This paper proposes a generic formulation that significantly expedites the training and deployment of image classification models, particularly under the scenarios of many image categories and high feature dimensions. As a defining property, our method represents both the images and learned classifiers using binary hash codes, which are simultaneously learned from the training data. Classifying an image thereby reduces to computing the Hamming distance between the binary codes of the image and classifiers and selecting the class with minimal Hamming distance. Conventionally, compact hash codes are primarily used for accelerating image search. Our work is first of its kind to represent classifiers using binary codes. Specifically, we formulate multi-class image classification as an optimization problem over binary variables. The optimization alternatively proceeds over the binary classifiers and image hash codes. Profiting from the special property of binary codes, we show that the sub-problems can be efficiently solved through either a binary quadratic program (BQP) or linear program. In particular, for attacking the BQP problem, we propose a novel bit-flipping procedure which enjoys high efficacy and local optimality guarantee. Our formulation supports a large family of empirical loss functions and is here instantiated by exponential hinge losses. Comprehensive evaluations are conducted on several representative image benchmarks. The experiments consistently observe reduced complexities of model training and deployment, without sacrifice of accuracies.
@cite_18: This paper addresses the problem of learning similarity-preserving binary codes for efficient similarity search in large-scale image collections. We formulate this problem in terms of finding a rotation of zero-centered data so as to minimize the quantization error of mapping this data to the vertices of a zero-centered binary hypercube, and propose a simple and efficient alternating minimization algorithm to accomplish this task. This algorithm, dubbed iterative quantization (ITQ), has connections to multiclass spectral clustering and to the orthogonal Procrustes problem, and it can be used both with unsupervised data embeddings such as PCA and supervised embeddings such as canonical correlation analysis (CCA). The resulting binary codes significantly outperform several other state-of-the-art methods. We also show that further performance improvements can result from transforming the data with a nonlinear kernel mapping prior to PCA or CCA. Finally, we demonstrate an application of ITQ to learning binary attributes or "classemes" on the ImageNet data set.
@cite_4: 
@cite_22: Fast retrieval methods are increasingly critical for many large-scale analysis tasks, and there have been several recent methods that attempt to learn hash functions for fast and accurate nearest neighbor searches. In this paper, we develop an algorithm for learning hash functions based on explicitly minimizing the reconstruction error between the original distances and the Hamming distances of the corresponding binary embeddings. We develop a scalable coordinate-descent algorithm for our proposed hashing objective that is able to efficiently learn hash functions in a variety of settings. Unlike existing methods such as semantic hashing and spectral hashing, our method is easily kernelized and does not require restrictive assumptions about the underlying distribution of the data. We present results over several domains to demonstrate that our method outperforms existing state-of-the-art techniques.
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@cite_21: Supervised hashing aims to map the original features to compact binary codes that are able to preserve label based similarity in the Hamming space. Non-linear hash functions have demonstrated their advantage over linear ones due to their powerful generalization capability. In the literature, kernel functions are typically used to achieve non-linearity in hashing, which achieve encouraging retrieval perfor- mance at the price of slow evaluation and training time. Here we propose to use boosted decision trees for achieving non-linearity in hashing, which are fast to train and evalu- ate, hence more suitable for hashing with high dimensional data. In our approach, we first propose sub-modular for- mulations for the hashing binary code inference problem and an efficient GraphCut based block search method for solving large-scale inference. Then we learn hash func- tions by training boosted decision trees to fit the binary codes. Experiments demonstrate that our proposed method significantly outperforms most state-of-the-art methods in retrieval precision and training time. Especially for high- dimensional data, our method is orders of magnitude faster than many methods in terms of training time.
@cite_1: We propose a method for learning similarity-preserving hash functions that map high-dimensional data onto binary codes. The formulation is based on structured prediction with latent variables and a hinge-like loss function. It is efficient to train for large datasets, scales well to large code lengths, and outperforms state-of-the-art methods.
@cite_19: 
@cite_2: The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50).
@cite_17: Learning short binary codes is challenged by the inherent discrete nature of the problem. The graph cuts algorithm is a well-studied discrete label assignment solution in computer vision, but has not yet been applied to solve the binary coding problems. This is partially because it was unclear how to use it to learn the encoding (hashing) functions for out-of-sample generalization. In this paper, we formulate supervised binary coding as a single optimization problem that involves both the encoding functions and the binary label assignment. Then we apply the graph cuts algorithm to address the discrete optimization problem involved, with no continuous relaxation. This method, named as Graph Cuts Coding (GCC), shows competitive results in various datasets.
The seminal work of LSH @cite_2 sheds a light on fast image search with theoretic guarantee. In a typical pipeline of hash code based image search @cite_4 @cite_9 , a set of hashing functions are generated either in an unsupervised manner or learned by perfectly separating similar dissimilar data pairs on the training set. The latter is also called supervised hashing since it often judges the similarity of two samples according to their semantic labels. For an unseen image, it finds the most similar images in the database by efficiently comparing the corresponding hash codes. It can be accomplished in sub-linear time using hash buckets @cite_2 . Representative methods in this line include Binary Reconstructive Embedding (BRE @cite_22 ), Minimal Loss Hashing (MLH @cite_1 ), Kernel-Based Supervised Hashing (KSH @cite_19 ), CCA based Iterative Quantization (CCA-ITQ @cite_18 ), FastHash @cite_21 , Graph Cuts Coding (GCC @cite_17 ), .