标题： 通过多惩罚正则化解决混合问题时最优支持恢复的条件 显示英文标题

标题： Conditions on optimal support recovery in unmixing problems by means of multi-penalty regularization

摘要： 受音频处理和医学图像分析中多个实际应用的启发，在这些应用中，感兴趣量由多个来源生成，需要被精确建模和分离，并结合正则化理论和优化领域的最新进展，我们通过多惩罚正则化方法研究了反问题中最佳支撑恢复的条件，这些问题属于解混类型。 我们考虑并分析了一个正则化泛函，它由一个数据拟合项（信号与噪声以加性方式混合）、一个非光滑、凸、稀疏促进项以及一个二次惩罚项组成，用于对噪声进行建模。 我们证明了单参数情况下已建立的稀疏恢复理论不仅可以在多惩罚设置下推广，而且我们还展示了多惩罚正则化在支撑识别方面相较于单一 $\ell^1$-最小化的优势。 此外，我们通过广泛的数值模拟验证并支持了理论结果，这些模拟给出了多惩罚正则化方案相对于单参数对应方案的鲁棒性统计信息。 最终，我们确认在实验中考虑的压缩感知问题中，与标准 $\ell^1$-正则化相比，性能有了显著提升。 显示英文摘要

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.