The paper introduces a general framework for solving linear, underdetermined inverse problems by converting notions of simplicity into convex penalty functions. It defines atomic norms, which are derived from convex hulls of atomic sets, and shows how these norms can be used to recover simple models from limited linear measurements. The paper discusses various examples of atomic norms, including those for sparse vectors, low-rank matrices, permutation matrices, and orthogonal matrices. It also provides conditions for exact and robust recovery using these norms, based on Gaussian widths of tangent cones. The paper highlights the connection between Gaussian widths and symmetry, and discusses the computational challenges of solving atomic norm minimization problems, particularly when the atomic set has algebraic structure. It proposes semidefinite programming relaxations for these problems and analyzes the trade-off between computational complexity and the number of measurements required for recovery. The paper concludes with a discussion of the broader implications of these results for the recovery of simple models from limited linear information.The paper introduces a general framework for solving linear, underdetermined inverse problems by converting notions of simplicity into convex penalty functions. It defines atomic norms, which are derived from convex hulls of atomic sets, and shows how these norms can be used to recover simple models from limited linear measurements. The paper discusses various examples of atomic norms, including those for sparse vectors, low-rank matrices, permutation matrices, and orthogonal matrices. It also provides conditions for exact and robust recovery using these norms, based on Gaussian widths of tangent cones. The paper highlights the connection between Gaussian widths and symmetry, and discusses the computational challenges of solving atomic norm minimization problems, particularly when the atomic set has algebraic structure. It proposes semidefinite programming relaxations for these problems and analyzes the trade-off between computational complexity and the number of measurements required for recovery. The paper concludes with a discussion of the broader implications of these results for the recovery of simple models from limited linear information.