Eigenvectors from quadratic form of a low-rank matrix


I would like to compute the leading eigenvectors of a positive semi-definite matrix A with a low-rank structure. Are you aware of method that only rely on evaluations of y^\top A x for arbitrary y, x vectors?

Why not use standards iterative methods (e.g. in KrylovKit, ARPACK, or IterativeSolvers), which require only Ax operations? Matrix-vector operations are efficient for matrices represented as a low-rank outer product, if that’s what you have.

Thank you for your answer. In my case A is given by E[\nabla f(u)^\top \nabla f(u)] : the expectation of \nabla f(u)^\top \nabla f(u) over some distribution for u, and f is a vector valued function. Thus y^{\top} Ax is cheap to compute, as it only requires two directional derivatives, while computing Ax requires an adjoint solver that is too costly for my application.