Non-Asymptotic Statistical Similarity Measures

本文研究的相似性度量可以直观地理解为距离度量。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离市中心“15分钟”或“5个公交站”。根据用例的不同，距离度量也可以明显地更加抽象。继续上面的例子，假设有人在城市里找房子。除了到市中心的距离，还有其他重要的标准，如公寓的大小或月租金。当用数学方法表述“寻平问题”时，将所有这些方面结合成一个函数来定义到理想平面的距离是很方便的。然后从所有可用的平面中选择距离最小的平面，即最接近理想的平面。通过最小化合适的距离来解决决策问题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，寻找公寓的人不会考虑抽象的距离，而是考虑目标和约束：他们需要一个最小的公寓，每月租金最高，尽可能靠近超市、公交车站、学校等。在数学上，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，这个项目的目的是结合这两种方法，以获得两全其美。更具体地说，将开发一种系统的方法，允许构造距离度量，使距离的最小化等同于一个定义良好的优化问题的解决方案。提出的研究重点是统计推断问题，即从噪声观测中以最佳方式获取有关系统状态的信息。相应的距离在概率分布空间上定义。现有的统计距离，如Kullback- Leibler散度或alpha散度，要么基于无限（渐近）样本量，要么基于公理，因此它们需要强有力的假设，并且不适合清晰的解释。这个项目的成功完成将使从定义良好的非渐近推理问题中构造统计距离成为可能。这将反过来允许对统计信号处理和相关领域的开放问题有新的见解，并基于透明的理论基础扩展现有结果。

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.