Foundations of Computational Mathematics

, Volume 19, Issue 2, pp 375–434 | Cite as

Learning Semidefinite Regularizers

  • Yong Sheng SohEmail author
  • Venkat Chandrasekaran


Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function, which is specified based on prior domain-specific expertise to induce a desired structure in the solution. We consider the problem of learning suitable regularization functions from data in settings in which precise domain knowledge is not directly available. Previous work under the title of ‘dictionary learning’ or ‘sparse coding’ may be viewed as learning a regularization function that can be computed via linear programming. We describe generalizations of these methods to learn regularizers that can be computed and optimized via semidefinite programming. Our framework for learning such semidefinite regularizers is based on obtaining structured factorizations of data matrices, and our algorithmic approach for computing these factorizations combines recent techniques for rank minimization problems along with an operator analog of Sinkhorn scaling. Under suitable conditions on the input data, our algorithm provides a locally linearly convergent method for identifying the correct regularizer that promotes the type of structure contained in the data. Our analysis is based on the stability properties of Operator Sinkhorn scaling and their relation to geometric aspects of determinantal varieties (in particular tangent spaces with respect to these varieties). The regularizers obtained using our framework can be employed effectively in semidefinite programming relaxations for solving inverse problems.


Atomic norm Convex optimization Low-rank matrices Nuclear norm Operator scaling Representation learning 

Mathematics Subject Classification

Primary 90C25 Secondary 90C22 15A24 41A45 52A41 



The authors were supported in part by NSF Career award CCF-1350590, by Air Force Office of Scientific Research Grants FA9550-14-1-0098 and FA9550-16-1-0210, by a Sloan research fellowship, and an A*STAR (Agency for Science, Technology, and Research, Singapore) fellowship. The authors thank Joel Tropp for a helpful remark that improved the result in Proposition 17.


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Copyright information

© SFoCM 2018

Authors and Affiliations

  1. 1.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of Electrical EngineeringCalifornia Institute of TechnologyPasadenaUSA

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