An improved ISOMAP for visualization and classification of multiple manifolds

Abstract

The classical algorithm ISOMAP can find the intrinsic low-dimensional structures hidden in high-dimensional data uniformly distributed on or around a single manifold. But if the data are sampled from mul-ti- class, each of which corresponds to an independent manifold, and clusters formed by data points belonging to each class are separated away, several disconnected neighborhood graphs will occur, which leads to the failure of ISOMAP. Moreover, ISOMAP behaves in an unsupervised manner and therefore works less effectively for classification. In this paper, two improved versions of ISOMAP, namely Multi-Class Multi-Manifold ISOMAP (MCMM-ISOMAP) for data visualization and ISOMAP for Classification (ISOMAP-C), are proposed respectively. MCMM-ISOMAP constructs a single neighborhood graph, named a between-class neighborhood graph by connection of between-class points with shortest distance of each within-class neighborhood graph, and then ISOMAP algorithm is applied to find the intrinsic low-dimensional embedding structure. ISOMAP-C is essentially an extension of MCMM-ISOMAP to a supervised manner, which is multiplied by scaling factor greater than one so that low dimensional data set after mapping become more compact within class and more separate between classes. Finally, the mapping function from original high dimensional space to low dimensional space can be approximately modeled using Back-Propagation neural network combined with genetic algorithm. Experimental results using MCMM-ISOMAP on synthetic and real data reveal its effectiveness and ones using ISOMAP-C show that the performance is greatly enhanced and robust to noisy data. © Springer-Verlag 2013.