Similarity search is the task of identifying, in a collection of items, the ones that are “similar” to a givenquery item. This task has a range of important applications (e.g. in information retrieval, patternrecognition, statistics, and machine learning) where data sets are often big, high dimensional, andpossibly noisy. State-of-the-art methods for similarity search offer only weak guarantees when faced withbig data. Either the space overhead is excessive (1000s of times larger than the space for the data itself),or the work needed to report the similar items may be comparable to the work needed to go through allitems (even if just a tiny fraction of the items are similar). As a result, many applications have to resort tothe use of ad-hoc solutions with only weak theoretical guarantees.This proposal aims at strengthening the theoretical foundation of scalable similarity search, anddeveloping novel practical similarity search methods backed by theory. In particular we will:- Leverage new types of embeddings that are kernelized, asymmetric, and complex-valued.- Consider statistical models of noise in data, and design similarity search data structures whoseperformance guarantees are phrased in statistical terms.- Build a new theory of the communication complexity of distributed, dynamic similarity search,emphasizing the communication bottleneck present in modern computing infrastructures.The objective is to produce new methods for similarity search that are: 1) Provably robust, 2) scalableto large and high-dimensional data sets, 3) substantially more resource efficient than current state-ofthe-art solutions, and 4) able to provide statistical guarantees on query answers.The study of similarity search has been an incubator for techniques (e.g. locality-sensitive hashing andrandom projections) that have wide-ranging applications. The new techniques developed in this projectare likely to have significant impacts beyond similarity search.