通过多惩罚正则化方法研究混叠问题中最佳支持恢复的条件
Conditions on optimal support recovery in unmixing problems by means of multi-penalty regularization
AI总结：
本文研究了通过多惩罚正则化方法在混叠逆问题中实现最佳支持恢复的条件，通过数值模拟验证了其在稀疏恢复中的优越性。
摘要：
受音频处理和医学图像分析等实际应用的启发，以及正则化理论和优化的最新进展，我们研究了通过多惩罚正则化方法在混叠类型逆问题中实现最佳支持恢复的条件。我们考虑并分析了一个由数据保真项、非光滑凸稀疏促进项和二次惩罚项组成的正则化功能，证明了稀疏恢复单参数情况下的经典理论可推广到多惩罚设置，并展示了多惩罚正则化在支持识别方面的增强特性。通过广泛的数值模拟验证了理论结果，并确认了与单参数方法相比，多惩罚正则化在压缩感知问题中的显著性能提升。
英文摘要：
Inspired by several real-life applications in audio processing and medical image analysis, where the quantity of interest is generated by several sources to be accurately modeled and separated, as well as by recent advances in regularization theory and optimization, we study the conditions on optimal support recovery in inverse problems of unmixing type by means of multi-penalty regularization. We consider and analyze a regularization functional composed of a data-fidelity term, where signal and noise are additively mixed, a non-smooth, convex, sparsity promoting term, and a quadratic penalty term to model the noise. We prove not only that the well-established theory for sparse recovery in the single parameter case can be translated to the multi-penalty settings, but we also demonstrate the enhanced properties of multi-penalty regularization in terms of support identification compared to sole $\ell^1$-minimization. We additionally confirm and support the theoretical results by extensive numerical simulations, which give a statistics of robustness of the multi-penalty regularization scheme with respect to the single-parameter counterpart. Eventually, we confirm a significant improvement in performance compared to standard $\ell^1$-regularization for compressive sensing problems considered in our experiments.
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