  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.
