The convex algebraic geometry of linear inverse problems

• Main Authors: Chandrasekaran, Venkat (Contributor), Recht, Benjamin (Author), Parrilo, Pablo A. (Contributor), Willsky, Alan S. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Laboratory for Information and Decision Systems (Contributor)
• Format: Article
• Language: English
• Published: Institute of Electrical and Electronics Engineers (IEEE), 2012-09-14T15:56:55Z.
• Subjects: Article
• Online Access: Get fulltext

 We study a class of ill-posed linear inverse problems in which the underlying model of interest has simple algebraic structure. We consider the setting in which we have access to a limited number of linear measurements of the underlying model, and we propose a general framework based on convex optimization in order to recover this model. This formulation generalizes previous methods based on ℓ[subscript 1]-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices from a small number of linear measurements. For example some problems to which our framework is applicable include (1) recovering an orthogonal matrix from limited linear measurements, (2) recovering a measure given random linear combinations of its moments, and (3) recovering a low-rank tensor from limited linear observations.