Low-Rank Matrix Recovery via Rank One Tight Frame Measurements

• Published in: The Journal of fourier analysis and applications Vol. 25; no. 2; pp. 588 - 593
• Main Authors: Rauhut, Holger, Terstiege, Ulrich
• Format: Journal Article
• Language: English
• Published: New York Springer US 15-04-2019; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Convexity; Fourier Analysis; Machine learning; Mathematical analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Matrix methods; Optimization; Partial Differential Equations; Phase retrieval; Signal,Image and Speech Processing; Nuclear norm minimization; Convex optimization; 90C25; Phase retrieval; Positive semidefinite least squares problem; Random measurements; 60B20; 94A12; Quantum state tomography; Low rank matrix recovery; 94A20
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-017-9579-x

The task of reconstructing a low rank matrix from incomplete linear measurements arises in areas such as machine learning, quantum state tomography and in the phase retrieval problem. In this note, we study the particular setup that the measurements are taken with respect to rank one matrices constructed from the elements of a random tight frame. We consider a convex optimization approach and show both robustness of the reconstruction with respect to noise on the measurements as well as stability with respect to passing to approximately low rank matrices. This is achieved by establishing a version of the null space property of the corresponding measurement map.