We prove that any real matrix $A$ contains a subset of at most
$4k/\eps + 2k \log(k+1)$ rows whose span ``contains" a matrix of
rank at most $k$ with error only $(1+\eps)$ times the error of the
best rank-$k$ approximation of $A$. This leads to an algorithm to
find such ... more >>>

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These ... more >>>

A recommendation system suggests products to users based on data about user preferences. It is typically modeled by a problem of completing an $m\times n$ matrix of small rank $k$. We give the first classical algorithm to produce a recommendation in $O(\text{poly}(k)\text{polylog}(m,n))$ time, which is an exponential improvement on previous ... more >>>