Binary embedding of high-dimensional data requires long codes to preserve the discriminative power of the input space. Traditional binary coding methods often suffer from very high computation and storage costs in such a scenario. To address this problem, we propose Circulant Binary Embedding (CBE) which generates binary codes by projecting the data with a circulant matrix. The circulant structure enables the use of Fast Fourier Transformation to speed up the computation. Compared to methods that use unstructured matrices, the proposed method improves the time complexity from $\mathcal{O}(d^2)$ to $\mathcal{O}(d\log{d})$, and the space complexity from $\mathcal{O}(d^2)$ to $\mathcal{O}(d)$ where $d$ is the input dimensionality. We also propose a novel time-frequency alternating optimization to learn data-dependent circulant projections, which alternatively minimizes the objective in original and Fourier domains. We show by extensive experiments that the proposed approach gives much better performance than the state-of-the-art approaches for fixed time, and provides much faster computation with no performance degradation for fixed number of bits.

This paper focuses on the problem of learning binary codes for efficient retrieval of high-dimensional nonnegative data that arises in vision and text applications where counts or frequencies are used as features. The similarity of such feature vectors is commonly measured using the cosine of the angle between them. In this work, we introduce a novel angular quantization-based binary coding (AQBC) technique for such data and analyze its properties. In its most basic form, AQBC works by mapping each nonnegative feature vector onto the vertex of the binary hypercubewith which it has the smallest angle. Even though the number of vertices (quantization landmarks) in this scheme grows exponentially with data dimensionalityd, we propose a method for mapping feature vectors to their smallest-angle binary vertices that scales as O(d log d). Further, we propose a method for learning a linear transformation of the data to minimize the quantization error,and show that it results in improved binary codes. Experiments on image and text datasets show that the proposed AQBC method outperforms the state of the art.