Fast Decentralized Federated Low Rank Matrix Recovery from Column-wise Linear Projections

—This work develops an alternating projected gradi- ent descent and minimization algorithm for recovering a low rank (LR) matrix from mutually independent linear projections of each of its columns in a “decentralized federated” fashion. This means that the data is federated but there is no central coordinating node. Some works refer to this setting as “fully-decentralized”. To be precise, our goal is to recover an unknown n × q rank- r matrix X (cid:63) = [ x (cid:63) 1 , x (cid:63) 2 , . . . x (cid:63)k , . . . x (cid:63)q ] from y k := A k x (cid:63)k , k = 1 , 2 , ...q , when each y k is an m -length vector with m < n ; different subsets of y k s are acquired at different nodes (vertical federation); and there is no central coordinate node (each node can only exchange information with its neighbors). We obtain constructive provable guarantees that provide a lower bound on the required sample complexity and an upper bound on the iteration complexity (total number of iterations needed to achieve a certain error level) of our proposed algorithm. This latter bound allows us to bound the time and communication complexity of our algorithm and argue that is fast and communication-efficient.