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

A Fast Interior Point Method for Atomic Norm Soft Thresholding