  Locality-sensitive hashing (LSH) is a popular data-independent indexing
method for approximate similarity search, where random projections followed by
quantization hash the points 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. Most of high-dimensional visual
descriptors for images exhibit a natural matrix structure. When visual
descriptors are represented by high-dimensional feature vectors and long binary
codes are assigned, a random projection matrix requires expensive complexities
in both space and time. In this paper we analyze a bilinear random projection
method where feature matrices are transformed to binary codes by two smaller
random projection matrices. We base our theoretical analysis on extending
Raginsky and Lazebnik's result where random Fourier features are composed with
random binary quantizers to form locality sensitive binary codes. To this end,
we answer the following two questions: (1) whether a bilinear random projection
also yields similarity-preserving binary codes; (2) whether a bilinear random
projection yields performance gain or loss, compared to a large linear
projection. Regarding the first question, we present upper and lower bounds on
the expected Hamming distance between binary codes produced by bilinear random
projections. In regards to the second question, we analyze the upper and lower
bounds on covariance between two bits of binary codes, showing that the
correlation between two bits is small. Numerical experiments on MNIST and
Flickr45K datasets confirm the validity of our method.
