Locally Nonlinear Embedding Dimensionality Reduction Method For High Dimensional Big Data

With the rapid development and wide application of information technology,a large number of high-dimensional data have been produced in many fields.It is very difficult to analyze and use these high-dimensional data directly.Dimensionality reduction can solve this kind of difficulty to a large extent.Locally linear embedding(LLE)is a nonlinear dimensionality reduction algorithm,which has become a research hotspot since it was proposed in 2000.LLE can find low-dimensional structure hidden in high-dimensional data,and has low computational complexity.However,LLE requires points to have linear relationship locally,which is difficult to meet in the real world.In the real world,data is often noisy or sparse,at this time,locally linear relationship is easy to be destroyed,so the effect of LLE is not good.Based on LLE,locally nonlinear embedding(LNE)algorithm is proposed.LNE is the extension and improvement of LLE.It not only has the advantages of low complexity of LLE,but also extends the application scope of LLE.The main idea is that if the local points do not meet the linear relationship,it maintains a certain nonlinear relationship,which is realized by applying mapping to the nearest neighbor of each point.We can also understand LNE from another aspect.After mapping the nearest neighbor points of each point,these points become new points,among the new points,the local linear relationship is just satisfied.Compared with various improved algorithms of LLE,LNE has two obvious advantages:(1)it is more convenient to implement without introducing redundant parameters;(2)it has a wider application range,can be used in noisy data,sparse data and other data with complex structure,the results are excellent.One of the difficulties of LNE algorithm is how to choose the right mapping.In this paper,based on repeated experiments,we give a class of available mappings and summarize the necessary conditions that the mappings should meet.At the same time,we make a theoretical analysis of LNE,which gives the reason why LNE is suitable for noisy data to some extent.Finally,experiments are carried out on several datasets,including S surface,Swiss roll surface,a variant of Swiss roll surface and MNIST handwritten digits,to verify the effectiveness of the proposed algorithm,which provides a new idea for the actual dimensionality reduction of high-dimensional big data.