We present an iterative flat clustering algorithm designed to operate on arbitrary similarity matrices, with the only constraint that these matrices be symmetrical. Although functionally very close to kernel k-means, our proposal performs an maximization of average intra-class similarity, instead of a squared distance minimization, in order to remain closer to the semantics of similarities. We show that this approach allows relaxing the conditions on usable matrices, as well as opening better optimization possibilities. Systematic evaluation on a variety of data sets shows that the proposed approach outperforms or equals kernel k-means in a large majority of cases, while running much faster. Most notably, it significantly reduces memory access, which makes it a good choice for large data collections.