A Fast Interior Point Method for Atomic Norm Soft Thresholding

Thomas Lundgaard Hansen, Tobias Lindstrøm Jensen

The atomic norm provides a generalization of the $\ell_1$-norm to continuous parameter spaces. When applied as a sparse regularizer for line spectral estimation the solution can be obtained by solving a convex optimization problem. This problem is known as atomic norm soft thresholding (AST). It can be cast as a semidefinite program and solved by standard methods. In the semidefinite formulation there are $O(N^2)$ dual variables which complicates the implementation of a standard primal-dual interior-point method based on symmetric cones. That has lead researcher to consider alternating direction method of multipliers (ADMM) for the solution of AST, but this method is still somewhat slow for large problem sizes. To obtain a faster algorithm we reformulate AST as a non-symmetric conic program. That has two properties of key importance to its numerical solution: the conic formulation has only $O(N)$ dual variables and the Toeplitz structure inherent to AST is preserved. Based on it we derive FastAST which is a primal-dual interior point method for solving AST. Two variants are considered with the fastest one requiring only $O(N^2)$ flops per iteration. Extensive numerical experiments demonstrate that FastAST solves AST significantly faster than a state-of-the-art solver based on ADMM.

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