  Thus far, sparse representations have been exploited largely in the context
of robustly estimating functions in a noisy environment from a few
measurements. In this context, the existence of a basis in which the signal
class under consideration is sparse is used to decrease the number of necessary
measurements while controlling the approximation error. In this paper, we
instead focus on applications in numerical analysis, by way of sparse
representations of linear operators with the objective of minimizing the number
of operations needed to perform basic operations (here, multiplication) on
these operators. We represent a linear operator by a sum of rank-one operators,
and show how a sparse representation that guarantees a low approximation error
for the product can be obtained from analyzing an induced quadratic form. This
construction in turn yields new algorithms for computing approximate matrix
products.
