Guide Manifold Alignment by Relative Comparisons

When we are faced with data, one common task is to learn the correspondence relationship between different data sets. More concretely, by learning data correspondence, samples that share similar intrinsic parameters, which are often hard to estimate directly, can be discovered. For example, given some face image data, an alignment algorithm is able to find images of two different persons with similar poses or expressions. We call this technique the alignment of data. Besides its usage in data analysis and visualization, this problem also has wide potential applications in various fields. For instance, in facial expression recognition, one may have a set of standard labeled images with known expressions, such as happiness, sadness, surprise, anger and fear, of a particular person. Then we can recognize the expressions of another person just by aligning his/her facial images to the standard image set. Its application can also be found directly in pose estimation. One can refer to (Ham, Lee & Saul, 2005) for more details. Although intuitive, without any premise this alignment problem can be very difficult. Usually, the samples are distributed in high-dimensional observation spaces, and the relation between features and samples’ intrinsic parameters can be too complex to be modeled explicitly. Therefore, some hypotheses about the data distribution are made. In the recent years, the manifold assumption of data distribution has been very popular in the field of data mining and machine learning. Researchers have realized that in many applications the samples of interest are actually confined to particular subspaces embedded in the high-dimensional feature space (Seung & Lee, 2000; Roweis & Saul, 2000). Intuitively, the manifold assumption means that certain groups of samples are lying in a non-linear low-dimensional subspace embedded in the observation space. This assumption has been verified to play an important role in human perception (Seung & Lee, 2000), and many effective algorithms are developed under it in the recent years. Under the manifold assumption, structural information of data can be utilized to facilitate the alignment.