RI: Small: Algebraic and Spectral Structure of Data in High Dimension

从数据中获取信息是现代科学技术最基本的问题之一。机器学习的目标是开发算法，从复杂的高维数据中自动提取有用的信息。 朝着这一目标取得进展需要发展对数据方面的理解，这些方面可以进行分析，并且可以使用计算效率高的方法进行学习。特别是，高维数据中的非线性结构建模已经成为非常具有挑战性和活跃的研究方向之一，在过去的十年中取得了重大进展。本项目的目标是开发和分析基于谱和代数方法的数据的新数学表示。我们将探索数据中的不同结构，如聚类，流形或参数模型结构，如何反映在它们的光谱和代数属性中，以及如何从数据中提取算法，特别注意高维和非线性问题。这些见解将用于为推理和数据分析任务构建更好、更自适应的算法。我们还将从实验和理论上分析这些算法的性能，当数据偏离假定的模型结构。 这是实际应用中的一个关键问题，几乎总是涉及不确定性和噪声。

Obtaining information from data is one of the most fundamental problems of modern science and technology. The aim of machine learning is to develop algorithms to automatically extract useful information from complex, high-dimensional data. Making progress toward this aim requires developing an understanding of the aspects of data, which are amenable to analysis and can be learned using computationally efficient methods. In particular, modeling non-linear structures in high-dimensional data has become one of the very challenging and active lines of research, which has seen significant progress over the last ten years.The goal of this project is to develop and analyze new mathematical representations for data, based on spectral and algebraic methods. We will explore how different structures in the data, such as cluster, manifold or parametric model structures, are reflected in their spectral and algebraic properties and how they can be extracted algorithmically from data, paying particular attention to the issues of high dimensionality and non-linearity. These insights will be used to build better and more adaptive algorithms for inference and data analysis tasks. We will also analyze experimentally and theoretically properties of these algorithms, when data deviates from the posited model structure. This is a key issue in practical applications, which nearly always involve uncertainty and noise.