We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of the solution. We establish general conditions for existence and derive the generic parametric representation of the solution components. These representations fall into three categories depending on the underlying regularization norm: (i) a linear expansion in terms of predefined “kernels” when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a non-linear (duality) mapping of a linear combination of measurement functionals when the component Banach space is strictly convex, and, (iii) an adaptive expansion in terms of a small number of atoms within a larger dictionary when the component Banach space is not strictly convex. Our approach generalizes and unifies a number of multi-kernel (RKHS) and sparse-dictionary learning techniques for compressed sensing available in the literature. It also yields the natural extension of the classical spline-fitting techniques in (semi-)RKHS to the abstract Banach-space setting.

我们描述了一大类凸优化问题的解决方案，这些问题解决了从有限数量的线性测量中重建函数的问题。潜在的假设是，该解决方案可分解为一个有限的分量之和，其中每个分量都属于它自己规定的 Banach 空间；此外，通过惩罚解决方案的某些复合范数来使问题正则化。我们建立存在的一般条件并推导出解决方案组件的通用参数表示。这些表示根据潜在的正则化范数分为三类：（i）当组件空间是再生内核希尔伯特空间（RKHS）时，根据预定义的“内核”进行线性扩展，(ii) 当分量 Banach 空间是严格凸的时，测量泛函的线性组合的非线性（对偶）映射，以及，（iii）当分量为较大字典中的少量原子时，自适应扩展Banach 空间不是严格凸的。我们的方法概括并统一了许多用于文献中可用的压缩感知的多内核 (RKHS) 和稀疏字典学习技术。它还产生了（半）RKHS 中经典样条拟合技术到抽象 Banach 空间设置的自然扩展。我们的方法概括并统一了许多用于文献中可用的压缩感知的多内核 (RKHS) 和稀疏字典学习技术。它还产生了（半）RKHS 中经典样条拟合技术到抽象 Banach 空间设置的自然扩展。我们的方法概括并统一了许多用于文献中可用的压缩感知的多内核 (RKHS) 和稀疏字典学习技术。它还产生了（半）RKHS 中经典样条拟合技术到抽象 Banach 空间设置的自然扩展。