Efficient implementation of covariance updates

[ \sum_{i=1}^N x_{ij} r_i = (x_j,y) - \sum_{k:|\tilde{\beta}_k | greater 0 }} (x_j, x_k) \tilde{\beta}_k ] (9) How do I implement the following efficiently? [X \in R^{N \times p}, y, \tilde{\beta} \in R^{p}, p \gg N ]