The Hierarchical Isometric Self-Organizing Map for Manifold Representation

Abstract

We present an algorithm, Hierarchical ISOmetric Self-Organizing Map (H-ISOSOM), for a concise, organized manifold representation of complex, non-linear, large scale, high-dimensional input data in a low dimensional space. The main contribution of our algorithm is threefold. First, we modify the previous ISOSOM algorithm by a local linear interpolation (LLl) technique, which maps the data samples from low dimensional space back to high dimensional space and makes the complete mapping pseudo-invertible. The modified-ISOSOM (M-ISOSOM) follows the global geometric structure of the data, and also preserves local geometric relations to reduce the nonlinear mapping distortion and make the learning more accurate. Second, we propose the H-ISOSOM algorithm for the computational complexity problem of Isomap, SOM and LLI and the nonlinear complexity problem of the highly twisted manifold. H-ISOSOM learns an organized structure of a non-convex, large scale manifold and represents it by a set of hierarchical organized maps. The hierarchical structure follows a coarse-to-fine strategy. According to the coarse global structure, it "unfolds " the manifold at the coarse level and decomposes the sample data into small patches, then iteratively learns the nonlinearity of each patch in finer levels. The algorithm simultaneously reorganizes and clusters the data samples in a low dimensional space to obtain the concise representation. Third, we give quantitative comparisons of the proposed method with similar methods on standard data sets. Finally, we apply H-ISOSOM to the problem of appearance-based hand pose estimation. Encouraging experimental results validate the effectiveness and efficiency of H-ISOSOM.