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arXiv 1809.05045math.OC
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有限维变分反问题解的稀疏性

Sparsity of solutions for variational inverse problems with finite-dimensional data

Kristian Bredies, Marcello Carioni

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摘要：

在本文中，我们刻画了变分问题的稀疏解的形式$\min_{u\in X} ϕ(u) + F(\mathcal{A} u)$，其中$X$是一个局部凸空间，$\mathcal{A}$是一个线性连续算子，映射到有限维希尔伯特空间和$ϕ$是一个矩阵。更准确地说，我们证明了存在一个极小是“稀疏”的意义上说，它是表示为一个线性组合的极值点的单位球与正则化$ϕ$（可能翻译的零空间中的元素$ϕ$）。我们将这一结果应用到相关的正则化，如全变差范数和Radon范数的标量线性微分算子。在第一个例子中，我们提供了所谓的阶梯效应的理论证明，在第二个例子中，我们在较弱的假设下恢复了[31]中的结果。

英文摘要：

In this paper we characterize sparse solutions for variational problems of the form $\min_{u\in X} ϕ(u) + F(\mathcal{A} u)$, where $X$ is a locally convex space, $\mathcal{A}$ is a linear continuous operator that maps into a finite dimensional Hilbert space and $ϕ$ is a seminorm. More precisely, we prove that there exists a minimizer that is `sparse' in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer $ϕ$ (possibly translated by an element in the null space of $ϕ$). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear differential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in [31] under weaker hypotheses.

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