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