Implements the method of [1] for line spectral estimation via atomic norm minimization. If you use this code, please cite this work.

[1] T. L. Hansen and T. L. Jensen, "A Fast Interior Point Method for Atomic Norm Soft Thresholding," submitted to IEEE Transactions on Signal Processing, 2018, preprint available on arXiv.

Abstract:

The atomic norm provides a generalization of the l_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 and a standard primal-dual interior point method requires at least O(N^6) flops per iteration. That has lead researchers to consider the 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 both variants of FastAST solve AST significantly faster than a state-of-the-art solver based on ADMM.

A significant speedup of the code is obtained by building a mex version of the generalized Schur algorithm which is used internally. Do so by running buildmex in the MATLAB prompt. The MATLAB codegen feature is used to generate mex files.

To allow static memory allocation, the largest N = length(y) that the generated mex files will support is specified in buildmex.m.

MATLAB did not support recursion in codegen prior to version 9.0 (R2016a). On these earlier versions a slower fallback approach is used which only uses mex for the innermost iteration of the generalized Schur algorithm.

A simple example is provided in example.m. Type help solve_with_fastast in the MATLAB prompt for usage details.

This package redistributes the ADMM solver from https://github.com/badrinarayan/astlinespec.