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ABSTRACT Diffusion maps are an emerging data-driven technique for nonlinear dimensionality reduction, which are especially useful for the analysis of coherent structures and nonlinear embeddings of dynamical systems. Thus, long time-series data presents a significant challenge for fast and efficient embedding. We propose integrating the Nystr om method with diffusion maps in order to ease the computational demand. We achieve a speedup of roughly two to four times when approximating the dominant diffusion map components. Index Terms-- Dimension Reduction, Nystr om method 1. MOTIV A TION In the era of'big data', dimension reduction is critical for data science.

Dimensionality reduction is the process by which a set of data points in a higher dimensional space are mapped to a lower dimension while maintaining certain properties of these points relative to each other. One important property is the preservation of the three angles formed by a triangle consisting of three neighboring points in the high dimensional space. If this property is maintained for those same points in the lower dimensional embedding then the result is a conformal map. However, many of the commonly used nonlinear dimensionality reduction techniques, such as Locally Linear Embedding (LLE) or Laplacian Eigenmaps (LEM), do not produce conformal maps. Post-processing techniques formulated as instances of semi-definite programming (SDP) problems can be applied to the output of either LLE or LEM to produce a conformal map. However, the effectiveness of this approach is limited by the computational complexity of SDP solvers. This paper will propose an alternative post-processing algorithm that produces a conformal map but does not require a solution to a SDP problem and so is more computationally efficient thus allowing it to be applied to a wider selection of datasets. Using this alternative solution, the paper will also propose a new algorithm for 3D object classification. An interesting feature of the 3D classification algorithm is that it is invariant to the scale and the orientation of the surface.