Topological Deformation Learning: Nonlinear Dimension Reduction by Autonomous Evolving of Data Manifold

To provide a geometric and self-organized model of dimension reduction in machine learning, a novel learning mechanism is proposed for data set dimension reduction by autonomous deforming of data manifolds. The deformation of the data manifold is guided by the proposed deforming vector field, which is defined by two kinds of virtual interactions between data points. In the deformation mechanism, the flattening of the data manifold is achieved as an emergent behavior under the elastic and repelling interactions between data points, meanwhile, the topological structure of the manifold is preserved. To overcome the uneven sampling (or “short-cut edge”) problem in many learning tasks, the soft neighborhood is proposed, in which the neighbor degree is defined and adaptive interactions between neighbor points are achieved. The proposed Topological Deformation Learning provides a novel geometric viewpoint on dimension reduction. Experimental results of image data sets prove the effectiveness of the proposed method. The results also indicate that implicit features of data sets may be revealed.