance degree of the bit for the samples is . A smaller balance degree indicates better code balance.
We now present two theorems and their corresponding proofs, which are utilized in the proposed fusion strategies.
Theorem 1. Given a hash matrix , duplicate rows can be removed from the hash matrix because they have no influence on the preservation of semantics.
Proof: Assume there are two hash matrices and . Compared to , there is one duplicate row in . One can see that . That is to say, after adding a duplicate row to the hash matrix , the similarity between different hash codes remained unchanged. Therefore, there is no influence on the Hamming distance between different samples. In other words, duplicate hash bits can also be removed without any influence on the preservation of semantics.
Furthermore, assume is a projection between an original feature and hash code. The loss for hash learning can be simply calculated as follows:
(3) 
Set the derivative of the objective function in Equation (3) w.r.t to 0. Then,
(4) 
The closedform solution of can be derived as
(5) 
For hash matrices and , we have and , respectively. We define . Then, we have and . Finally,
(6) 
One can see that the fitting error is unchanged.
In conclusion, there is no influence on semantic preservation when duplicate rows are removed from the hash matrix.
Theorem 2. Given a hash matrix , the hash bit rows of can be out of order because ordering has no influence on semantic preservation.
Proof: For the hash matrix , is a hash matrix whose hash bit rows are a random permutation of . One can see that . Therefore, the semantics can be preserved, even if the hash bits are out of order.
Furthermore, according to Eqs. (3) and (4), for hash matrices and , we have and , respectively. One can see that . We define . Then, we have
(7) 
where . Therefore, we have because . That is to say, . One can see that the fitting error remains unchanged.
In conclusion, there is no influence on semantic preservation when the hash bit rows of are out of order.
We now propose two novel fusion strategies to obtain more accurate and stable hash codes for training samples.
IiB1 Bitbybit fusion
Given a hashing method , after executing it times in the training set, we obtain different hash matrices , where and . One can see that is the row of hash matrix .
The goal of hash fusion is to obtain an accurate and stable hash matrix for all training samples from the hash matrices . Here, we propose a bitbybit fusion strategy based on the code balance condition. For the bit (i.e., the row in ) of all training samples, we first compute the balance degrees of the bit in all hash matrices , then select the row whose balance degree is the smallest among all hash matrices , meaning we find the most balanced bit row among all hash matrices. If there are two or more rows with the same minimum balance degree, we empirically select the row in the first hash matrix. It should be noted that this phenomenon is rarely seen when the number of samples is large. We repeat this process for all rows to obtain a final hash matrix . Additionally, if there are duplicate rows in hash matrix , they are removed according to Theorem 1 to obtain more compact hash codes.
To demonstrate the bitbybit strategy, we presented an example in Fig. 2, where =3, =3, and =6. For each row in the three hash matrices , we compute a balance degree. The row with the smallest balance degree among the three hash matrices is selected as a row of .
IiB2 Codebycode fusion
In this strategy, given hash matrices , in contrast to bitbybit fusion, we randomly concatenate all hash matrices in the column direction. According to Theorem 2, random ordering has no impact on semantic preservation and we obtain a new matrix . If we wish to obtain a hash code of length , we then select the first rows from matrix with minimum balance degrees to construct the final hash matrix . If there are duplicate rows in hash matrix , they are removed according to Theorem 1 to obtain more compact hash codes.
We present an example in Fig. 3, where =3, =3, and =6. In this example, we first concatenate the hash matrices , , and in the column direction, then select the first three rows with minimum balance degrees to construct the final hash matrix .
IiB3 Discussion
To adequately describe the motivation for hash fusion strategies, we can consider a hash bit as a binary feature of the training samples, where a more balanced bit represents a better feature. Therefore, the goal of the two proposed fusion strategies is to replace bad binary features with good binary features. Additionally, according to Theorem 2, we can neglect the order of the hash bits. Therefore, good binary features (which have minimum balance degrees) can be sampled more than once, which is another motivation for the codebycode fusion strategy.
