Compressive Online Decomposition of Dynamic Signals Via N-ℓ1 Minimization With Clustered Priors

We introduce a compressive online decomposition via solving an ${n}$-$\ell _{1}$ cluster-weighted minimization to decompose a sequence of data vectors into sparse and low-rank components. In contrast to conventional batch Robust Principal Component Analysis (RPCA)—which needs to access full data—our method processes a data vector of the sequence per time instance from a small number of measurements. The $n-\ell _{1}$ cluster-weighted minimization promotes (i) the structure of the sparse components and (ii) their correlation with multiple previously-recovered sparse vectors via clustering and re-weighting iteratively. We establish guarantees on the number of measurements required for successful compressive decomposition under the assumption of slowly-varying low-rank components. Experimental results show that our guarantees are sharp and the proposed algorithm outperforms the state of the art.