  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.