IiC Hash Learning
After obtaining a desirable hash matrix of training samples, we then learn a hash function for outofsample inputs. Generally, any type of hash function, such as a kernel, spherical function, neural network, or nonparametric function, can be utilized in this step. However, we utilized a linear hash function in this study. A simple form of the relevant optimization problem can be written as follows:
(8) 
The matrix is obtained following hash code fusion and the solution of can be easily obtained by utilizing Eqs. (4) and (5). Based on the learned projection , we can obtain hash codes for outofsample inputs by utilizing a sign function.
In summary, the proposed FH is presented in Algorithm 1.
IiD Time complexity analysis
We assume that the time complexity for a given hash algorithm is . Then, the time complexity for generating hash codes is . Additionally, the time complexity for solving the linear projection is , where is the dimension of the input features. For hash fusion, the time complexity depends on the fusion strategy. In the balance degree sorting process, the time complexity is typically no greater than . In the process of balance degree computation, the time complexity is . Therefore, the time complexity for FH during training is . Because and are much smaller than and , the time complexity for FH during training can be rewritten as . In general, compared to the time complexity of the hash algorithm , the time complexity for learning the linear projection is negligible. Therefore, the proposed FH method is only times as complex as the original hashing method, but achieves superior precision. Additionally, the value is always small because we found that a small value is acceptable based on our experiments. Precision only increases very slowly with an increase in . Therefore, the proposed framework does not require significant extra expenditures in terms of time and space to achieve superior precision.
Iii Experiments
In this section, we present our experimental settings and results. Three image datasets were utilized to evaluate the performance of the proposed method. Extensive experiments were conducted to evaluate the proposed framework. Our experiments were conducted on a computer with an Intel(R) Core(TM) i74790 CPU and 16 GB of RAM. The hyperparameter settings employed are listed in the experimental settings section.
Iiia Experimental Settings
Method  CIFAR10  MSCOCO  NUSWIDE  
24 bits  48 bits  64 bits  24 bits  48 bits  64 bits  24 bits  48 bits  64 bits  
LSH  0.2604  0.2942  0.3101  0.6093  0.7121  0.7145  0.4095  0.5968  0.5934 
LSH  0.2600  0.2901  0.3027  0.6338  0.6715  0.6558  0.5407  0.6109  0.5903 
LSH  0.2704  0.2908  0.2898  0.6404  0.6548  0.7051  0.4754  0.5814  0.5969 
FHBB  0.3046  0.3441  0.3656  0.7224  0.7629  0.7576  0.5790  0.7027  0.6853 
FHCC  0.3041  0.3450  0.3753  0.7260  0.7677  0.7712  0.5946  0.7096  0.6831 
Method  CIFAR10  MSCOCO  NUSWIDE  
24 bits  48 bits  64 bits  24 bits  48 bits  64 bits  24 bits  48 bits  64 bits  
PCAITQ  0.3418  0.3502  0.3547  0.6273  0.6967  0.7166  0.4048  0.6169  0.6390 
PCAITQ  0.3429  0.3539  0.3555  0.6276  0.6917  0.7154  0.4054  0.6171  0.6471 
PCAITQ  0.3354  0.3461  0.3521  0.6330  0.6907  0.7152  0.4057  0.6109  0.6382 
FHBB  0.3172  0.3570  0.3725  0.6395  0.6824  0.7095  0.3846  0.6190  0.6532 
FHCC  0.3296  0.3641  0.3747  0.6665  0.6842  0.7062  0.3847  0.6196  0.6552 
PCARR  0.2987  0.3114  0.3222  0.6596  0.6781  0.6970  0.4596  0.5941  0.5961 
PCARR  0.2787  0.3221  0.3285  0.6205  0.6865  0.6863  0.5001  0.6059  0.5573 
PCARR  0.3017  0.3204  0.3235  0.5561  0.7194  0.7397  0.4960  0.5990  0.6238 
FHBB  0.3072  0.3456  0.3691  0.7223  0.7692  0.7783  0.6013  0.6886  0.7018 
FHCC  0.3312  0.3483  0.3776  0.7237  0.7754  0.7794  0.5981  0.6983  0.7037 
