Abstract: ISOMAP is a widely used nonlinear unsupervised dimensionality reduction algorithm,and preserves the coor-dinate transformation from high-dimensional space to low-dimensional space by keeping the geodesic distance between the observation samples.However,practical data are inevitably corrupted by noise.Since the calculation of geodesic distance is sensitive to noise and does not consider the density distribution of dataset,the geometric deformation of the ISOMAP algorithm is happening after dimensionality reduction.For this shortcoming,according to the idea of local density,a density scaling factor based ISOMAP algorithm called D-ISOMAP was proposed.In the framework of traditional ISOMAP algorithm,local density scaling factor is first calculated for each observation sample.And then the original geo-desic distance of two samples is divided by the product of their density scaling factors.Finally,the improved distance matrix is obtained by the shortest path algorithm.Since the improved geodesic distance is reduced in the larger density region and enlarged in the smaller density region,it can achieve a good effect in noisy situations.The experimental results on artificial dataset and UCI dataset show that the D-ISOMAP algorithm is better than the classical unsupervised dimensionality reduction algorithms in terms of visualization and clustering of data sets.

Key words: Density scaling factor, Dimensionality reduction, ISOMAP, Manifold learning, Noise data