We consider the constrained Linear Inverse Problem (LIP), where a certain atomic norm (like the $\ell_1 $ norm) is minimized subject to a quadratic constraint. Typically, such cost functions are non-differentiable which makes them not amenable to the fast optimization methods existing in practice. We propose two equivalent reformulations of the constrained LIP with improved convex regularity: (i) a smooth convex minimization problem, and (ii) a strongly convex min-max problem. These problems could be solved by applying existing acceleration-based convex optimization methods which provide better $ O \left( \frac{1}{k^2} \right) $ theoretical convergence guarantee, improving upon the current best rate of $ O \left( \frac{1}{k} \right) $. We also provide a novel algorithm named the Fast Linear Inverse Problem Solver (FLIPS), which is tailored to maximally exploit the structure of the reformulations. We demonstrate the performance of FLIPS on the classical problems of Binary Selection, Compressed Sensing, and Image Denoising. We also provide open source \texttt{MATLAB} package for these three examples, which can be easily adapted to other LIPs.

翻译：我们考虑约束线性逆问题（Linear Inverse Problem, LIP），其中在二次约束下最小化某种原子范数（如$\ell_1$范数）。通常，此类代价函数不可微，因此难以应用实践中存在的快速优化方法。我们提出了约束LIP的两种等价重构形式，其凸正则性得到改善：（i）光滑凸最小化问题，以及（ii）强凸最小-最大问题。这些问题可通过现有基于加速的凸优化方法求解，这些方法提供了更好的$ O \left( \frac{1}{k^2} \right) $理论收敛保证，优于当前最优的$ O \left( \frac{1}{k} \right) $收敛速率。我们还提出了一种名为快速线性逆问题求解器（FLIPS）的新型算法，该算法专门设计用于最大化利用重构形式的结构。我们通过二元选择、压缩感知和图像去噪等经典问题展示了FLIPS的性能。同时，我们为这三个示例提供了开源的\texttt{MATLAB}程序包，该程序包可轻松适配于其他LIP问题。