The problem of finding and exploiting low-dimensional structures in high-dimensional data is taking on increasing importance in image, video, or audio processing; Web data analysis/search; and bioinformatics, where data sets now routinely lie in observational spaces of thousands, millions, or even billions of dimensions. The curse of dimensionality is in full play here: We often need to conduct meaningful inference with a limited number of samples in a very high-dimensional space. Conventional statistical and computational tools have become severely inadequate for processing and analyzing such high-dimensional data. © 2006 IEEE.