Abstract

We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex 1 penalty and many of its non-convex surrogates that have been well studied in the literature. We establish the exact and stable recovery theory for the recovery model using the springback penalty for both sparse and nearly sparse signals, respectively, and derive an easily implementable difference-of-convex algorithm. In particular, we show its theoretical superiority to some existing models with a sharper recovery bound for some scenarios where the level of measurement noise is large or the amount of measurements is limited. We also demonstrate its numerical robustness regardless of the varying coherence of the sensing matrix. The springback penalty is particularly favorable for the scenario where the incomplete and inaccurate measurements are collected by coherence-hidden or -static sensing hardware due to its theoretical guarantee of recovery with severe measurements, computational tractability, and numerical robustness for ill-conditioned sensing matrices.

Biography

安聪沛, 本科、硕士毕业于中南大学，博士毕业于香港理工大学，现为西南财经大学数学学院副教授、博导。入选四川省"天府峨眉计划",最近又获得2023年四川省数学会应用数学一等奖。美国《数学评论》评论员，主持过三项国家自然科学基金。 在构造逼近，球面t设计，反问题计算等领域取得了国际同行关注的结果，例如2022年菲尔兹奖得主Maryna Viasovska就证明过安聪沛与和作者提出的关于球t-设计猜想。