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The underlying variability of many high-dimensional data sets can be characterized by very few degrees of freedom. For instance, for a collection of images,
these degrees of freedom may correspond to the lighting conditions, pose and distances of
the viewed objects. In the framework of manifold learning, we imagine data lying on a low
dimensional (nonlinear) manifold embedded in a high dimensional space.
W can discover the manifolds - hence the intrinsic structures - using only limited
a prior knowledge about the data, for example, nearest-neighborhood relations. Manifold learning algorithms have found many applications in information visualization and navigation
of texts and images, robotics, language processing and etc. Our
research builds on top of current manifold learning approaches and significantly extends them.
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Left.We embed 2000 images in 3-dimensional space with Conformal Component Analysis, a manifold learning algorithm that preserves local angles. A few representative images are overlaid on the top of these 2000 3-d dots. They show interesting and interpretably clustering patterns. 
Above.We discover geographical locations of US cities from incomplete pairwise distances among them, eg., reported by sensors in a sensor network. The key step is to preserve pairwise distances while looking for a low-rank embedding solution (specifically, 2-d coordinates). |
| 1. |
Kilian Q. Weinberger, Fei Sha, Qihui Zhu, and Lawrence K. Saul.Graph regularization for maximum variance unfolding, with an application to sensor
localization. Advances in Neural Information Processing Systems 19, pages 1489-1496. B. Schölkopf, J.C. Platt, and T. Hofmann. Cambridge, MA, 2007. MIT Press. [ PDF ] |
| 2. |
Lawrence K. Saul, Kilian Q. Weinberger, Fei Sha, Jihun Hamm, and Daniel D.
Lee. Spectral methods for dimensionality reduction.
Semi-supervised Learning, pages 293-308. Olivier Chapelle,
Bernhard Schöolkopf, and Alexander Zien. MIT Press, Cambridage, MA, 2006. |
| 3. |
Fei Sha and Lawrence K. Saul.Analysis and extension of spectral methods for
nonlinear dimensionality reduction. Proceedings of the Twenty-second
International Conference of Machine Learning (ICML 2005), pages 784-791. Bonn, Germany, 2005. ACM (New York, NY). [ GZipped PDF ] |
| 4. |
Kilian Q. Weinberger, Fei Sha, and Lawrence K. Saul. Learning a kernel matrix
for nonlinear dimensionality reduction.Proceedings of the Twenty First International Conference on Machine Learning (ICML 2004), pages 839-
846, Banff, Canada, 2004. [ PDF ] Outstanding Student Paper Award |
In this project, we investigate the problem of regression analysis while the covariates live on low dimensional submanifolds. Our primary goal is to use the response variable as side information to guide the search for the low dimensional structure without computing the regression surface.
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| Left.Temperatures on the Earth globe. Right.A near-linear relationship between the 1-d low dimensional embedding we computed and the temperature.
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| 5. |
Jens Nilsson, Fei Sha, and Michael I. Jordan.Regression of data on manifold
with kernel dimension reduction. Proceedings of the Twenty-Fourth Annual International Conference on Machine Learning(ICML 2007), pages 697-704. Z. Ghahramani. Corvallis,OR, 2007. MIT Press. [ PDF ] |
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941 West 37th Place,
Los Angeles, CA 90089
Tel: (213) 740-5924
Fax: (213) 740-7512
Office: RTH 403
Email: feisha@usc.edu
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