Tangent Space To Keep The Manifold Non-linear Structure, Linear Dimensionality Reduction Method

As a technology for obtaining low dimensional representation of extremely high dimension data, dimensionality reduction plays an important role in many fields such as pattern recognition, machine learning, image processing. It is frequently used by human to discover the essential principal which is deeply hidden in complicated phenomenon. With the purpose of preserving several certain properties, dimensionality reduction generate a function mapping the sample points in a high dimension space to a subspace with much lower dimension. So it can simplify the successive procedure. According to different properties we wish to preserve, scientists have proposed many algorithms for dimensionality reduction, such as PCA, LLE, they all act well in their own fields, and become useful methods for data mining.In this paper I will give a brief introduction about the background and the existing research work of dimensionality reduction at first, then I will review some classical algorithms by analyzing their idea and approach, I even present the detail experiment results of these methods. After this, I focus on the nonlinear structure of high dimension manifolds and define a concept called linearization degree on different directions of a manifold. Then I fix a subspace with high linearization degree via the manifold tangent space. At last, I will propose a linear dimensionality reduction based on preserving manifold's the nonlinear structure.