Consistent Multigroup Low-Rank Approximation

We consider the problem of consistent low-rank approximation for multigroup data: we ask for a sequence of $k$ basis vectors such that projecting the data onto their spanned subspace treats all groups as equally as possible, by minimizing the maximum error among the groups. Additionally, we require that the sequence of basis vectors satisfies the natural consistency property: when looking for the best $k$ vectors, the first $d<k$ vectors are the best possible solution to the problem of finding $d$ basis vectors. Thus, this multigroup low-rank approximation method naturally generalizes \svd and reduces to \svd for data with a single group. We give an iterative algorithm for this task that sequentially adds to the basis the vector that gives the best rank$-1$ projection according to the min-max criterion, and then projects the data onto the orthogonal complement of that vector. For finding the best rank$-1$ projection, we use primal-dual approaches or semidefinite programming. We analyze the theoretical properties of the algorithms and demonstrate empirically that the proposed methods compare favorably to existing methods for multigroup (or fair) PCA.