  This paper describes a suite of algorithms for constructing low-rank
approximations of an input matrix from a random linear image of the matrix,
called a sketch. These methods can preserve structural properties of the input
matrix, such as positive-semidefiniteness, and they can produce approximations
with a user-specified rank. The algorithms are simple, accurate, numerically
stable, and provably correct. Moreover, each method is accompanied by an
informative error bound that allows users to select parameters a priori to
achieve a given approximation quality. These claims are supported by numerical
experiments with real and synthetic data.
