Data analysis by learning manifolds from high-dimensional data

多维性约简（流形学习）通过将高维数据映射到更少的维度来解决处理复杂数据的问题。许多需要分析非常大和高维数据集的科学问题可以从降维技术中受益。大多数现有的降维技术忽略了关于数据点之间的观测和动作序列的信息。 我最近共同开发了一种新的降维技术，称为Action Respecting Embedding（ARE），它利用了这些额外的信息，成功地将动作转换为有意义的和可解释的低维表示。在我提出的研究中，我打算改进这种新技术，使其更快，更有效。这可能会导致新的解决方案，以现实世界的顺序决策问题，如规划和定位机器人。与现有技术不同，ARE方法不需要有关该领域的专业知识来找到有效的解决方案。在更多的理论方面，我建议探索和形式化的非线性降维技术的概率模型。我将讨论如何构建这样的模型，以及当数据缺失时它们应该如何响应。这在物理学、经济学和医学等领域有许多潜在的用途，这些领域必须从大型数据集中提取有意义的信息。

Dimensionality reduction (manifold learning) addresses the problem of dealing with complex data by mapping high-dimensional data into fewer dimensions. Many problems of scientific interest that require the analysis of very large and high-dimensional data sets can benefit from dimensionality reduction techniques. Most existing dimensionality reduction techniques ignore information about the sequence of observations and actions between data points.  I recently co-developed a new dimensionality reduction technique called Action Respecting Embedding (ARE) which exploits this additional information, successfully translating actions into meaningful and interpretable low-dimensional representations. In my proposed research, I intend to refine this new technique to make it faster and more efficient. This could lead to novel solutions to real-world sequential decision problems such as planning and localization for robots. Unlike existing techniques, an ARE approach would require no expert knowledge about the domain to find effective solutions. On the more theoretical side, I propose to explore and formalize non-linear dimensionality reduction techniques as probabilistic models. I will address the problem of how such models should be constructed, and how they should respond when data is missing. This has many potential uses in fields such as physics, economics, and medicine, where meaningful information must be extracted from large data sets.