Maximum inner-product search

Maximum inner-product search (MIPS) is a search problem, with a corresponding class of search algorithms which attempt to maximise the inner product between a query and the data items to be retrieved. MIPS algorithms are used in a wide variety of big data applications, including recommendation algorithms and machine learning.^[1]

Formally, for a database of vectors ${\displaystyle x_{i}}$ defined over a set of labels ${\displaystyle S}$ in an inner product space with an inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ defined on it, MIPS search can be defined as the problem of determining

${\displaystyle {\underset {i\in S}{\operatorname {arg\,max} }}\ \langle x_{i},q\rangle }$

for a given query ${\displaystyle q}$.

Although there is an obvious linear-time implementation, it is generally too slow to be used on practical problems. However, efficient algorithms exist to speed up MIPS search.^[1][2]

Under the assumption of all vectors in the set having constant norm, MIPS can be viewed as equivalent to a nearest neighbor search (NNS) problem in which maximizing the inner product is equivalent to minimizing the corresponding distance metric in the NNS problem.^[3] Like other forms of NNS, MIPS algorithms may be approximate or exact.^[4]

MIPS search is used as part of DeepMind's RETRO algorithm.^[5]