  A private information retrieval (PIR) protocol guarantees that a user can
privately retrieve files stored in a database without revealing any information
about the identity of the requested file. Existing information-theoretic PIR
protocols ensure perfect privacy, i.e., zero information leakage to the servers
storing the database, but at the cost of high download. In this work, we
present weakly-private information retrieval (WPIR) schemes that trade off
perfect privacy to improve the download cost when the database is stored on a
single server. We study the tradeoff between the download cost and information
leakage in terms of mutual information (MI) and maximal leakage (MaxL) privacy
metrics. By relating the WPIR problem to rate-distortion theory, the
download-leakage function, which is defined as the minimum required download
cost of all single-server WPIR schemes for a given level of information leakage
and a fixed file size, is introduced. By characterizing the download-leakage
function for the MI and MaxL metrics, the capacity of single-server WPIR is
fully described.
