Sparse PCA from Inaccurate and Incomplete Measurements

We consider the problem of recovering an unknown effectively sparse low-rank matrix from a small set of incomplete and inaccurate linear measurements of the form $y = \mathcal A (X) + η$, where $η$ is an {\it ineliminable} noise. This problem encompasses and fuses two important classes of machine learning and signal processing challenges: sparse principal component analysis and compressed sensing. More specifically, we aim at recovering low-rank-$R$ matrices with effectively $(s_1,s_2)$-sparse non-orthogonal rank-$1$ decompositions. We describe an optimization problem for matrix recovery under the considered model and propose a novel algorithm, called {\it \textbf{A}lternating \textbf{T}ikhonov regularization and \textbf{Las}so} (A-T-LA$\rm{S}_{2,1}$), to solve it. The algorithm is based on a multi-penalty regularization, which is able to leverage both structures (low-rankness and sparsity) simultaneously. The algorithm is a fast first order method, and straightforward to implement. We prove global convergence for {\it any} linear measurement model to stationary points and local convergence to global minimizers of the multi-penalty objective functional. Global minimizers balance effective sparsity of their rank-$1$ decompositions and the fidelity to data, up to noise level. By adapting the concept of restricted isometry property from compressed sensing to our novel model class, we prove error bounds between global minimizers and ground truth, up to noise level, from a number of subgaussian measurements scaling as $R(s_1+s_2)$, up to log-factors in the dimension, and relative-to-diameter distortion. Simulation results demonstrate both the accuracy and efficacy of the algorithm, as well as its superiority to the state-of-the-art algorithms in strong noise regimes and for matrices, whose singular vectors do not possess exact (joint-) sparse support.