SH  0.2908  0.2961  0.2992  0.6616  0.6501  0.6659  0.6070  0.5986  0.5934 
SH  0.2908  0.2961  0.2992  0.6616  0.6501  0.6659  0.6070  0.5986  0.5934 
SH  0.2908  0.2961  0.2992  0.6616  0.6501  0.6659  0.6070  0.5986  0.5934 
FHBB  0.3191  0.3323  0.3430  0.7130  0.7173  0.7355  0.6501  0.6690  0.6661 
FHCC  0.2689  0.2665  0.2760  0.5536  0.6051  0.6343  0.5210  0.5521  0.6084 
Method  CIFAR10  MSCOCO  NUSWIDE  
24 bits  48 bits  64 bits  24 bits  48 bits  64 bits  24 bits  48 bits  64 bits  
SDH  0.2333  0.4622  0.4007  0.8026  0.8413  0.5948  0.6837  0.7045  0.7036 
SDH  0.2236  0.5346  0.4797  0.6280  0.7204  0.8452  0.6888  0.7611  0.6844 
SDH  0.2640  0.4322  0.3089  0.8019  0.5156  0.8219  0.5004  0.6683  0.6926 
FHBB  0.2013  0.5004  0.3618  0.8101  0.6355  0.7629  0.6303  0.7410  0.7367 
FHCC  0.2124  0.4090  0.2932  0.8120  0.6678  0.7836  0.6648  0.7528  0.7389 
COSDISH  0.4566  0.5034  0.5269  0.5082  0.5900  0.6563  0.3633  0.4192  0.4007 
COSDISH  0.4795  0.5034  0.5143  0.5850  0.6890  0.6505  0.4351  0.4830  0.4531 
COSDISH  0.4817  0.5268  0.5184  0.4967  0.6006  0.6078  0.4452  0.4555  0.4894 
FHBB  0.5559  0.6112  0.6021  0.6011  0.7657  0.7399  0.5181  0.5826  0.5458 
FHCC  0.5827  0.6310  0.6255  0.6011  0.7646  0.7485  0.5181  0.5845  0.5455 
FSDH  0.6444  0.6798  0.6838  0.8122  0.8246  0.8209  0.7750  0.7756  0.7866 
FSDH  0.6443  0.6687  0.6872  0.7810  0.8232  0.8371  0.7750  0.7865  0.7878 
FSDH  0.6324  0.7006  0.7019  0.8151  0.8218  0.8234  0.7705  0.7798  0.7873 
FHBB  0.6682  0.6914  0.7004  0.8296  0.8394  0.8512  0.7713  0.7876  0.7895 
FHCC  0.6657  0.7006  0.7019  0.8367  0.8426  0.8536  0.7840  0.7913  0.7942 
IiiA1 Datasets
We utilized three different image datasets, namely CIFAR10 [16], MSCOCO [17], and NUSWIDE [18], in our experiments. These datasets are widely used in image retrieval studies. CIFAR10 is a singlelabel dataset containing 60,000 images that belong to 10 classes, with 6,000 images per class. We randomly selected 5,000 and 1,000 images (100 images per class) from the dataset as our training and testing sets, respectively.
The MSCOCO dataset is a multilabel dataset containing 82,783 images that belong to 91 categories. For the training image set, images with no category information were discarded and 82,081 remained. For the MSCOCO dataset, two images were defined as a similar pair if they shared at least one common label. We randomly selected 10,000 and 5,000 images from the dataset as our training and testing sets, respectively.
The NUSWIDE dataset contains 269,648 web images associated with 1,000 tags. In this multilabel dataset, each image may be annotated with multiple labels. We only selected 195,834 images belonging to the 21 most frequent concepts. For the NUSWIDE dataset, two images were defined as a similar pair if they shared at least one common label. We randomly selected 10,500 (500 from each concept) and 2,100 (100 from each concept) images from the dataset as our training and testing sets, respectively.
In this study, we employed a convolutional neural network (CNN) model called the CNNF model [19] to perform feature learning. The CNNF model has also been applied in deep pairwisesupervised hashing [20]and asymmetric deep supervised hashing [21] for feature learning. The CNNF model contains five convolutional layers and three fullyconnected layers. Their details are provided in [19]. It should be noted that the FH framework is sufficiently general to allow other deep neural networks to replace the CNNF model for feature learning. In this study, we only employed the CNNF model for illustrative purposes. Additionally, a radial basis function was utilized to reduce the number of parameters. The 4,096 deep features extracted by the CNNF model were mapped to 1,000 features.
