Implicit Multidimensional Projection Of Local Subspaces

With the advent of the era of artificial intelligence and big data,a large amount of multi-dimensional data is produced in people's daily lives.How to better visualize these data to help people dig out valuable information has become an important research direction.The dimensionality reduction scatter plot is one of the important tools for visualizing these multi-dimensional data.However,the traditional dimensionality reduction scatter plot can only represent the density distribution of the data,and some inherent manifold features in the data are often difficult to be presented.To solve this problem,this paper proposes a method framework for implicit subspace projection,which is suitable for analyzing the results of derivable dimensionality reduction algorithms.We use local PC A to represent the local subspace information around the sample,project it into a scatter plot,and calculate a closed Bspline curve based on the projection result to replace the traditional scatter.In order to calculate the projection of the local PCA,we regard the dimensionality reduction process as an implicit function,and calculate the derivative relationship between them according to the implicit function theorem,and then use this derivative to calculate the projection of the local PCA,making the calculation process of the projection more accurate.In addition,this article also designed a simple webpage interactive system to support users to adjust the visualization results and simple exploration.In order to verify the effectiveness of the method,we designed a numerical evaluation experiment to prove the accuracy of the method in this paper.At the same time,we used some commonly used machine learning data sets to do case analysis.The results show that through our method,we can indeed find some data structure features that cannot be displayed by traditional scatter plots.Inspired by the results of the implicit local subspace method in this paper,we then extended this idea to explore the introduction of local subspace information in the dimensionality reduction method,so that it can better identify the complex manifold structure in the data.As one of the most successful dimensionality reduction methods for manifold visualization,t-SNE has strong manifold structure recognition capabilities.However,when multiple manifold structures in the data intersect each other,the result of t-SNE is often unsatisfactory.In response to this problem,we modified the highdimensional probability calculation method of t-SNE,and replaced the original Euclidean distance with the weighted sum of the local PCA dissimilarity and the Euclidean distance.Our method can directly call the gradient method of t-SNE to calculate the dimensionality reduction result.Finally,through case analysis and quantitative evaluation experiments,this paper proves that the method in this paper can not only distinguish the samples that belong to different manifolds,but also maintain the structural information of each manifold.