Manifold Denoising As Preprocessing for Finding Natural Representations of Data

Matthias Hein, Markus Maier

A natural representation of data is given by the parameters which generated the data. If the space of parameters is continuous, then we can regard it as a manifold. In practice, we usually do not know this manifold but we just have some representation of the data, often in a very high-dimensional feature space. Since the number of internal parameters does not change with the representation, the data will effectively lie on a low-dimensional submanifold in feature space. However, the data is usually corrupted by noise, which particularly in high-dimensional feature spaces makes it almost impossible to find the manifold structure. This paper reviews a method called Manifold Denoising, which projects the data onto the submanifold using a diffusion process on a graph generated by the data. We will demonstrate that the method is capable of dealing with non-trival high-dimensional noise. Moreover, we will show that using the denoising method as a preprocessing step, one can significantly improve the results of a semi-supervised learning algorithm.

Subjects: 12. Machine Learning and Discovery; 9.3 Mathematical Foundations

Submitted: Apr 20, 2007

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