As science and engineering are increasingly data driven , The role of optimization has been extended to almost every stage of the data analysis pipeline , From signal and data acquisition to modeling and prediction . Optimization problems encountered in practice are often non convex . Although challenges vary from problem to problem , But a common source of nonconvex is nonlinearity in data or measurement models . The model is usually nonlinear , Create complex 、 Non convex objective perspective , Have multiple equivalent solutions . However , Simple method ( If the gradient falls ) It often performs very well in practice . The purpose of this review is to highlight a class of treatable nonconvex problems , It can be understood from the perspective of symmetry . These problems present a typical geometric structure : The local minimum is a single “ Truth value ” A symmetric copy of the solution , Other critical points appear at the equilibrium superposition of the symmetric copies of the truth solution , And show negative curvature in the direction of breaking symmetry . This structure enables an effective method to obtain the global minimum . We discussed imaging 、 Examples of this phenomenon arise from a wide range of problems in signal processing and data analysis . We emphasize the key role of symmetry in shaping an objective perspective , The different functions of rotational symmetry and discrete symmetry are discussed . There are many observed phenomena and outstanding problems in this area ; Last , We emphasize the direction of future research .

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