Low rank matrix regression?

I have a collection of samples x,y (both vectors of the same dimension) generated by an unknown symmetric matrix H s.t. Hx = y. I would like to approximate the values of this matrix but don't have enough samples to do a good job with least-squares regression. I know that H is positive semi-definite but I'm not sure if that constrains the problem enough to be useful. If I further assume that H is low rank, is there a good algorithm that can estimate it? Even better if I don't have to store H directly but rather a set of vectors v1, v2, etc.. s.t. v1 * v1^T + v2 * v2^T + ... = H. To get started I'd be happy to even get a rank-1 approximation of H. Is there a standard method for this?

tl;dr: I have sample of vectors x and y, can I estimate a v such that v * v^T * x = y?