Manifold learning is a hot research topic in the field of computer science. A crucial issue with current manifold learning methods is that they lack a natural quantitative measure to assess the quality of learned embeddings, which greatly limits their applications to real-world problems. In this paper, a new embedding quality assessment method for manifold learning, named as Normalization Independent Embedding Quality Assessment (NIEQA), is proposed. Compared with current assessment methods which are limited to isometric embeddings, the NIEQA method has a much larger application range due to two features. First, it is based on a new measure which can effectively evaluate how well local neighborhood geometry is preserved under normalization, hence it can be applied to both isometric and normalized embeddings. Second, it can provide both local and global evaluations to output an overall assessment. Therefore, NIEQA can serve as a natural tool in model selection and evaluation tasks for manifold learning. Experimental results on benchmark data sets validate the effectiveness of the proposed method.
We present the Procrustes measure, a novel measure based on Procrustes rotation that enables quantitative comparison of the output of manifold-based embedding algorithms such as LLE (Roweis and Saul, Science 290(5500), 2323---2326, 2000) and Isomap (, Science 290(5500), 2319---2323, 2000). The measure also serves as a natural tool when choosing dimension-reduction parameters. We also present two novel dimension-reduction techniques that attempt to minimize the suggested measure, and compare the results of these techniques to the results of existing algorithms. Finally, we suggest a simple iterative method that can be used to improve the output of existing algorithms.