Non-Asymptotic Statistical Similarity Measures

该方案中研究的相似性度量可以直观地理解为距离的度量。在日常生活中，两点之间的距离通常是物理长度。例如，一套公寓可能位于距离市中心3公里的地方。然而，这并不是测量距离的唯一方法。同样的公寓也可以被描述为距离市中心“15分钟”或“5个公交车站”。根据用例的不同，距离测量也可以明显更抽象。继续上面的例子，假设有人在城市里找一套公寓。除了到市中心的距离外，还有其他重要的标准，如公寓的大小或月租金。在数学上表述“找房问题”时，将所有这些方面组合成一个函数来定义离理想公寓的距离是很方便的。然后，从所有可用的单位中，挑选一个最接近理想距离的单位，通过最小化合适的距离来解决决策问题，这在概念上是一种优雅而强大的方法。然而，在实践中，如何选择正确的距离度量出现了问题。在我们的例子中，寻找公寓的人不是从抽象的距离出发，而是从目标和约束的角度来考虑：他们需要一套面积最小、月租金最高、离超市、公交车站、学校等尽可能近的公寓。从数学上讲，这对应于一个约束优化问题。这种表述通常不那么优雅，但允许对解决方案进行清晰的解释。简而言之，这个项目的目标是将这两种方法结合起来，以便两全其美。更具体地说，将开发一种系统的方法，允许构造距离度量，使得距离的最小化等价于明确定义的优化问题的解决。拟议的研究集中在统计推断问题上，即以最佳方式从噪声观测中获得关于系统状态的信息。在概率分布空间上定义了相应的距离。现有的统计距离，如Kullback-Leibler发散或阿尔法发散，要么基于无限(渐近)样本大小，要么基于公理，因此它们需要强有力的假设，不适合明确的解释。这个项目的成功完成将使我们有可能从定义明确的非渐近推理问题中构建统计距离。这反过来将使人们能够对统计信号处理和相关领域中的未决问题有新的见解，并在透明的理论基础上扩展现有的成果。

The similarity measures investigated in this proposal can be intuitively understood as measures for a distance. In daily life, the distance between two points is usually a physical length. For example, a flat might be located 3km from the city center. However, this is not the only way of measuring distances. The same flat might also be described as being "15min" or "5 bus stops" from the city center.Depending on the use-case, distance measures can also be significantly more abstract. Sticking with the above example, assume that someone is looking for a flat in a city. Apart from the distance to the city center, there are other important criteria such as the size of the flat or the monthly rent. When formulating the "flat-hunting problem'' mathematically, it is convenient to combine all these aspects into a single function that defines a distance from an ideal flat. From all available flats, one then picks the one that minimizes this distance, i.e., comes closest to the ideal.Solving decision making problems by minimizing a suitable distance is a conceptually elegant and powerful method. In practice, however, the problem of how to choose the correct distance measure arises. In our example, the person looking for a flat does not think in terms of abstract distances, but in terms of goals and constraints: they need a flat of a minimum size, at a maximum monthly rent, as close as possible to a supermarket, bus stop, school, etc. Mathematically, this corresponds to a constrained optimization problem. This formulation is often less elegant, but allows for a clear interpretation of the solution. In a nutshell, the aim of this project is to combine both approaches in order to get the best of both worlds. More specifically, a systematic method will be developed that allows to construct distance measures so that the minimization of a distance is equivalent to the solution of a well-defined optimization problem.The proposed research is focused on problems of statistical inference, i.e. obtaining information about the state of a system -- in an optimal manner -- from noisy observations. The corresponding distances are defined on the space of probability distributions. Existing statistical distances, such as the Kullback--Leibler divergence or the alpha divergence, are either based on infinite (asymptotic) sample sizes or axioms, so that they require strong assumptions and do not lend themselves to a clear interpretation. A successful completion of this project will make it possible to construct statistical distances from well-defined, non-asymptotic inference problems. This will in turn allow for new insights into open problems in statistical signal processing and related fields and for extensions of existing results based on transparent theoretical foundations.