A Fast Interior Point Method for Atomic Norm Soft Thresholding

TL;DR

This paper introduces FastAST, a novel primal-dual interior point method that significantly accelerates atomic norm soft thresholding by reformulating it as a non-symmetric conic program with fewer dual variables.

Contribution

The paper presents a new reformulation of AST as a non-symmetric conic program, enabling a faster interior point method called FastAST that outperforms existing ADMM-based solvers.

Findings

FastAST reduces computation time compared to ADMM-based methods.

The reformulation preserves Toeplitz structure and has fewer dual variables.

FastAST requires only O(N^2) flops per iteration for the fastest variant.

Abstract

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.