RI-Small: Learning from data of low intrinsic dimension

该项目从看似高维但实际上具有低内在维的数据中研究机器学习（例如，数据位于低维流形上）。许多应用程序中的物理限制恰恰会产生这样的情况。该项目正在开发机器学习系统，该系统使用的资源（例如，计算时间和空间）与内在维度而不是外在维度相匹配。数据分布在歧管上的想法很有吸引力，也很有启发性，并且已经成为最近机器学习领域许多令人兴奋的工作的灵感来源。通常目的是将这些数据嵌入到较低维空间中，之后应用标准方法消耗的资源更少。pi已经发展了一种精确的内在维度概念，它既能捕捉到多种直觉，又足够广泛，既能在统计上合理，又能在经验上得到验证。然后将该量视为一个基本参数，根据该参数可以评估各种新的非参数方法。第一个是k-d树的简单变体，它可以证明自适应于内在维数。pi还考虑了非参数分类和回归、流形学习和嵌入的方案。这些新的算法和想法将应用于各种领域的基本挑战，包括传感器网络、计算机视觉、蛋白质结构预测和机器人控制。

This project studies machine learning from data that appears high dimensional but, in fact, has low intrinsic dimension (e.g., the data lies on a low-dimensional manifold). Physical constraints in many applications produce exactly such a situation. The project is developing machine learning systems that use resources (e.g., compuational time and space) that scale with the intrinsic rather than the extrinsic dimension. The idea of data lying on a manifold is appealing and suggestive, and has been the inspiration of a lot of recent, exciting work in machine learning. Often the aim is to embed such data into a lower dimensional space, after which the application of standard methods consume less resources. The PIs have developed a precise notion of intrinsic dimension that captures the manifold intuition while being broad enough to both be statistically sensible and empirically verifiable. This quantity is then treated as a fundamental parameter in terms of which a variety of new nonparametric methods can be assessed. The first of these is a simple variant of the k-d tree that is provably adaptive to intrinsic dimension. The PIs also consider schemes for nonparametric classification and regression, for manifold learning, and for embedding. These new algorithms and ideas will be applied to fundamental challenges in a variety of domains, including sensor networks, computer vision, protein structure prediction, and robotic control.