IiiA2 Evaluation Metrics
To evaluate the proposed method, we utilized an evaluation metric known as mean average precision (MAP), which is used widely in image retrieval evaluation. MAP is the mean of the average precision values obtained for the top retrieved samples.
(9) 
where is the number of query images and is the AP of the instance. AP is defined as
(10) 
where is the number of relevant instances in the retrieved samples. Here, = 1 if the instance is relevant to the query. Otherwise, = 0.
IiiB Experimental Results and Analysis
We applied the proposed FH framework to the following methods: LSH [22], spectral hashing (SH) [23], principle component analysis (PCA)iterative quantization (PCAITQ) [24], PCArandom rotation [24], supervised discrete hashing (SDH) [3], column sampling based discrete supervised hashing [1], and fast supervised discrete hashing (FSDH) [25]. LSH is a dataindependent method. SH, PCAITQ, and PCARR are unsupervised hashing methods. All the other methods are supervised hashing methods. All of the hyperparameters were initialized as suggested in the original publications. The proposed FH is a datadependent framework. However, the proposed FH can be applied to dataindependent method such as LSH.
The proposed FH with the bitbybit strategy is denoted FHBB, whereas FH with the codebycode strategy is denoted FHCC. We executed the original methods three times each. In other words, we set .
Table II lists the MAP scores of the dataindependent LSH method with hash lengths ranging from 24–64 bits. One can see that the performance was improved by applying the proposed FH to LSH on all three benchmark datasets. FHBB and FHCC resulted in similar performance and both achieved 4%12% MAP score improvements for the three benchmark datasets.
Table III lists the MAP scores of the unsupervised methods with hash lengths ranging from 24–64 bits. One can see that the performances of the unsupervised methods were also improved by the proposed FHBB and FHCC in most cases. However, the proposed FHCC could not improve the performance of the SH method. One possible reason is that the performance of this method is so stable that the MAP does not change based on the number of executions. However, FHBB still improved the MAP performance of SH. This indicates that the proposed FHBB and FHCC can be applied in different scenarios with different results. We plan to investigate to best scenarios for each strategy in a future study.
Table IV lists the MAP scores of the supervised methods with hash lengths ranging from 24–64 bits. We can see similar results to those listed in Table III. When applying FHBB and FHCC, superior performance was achieved in most cases. It is worth mentioning that when applying the proposed framework to the SDH method, the performance was not always improved. The main reason for this is that SDH is very unstable, which leads to widely varying MAP performances for different executions. Then, a hash code with a bad MAP has a negative effect on hash code fusion. Compared to SDH, the FSDH method, which is a stable version of SDH, saw significant improvements with the application of FHBB and FHCC.
Fig. 4 presents the precision results for the three benchmark datasets with hash lengths ranging from 12–128 bits, where represents the running for the method, such as . Five methods were selected for tested and we executed the original methods three times. One can see that the proposed FHBB and FHCC almost produce superior precision, except for FHBB with the SH method.
Fig. 5 and Fig. 6 show the Precision@ 5000 on three benchmark datasets with the hash length ranging from 24–128 bits and the number of running times ranging from 2–6 bits by using the fusion strategy FHBB and FHCC, respectively. Five methods are selected limited by the space. It can be seen that the precisions of hashing methods are improved by using the proposed fusion strategies with the number of running times and hash bit being bigger. However, we can see that the precision increases slowly with bigger number of running times, which indicates that the proposed framework does not need too much extra expenditure in term of time and space to get superior precision.
Iv Conclusion
In this study, we proposed a general framework called FH to facilitate the selfimprovement of various hashing methods. Generally, the proposed framework can be applied to existing hashing methods without adding new constraint terms. In the proposed framework, we implemented two fusion strategies to obtain more accurate and stable hash codes from a given original hashing method. We then learn a simple linear projection for outofsample inputs. Experiments conducted on three benchmark datasets demonstrated the superior performance of the proposed framework.
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