In this paper, we address the problem of searching for semantically similar images from a large database. We present a compact coding approach, supervised quantization. Our approach simultaneously learns feature selection that linearly transforms the database points into a low-dimensional discriminative subspace, and quantizes the data points in the transformed space. The optimization criterion is that the quantized points not only approximate the transformed points accurately, but also are semantically separable: the points belonging to a class lie in a cluster that is not overlapped with other clusters corresponding to other classes, which is formulated as a classification problem. The experiments on several standard datasets show the superiority of our approach over the state-of-the art supervised hashing and unsupervised quantization algorithms.
Our approach is a supervised version of quantization. The quantizer we adopt is composite quantization @cite_15 , which is shown to be a generalized version of product quantization @cite_39 and cartesian k-means @cite_34 , and achieves better performance. Rather than performing the quantization in the input space, our approach conducts the quantization in a discriminative space, which is jointly learned with the composite quantizer.
@cite_15 @cite_34 @cite_39
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Unlike safety properties which require the absence of a “bad” program trace, k-safety properties stipulate the absence of a “bad” interaction between k traces. Examples of k-safety properties include transitivity, associativity, anti-symmetry, and monotonicity. This paper presents a sound and relatively complete calculus, called Cartesian Hoare Logic (CHL), for verifying k-safety properties. We also present an automated verification algorithm based on CHL and implement it in a tool called DESCARTES. We use DESCARTES to analyze user-defined relational operators in Java and demonstrate that DESCARTES is effective at verifying (or finding violations of) multiple k-safety properties. Product quantization (PQ) is an effective vector quantization method. A product quantizer can generate an exponentially large codebook at very low memory time cost. The essence of PQ is to decompose the high-dimensional vector space into the Cartesian product of subspaces and then quantize these subspaces separately. The optimal space decomposition is important for the PQ performance, but still remains an unaddressed issue. In this paper, we optimize PQ by minimizing quantization distortions w.r.t the space decomposition and the quantization codebooks. We present two novel solutions to this challenging optimization problem. The first solution iteratively solves two simpler sub-problems. The second solution is based on a Gaussian assumption and provides theoretical analysis of the optimality. We evaluate our optimized product quantizers in three applications: (i) compact encoding for exhaustive ranking [1], (ii) building inverted multi-indexing for non-exhaustive search [2], and (iii) compacting image representations for image retrieval [3]. In all applications our optimized product quantizers outperform existing solutions. Product quantization is an effective vector quantization approach to compactly encode high-dimensional vectors for fast approximate nearest neighbor (ANN) search. The essence of product quantization is to decompose the original high-dimensional space into the Cartesian product of a finite number of low-dimensional subspaces that are then quantized separately. Optimal space decomposition is important for the performance of ANN search, but still remains unaddressed. In this paper, we optimize product quantization by minimizing quantization distortions w.r.t. the space decomposition and the quantization codebooks. We present two novel methods for optimization: a non-parametric method that alternatively solves two smaller sub-problems, and a parametric method that is guaranteed to achieve the optimal solution if the input data follows some Gaussian distribution. We show by experiments that our optimized approach substantially improves the accuracy of product quantization for ANN search. Consider a pair of correlated Gaussian sources (X 1,X 2). Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination of X 1 and X 2 to within a mean-square distortion of D. We obtain an inner bound to the optimal rate-distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components X 1 and X 2 first. This results in a better rate region for certain parameter values. Our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory. We then consider the case of linear reconstruction of K sources and provide an inner bound to the optimal rate-distortion region. Some parts of the inner bound are achieved using the following coding structure: lattice vector quantization followed by ldquocorrelatedrdquo lattice-structured binning.