8,190 research outputs found
A scale-based approach to finding effective dimensionality in manifold learning
The discovering of low-dimensional manifolds in high-dimensional data is one
of the main goals in manifold learning. We propose a new approach to identify
the effective dimension (intrinsic dimension) of low-dimensional manifolds. The
scale space viewpoint is the key to our approach enabling us to meet the
challenge of noisy data. Our approach finds the effective dimensionality of the
data over all scale without any prior knowledge. It has better performance
compared with other methods especially in the presence of relatively large
noise and is computationally efficient.Comment: Published in at http://dx.doi.org/10.1214/07-EJS137 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Principal arc analysis on direct product manifolds
We propose a new approach to analyze data that naturally lie on manifolds. We
focus on a special class of manifolds, called direct product manifolds, whose
intrinsic dimension could be very high. Our method finds a low-dimensional
representation of the manifold that can be used to find and visualize the
principal modes of variation of the data, as Principal Component Analysis (PCA)
does in linear spaces. The proposed method improves upon earlier manifold
extensions of PCA by more concisely capturing important nonlinear modes. For
the special case of data on a sphere, variation following nongeodesic arcs is
captured in a single mode, compared to the two modes needed by previous
methods. Several computational and statistical challenges are resolved. The
development on spheres forms the basis of principal arc analysis on more
complicated manifolds. The benefits of the method are illustrated by a data
example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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