Fast and Sample-Efficient Federated Low Rank Matrix Recovery From Column-Wise Linear and Quadratic Projections

We study the following lesser-known low rank (LR) recovery problem: recover an $n \times q$ rank- $r$ matrix, ${ \boldsymbol {X}}^{\ast}=[\boldsymbol {x}^{\ast}_{1}, \boldsymbol {x}^{\ast}_{2}, \ldots, \boldsymbol {x}^{\ast}_{q}]$ , with $r \ll \min (n,q)$ , from $m$ independent linear projections of each of its $q$ columns, i.e., from $\boldsymbol {y}_{k}:= \boldsymbol {A}_{k} \boldsymbol {x}^{\ast}_{k}, k \in [q]$ , when $\boldsymbol {y}_{k}$ is an $m$ -length vector with $m < n$ . The matrices $\boldsymbol {A}_{k}$ are known and mutually independent for different $k$ . We introduce a novel gradient descent (GD) based solution called AltGD-Min. We show that, if the $\boldsymbol {A}_{k}\text{s}$ are i.i.d. with i.i.d. Gaussian entries, and if the right singular vectors of ${ \boldsymbol {X}}^{\ast}$ satisfy the incoherence assumption, then $\epsilon $ -accurate recovery of ${ \boldsymbol {X}}^{\ast}$ is possible with order $(n+q) r^{2} \log (1/\epsilon)$ total samples and order $mq nr \log (1/\epsilon)$ time. Compared with existing work, this is the fastest solution. For $\epsilon < r^{1/4}$ , it also has the best sample complexity. A simple extension of AltGD-Min also provably solves LR Phase Retrieval, which is a magnitude-only generalization of the above problem. AltGD-Min factorizes the unknown ${ \boldsymbol {X}}$ as ${ \boldsymbol {X}}= { \boldsymbol {U}} \boldsymbol {B} $ where ${ \boldsymbol {U}}$ and $\boldsymbol {B}$ are matrices with $r$ columns and rows respectively. It alternates between a (projected) GD step for updating ${ \boldsymbol {U}}$ , and a minimization step for updating $\boldsymbol {B}$ . Its each iteration is as fast as that of regular projected GD because the minimization over $\boldsymbol {B}$ decouples column-wise. At the same time, we can prove exponential error decay for it, which we are unable to for projected GD. Finally, it can also be efficiently federated with a communication cost of only $nr$ per node, instead of $nq$ for projected GD.