超分辨率满足机器学习：近似测量
Super-resolution meets machine learning: approximation of measures
摘要：
超分辨问题一般来说是恢复一个关于某个规范正交系的给定多个系数$\hatμ(k)$的非线性支撑测度$μ$。有趣的情况涉及的情况下，其中所需的系数的数量基本上小于在支持$μ$的点之间的最小分离的倒数的幂。在本文中，我们考虑更严重的问题，恢复$μ$约没有任何假设$μ$超过有一个有限的全变差。特别地，可以在连续体上支持$μ$，使得支持$μ$的点之间的最小间隔是$0$。这个问题的一个变体也是机器学习以及反卷积的逆问题。我们定义了一个适当的概念之间的距离的目标措施和它的恢复版本，给出一个明确的表达式的恢复运营商，并估计之间的距离$μ$和它的近似。我们表明，这些估计是最好的可能在许多不同的方式。我们还解释了为什么一个支持的措施的近似质量的恢复是有界的从下面，如果信息量是小于什么是要求在超分辨率问题。
英文摘要：
The problem of super-resolution in general terms is to recuperate a finitely supported measure $μ$ given finitely many of its coefficients $\hatμ(k)$ with respect to some orthonormal system. The interesting case concerns situations, where the number of coefficients required is substantially smaller than a power of the reciprocal of the minimal separation among the points in the support of $μ$. In this paper, we consider the more severe problem of recuperating $μ$ approximately without any assumption on $μ$ beyond having a finite total variation. In particular, $μ$ may be supported on a continuum, so that the minimal separation among the points in the support of $μ$ is $0$. A variant of this problem is also of interest in machine learning as well as the inverse problem of de-convolution. We define an appropriate notion of a distance between the target measure and its recuperated version, give an explicit expression for the recuperation operator, and estimate the distance between $μ$ and its approximation. We show that these estimates are the best possible in many different ways. We also explain why for a finitely supported measure the approximation quality of its recuperation is bounded from below if the amount of information is smaller than what is demanded in the super-resolution problem.
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