U. Özertem; "Locally Defined Principal Curves&Surfaces", Nov.28, 13:40

- FENS
- U. Özertem; "Locally Defined Principal Curves&Surfaces", Nov.28, 13:40

**Faculty of Engineering and Natural Sciences
FENS EE 551 SEMINARS**

**Locally Defined Principal Curves and Surfaces**

*Umut Özertem, Oregon Health & Science University*

**Abstract**:

Defined as self consistent smooth curves passing through the middle of the data, principal curves are used in many applications of machine learning as a generalization, dimensionality reduction and a feature extraction tool. The ill-posed definition of principal curves leads to practical difficulties, main causes of which are the desire to use global statistics such as conditional expectations to build a self consistent definition, and not decoupling the definition of principal curve from the data samples. We take a novel approach by introducing the concept of principal sets, defined as the union of all principal curves/surfaces with a particular intrinsic dimensionality. We characterize the principal set in terms of the gradient and the Hessian of the probability density estimate. The theory lays a geometric understanding of the principal curves and surfaces, and a unifying view for clustering, principal curve fitting and manifold learning. As well as the applications existing in the literature, we also apply our definition of principal curves into nonlinear independent components analysis and time warping, in which principal curves have never been used before.

**Biography**:

Umut Özertem received his B.S. in Electrical & Electronics Engineering in 2003 from the Middle East Technical University, Turkey. After working at TÜBİTAK-BILTEN between August 2003 and July 2004 under the supervision of Prof. A. Aydın Alatan, he joined the Adaptive Systems Lab in the CSEE Department of the Oregon Health & Science University as a PhD student. His research interests include statistical signal processing and machine learning with a focus on information theoretic methods.

**November 28th, 2007, 13:40, FENS G035**

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