Ordinal Constraint Binary Coding for Approximate Nearest Neighbor Search

Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsup...

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Published in:	IEEE transactions on pattern analysis and machine intelligence Vol. 41; no. 4; pp. 941 - 955
Main Authors:	Liu, Hong, Ji, Rongrong, Wang, Jingdong, Shen, Chunhua
Format:	Journal Article
Language:	English
Published:	United States IEEE 01.04.2019 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Anchors Binary code learning Binary codes Binary system Codes Data points discrete optimization Encoding Hash based algorithms hashing image retrieval Learning Manifolds Manifolds (mathematics) Measurement Optimization ordinal preserving Permutations Preservation Quantization (signal) Ranking Tensile stress tensor graph Tensors
ISSN:	0162-8828, 1939-3539, 2160-9292, 1939-3539
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Abstract	Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the <inline-formula><tex-math notation="LaTeX">n</tex-math> <inline-graphic xlink:href="ji-ieq1-2819978.gif"/> </inline-formula>-pair ordinal graph by <inline-formula><tex-math notation="LaTeX">L</tex-math> <inline-graphic xlink:href="ji-ieq2-2819978.gif"/> </inline-formula>-pair anchor-based ordinal graph, and reduce the corresponding complexity from <inline-formula><tex-math notation="LaTeX">O(n^4)</tex-math> <inline-graphic xlink:href="ji-ieq3-2819978.gif"/> </inline-formula> to <inline-formula><tex-math notation="LaTeX">O(L^3)</tex-math> <inline-graphic xlink:href="ji-ieq4-2819978.gif"/> </inline-formula> (<inline-formula><tex-math notation="LaTeX">L\ll n</tex-math> <inline-graphic xlink:href="ji-ieq5-2819978.gif"/> </inline-formula>). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches.
AbstractList	Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the n-pair ordinal graph by L-pair anchor-based ordinal graph, and reduce the corresponding complexity from O(n ) to O(L ) ( L << n). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches. Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the <inline-formula><tex-math notation="LaTeX">n</tex-math> <inline-graphic xlink:href="ji-ieq1-2819978.gif"/> </inline-formula>-pair ordinal graph by <inline-formula><tex-math notation="LaTeX">L</tex-math> <inline-graphic xlink:href="ji-ieq2-2819978.gif"/> </inline-formula>-pair anchor-based ordinal graph, and reduce the corresponding complexity from <inline-formula><tex-math notation="LaTeX">O(n^4)</tex-math> <inline-graphic xlink:href="ji-ieq3-2819978.gif"/> </inline-formula> to <inline-formula><tex-math notation="LaTeX">O(L^3)</tex-math> <inline-graphic xlink:href="ji-ieq4-2819978.gif"/> </inline-formula> (<inline-formula><tex-math notation="LaTeX">L\ll n</tex-math> <inline-graphic xlink:href="ji-ieq5-2819978.gif"/> </inline-formula>). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches. Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the n-pair ordinal graph by L-pair anchor-based ordinal graph, and reduce the corresponding complexity from O(n4) to O(L3) ( L << n). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches.Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the n-pair ordinal graph by L-pair anchor-based ordinal graph, and reduce the corresponding complexity from O(n4) to O(L3) ( L << n). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches. Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the [Formula Omitted]-pair ordinal graph by [Formula Omitted]-pair anchor-based ordinal graph, and reduce the corresponding complexity from [Formula Omitted] to [Formula Omitted] ([Formula Omitted]). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches.
Author	Ji, Rongrong Shen, Chunhua Wang, Jingdong Liu, Hong
Author_xml	– sequence: 1 givenname: Hong orcidid: 0000-0002-4295-2224 surname: Liu fullname: Liu, Hong email: lynnliu0207@163.com organization: Fujian Key Laboratory of Sensing and Computing for Smart City, School of Information Science and Engineering, Xiamen University, Xiamen Shi, Fujian Sheng, China – sequence: 2 givenname: Rongrong orcidid: 0000-0001-9163-2932 surname: Ji fullname: Ji, Rongrong email: rrji@xmu.edu.cn organization: Fujian Key Laboratory of Sensing and Computing for Smart City, School of Information Science and Engineering, Xiamen University, Xiamen Shi, Fujian Sheng, China – sequence: 3 givenname: Jingdong orcidid: 0000-0002-4888-4445 surname: Wang fullname: Wang, Jingdong email: jingdw@microsoft.com organization: Microsoft Research Asia, Beijing, China – sequence: 4 givenname: Chunhua orcidid: 0000-0002-8648-8718 surname: Shen fullname: Shen, Chunhua email: chunhua.shen@adelaide.edu.au organization: University of Adelaide, SA, Australia
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SubjectTerms	Anchors Binary code learning Binary codes Binary system Codes Data points discrete optimization Encoding Hash based algorithms hashing image retrieval Learning Manifolds Manifolds (mathematics) Measurement Optimization ordinal preserving Permutations Preservation Quantization (signal) Ranking Tensile stress tensor graph Tensors
Title	Ordinal Constraint Binary Coding for Approximate Nearest Neighbor Search
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