  Binary embeddings provide efficient and powerful ways to perform operations
on large scale data. However binary embedding typically requires long codes in
order to preserve the discriminative power of the input space. Thus binary
coding methods traditionally suffer from 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 allows us to use Fast Fourier Transform
algorithms to speed up the computation. For obtaining $k$-bit binary codes from
$d$-dimensional data, this improves the time complexity from $O(dk)$ to
$O(d\log{d})$, and the space complexity from $O(dk)$ to $O(d)$.
  We study two settings, which differ in the way we choose the parameters of
the circulant matrix. In the first, the parameters are chosen randomly and in
the second, the parameters are learned using the data. For randomized CBE, we
give a theoretical analysis comparing it with binary embedding using an
unstructured random projection matrix. The challenge here is to show that the
dependencies in the entries of the circulant matrix do not lead to a loss in
performance. In the second setting, we design a novel time-frequency
alternating optimization to learn data-dependent circulant projections, which
alternatively minimizes the objective in original and Fourier domains. In both
the settings, we show by extensive experiments that the CBE approach gives much
better performance than the state-of-the-art approaches if we fix a running
time, and provides much faster computation with negligible performance
degradation if we fix the number of bits in the embedding.
