Consistent Low-Rank Approximation

Abstract

We introduce and study the problem of consistent low-rank approximation, in which rows of an input matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ arrive sequentially and the goal is to provide a sequence of subspaces that well-approximate the optimal rank-$k$ approximation to the submatrix $\mathbf{A}^{(t)}$ that has arrived at each time $t$, while minimizing the recourse, i.e., the overall change in the sequence of solutions. We first show that when the goal is to achieve a low-rank cost within an additive $\varepsilon\cdot||\mathbf{A}^{(t)}||_F^2$ factor of the optimal cost, roughly $\mathcal{O}\left(\frac{k}{\varepsilon}\log(nd)\right)$ recourse is feasible. For the more challenging goal of achieving a relative $(1+\varepsilon)$-multiplicative approximation of the optimal rank-$k$ cost, we show that a simple upper bound in this setting is $\frac{k^2}{\varepsilon^2}\cdot\text{poly}\log(nd)$ recourse, which we further improve to $\frac{k^{3/2}}{\varepsilon^2}\cdot\text{poly}\log(nd)$ for integer-bounded matrices and $\frac{k}{\varepsilon^2}\cdot\text{poly}\log(nd)$ for data streams with polynomial online condition number. We also show that $\Omega\left(\frac{k}{\varepsilon}\log\frac{n}{k}\right)$ recourse is necessary for any algorithm that maintains a multiplicative $(1+\varepsilon)$-approximation to the optimal low-rank cost, even if the full input is known in advance. Finally, we perform a number of empirical evaluations to complement our theoretical guarantees, demonstrating the efficacy of our algorithms in practice.