  Given a large dataset of binary codes and a binary query point, we address
how to efficiently find $K$ codes in the dataset that yield the largest cosine
similarities to the query. The straightforward answer to this problem is to
compare the query with all items in the dataset, but this is practical only for
small datasets. One potential solution to enhance the search time and achieve
sublinear cost is to use a hash table populated with binary codes of the
dataset and then look up the nearby buckets to the query to retrieve the
nearest neighbors. However, if codes are compared in terms of cosine similarity
rather than the Hamming distance, then the main issue is that the order of
buckets to probe is not evident. To examine this issue, we first elaborate on
the connection between the Hamming distance and the cosine similarity. Doing
this allows us to systematically find the probing sequence in the hash table.
However, solving the nearest neighbor search with a single table is only
practical for short binary codes. To address this issue, we propose the angular
multi-index hashing search algorithm which relies on building multiple hash
tables on binary code substrings. The proposed search algorithm solves the
exact angular $K$ nearest neighbor problem in a time that is often orders of
magnitude faster than the linear scan baseline and even approximation methods.
