Abstract

Predictive Modeling problems deal with high-dimensional data; however, the curse of dimensionality presents an obstacle to the use of many methods for their solutions. In many applications, real-world data occupy only a very small part of high-dimensional observation space whose intrinsic dimension is essentially lower than that of the space. A popular model for such data is the manifold model, according to which the data lie on an unknown low-dimensional manifold (Data Manifold) embedded in the ambient high-dimensional space. Predictive modeling problems, which are studied under this assumption, are called manifold estimation problems. The general goal of such problems is to identify the low-dimensional structure of multidimensional data from a given dataset. If dataset points are sampled according to an unknown probability measure on the data manifold, there is a need to model manifolds when solving various machine learning problems. We provide a short survey of such problems and outline some approaches to their solution.

中文翻译：

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机器学习中的流形建模

摘要

预测建模问题处理高维数据；然而，维数灾难阻碍了许多方法的使用。在许多应用中，真实世界的数据只占据高维观测空间的很小一部分，其内在维数本质上低于空间的维数。此类数据的流行模型是流形模型，根据该模型，数据位于嵌入环境高维空间中的未知低维流形（Data Manifold）上。在此假设下研究的预测建模问题称为流形估计问题。此类问题的一般目标是从给定数据集中识别多维数据的低维结构。如果根据数据流形上的未知概率度量对数据集点进行采样，则在解决各种机器学习问题时需要对流形进行建模。我们对此类问题进行了简短的调查，并概述了解决这些问题的一些方法。