Data analysis by learning manifolds from high-dimensional data

降维（流形学习）通过将高维数据映射到更少维来解决处理复杂数据的问题。许多需要分析非常大的高维数据集的科学问题都可以从降维技术中受益。大多数现有的降维技术都忽略了数据点之间的观测序列和动作的信息。我最近与人合作开发了一种新的降维技术，称为动作尊重嵌入（Action respect Embedding， ARE），它利用这些额外的信息，成功地将动作转化为有意义和可解释的低维表示。在我提出的研究中，我打算改进这项新技术，使其更快、更有效。这可能会为现实世界的顺序决策问题（如机器人的规划和定位）带来新颖的解决方案。与现有技术不同的是，ARE方法不需要有关该领域的专家知识就可以找到有效的解决方案。在更理论化的方面，我建议将非线性降维技术作为概率模型进行探索和形式化。我将讨论如何构建这样的模型，以及当数据丢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.