| Tenenbaum, J.B. Mapping a Manifold of Perceptual Observations 1998 (NIPS-10) [pdf] |
| Nonlinear dimensionality reduction is formulated here as the problem of trying to find a Euclidean feature-space embedding of a set of observations that preserves as closely as possible their intrinsic metric structure the distances between points on the observation manifold as measured along geodesic paths. Our isometric feature mapping procedure, or isomap, is able to reliably recover low-dimensional nonlinear structure in realistic perceptual data sets, such as a manifold of face images, where conventional global mapping methods find only local minima. The recovered map provides a canonical set of globally meaningful features, which allows perceptual transformations such as interpolation, extrapolation, and analogy highly nonlinear transformations in the original observation space to be computed with simple linear operations in feature space. |
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| Tenenbaum, J.B., de Silva, V., Langford, J.C. Nonlinear Dimensionality Reduction by Locally Linear Embedding Science 2000 (290):2323-2326 [pdf] |
| Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood- preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text. |
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