Dimension reduction by local principal component analysis
Neural Computation
Nonlinear component analysis as a kernel eigenvalue problem
Neural Computation
Learning and Design of Principal Curves
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Practical Sequential Method for Principal Component Analysis
Neural Processing Letters
Finding Curvilinear Features in Spatial Point Patterns: Principal Curve Clustering with Noise
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Unified Model for Probabilistic Principal Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Piecewise Linear Skeletonization Using Principal Curves
IEEE Transactions on Pattern Analysis and Machine Intelligence
Mean Shift: A Robust Approach Toward Feature Space Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
Mean Shift, Mode Seeking, and Clustering
IEEE Transactions on Pattern Analysis and Machine Intelligence
An Algorithm for Data-Driven Bandwidth Selection
IEEE Transactions on Pattern Analysis and Machine Intelligence
A k-segments algorithm for finding principal curves
Pattern Recognition Letters
A Soft k-Segments Algorithm for Principal Curves
ICANN '01 Proceedings of the International Conference on Artificial Neural Networks
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
Principal curves: learning, design, and applications
Principal curves: learning, design, and applications
Improved Fast Gauss Transform and Efficient Kernel Density Estimation
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Pattern Classification (2nd Edition)
Pattern Classification (2nd Edition)
A kernel view of the dimensionality reduction of manifolds
ICML '04 Proceedings of the twenty-first international conference on Machine learning
Generalized Discriminant Analysis Using a Kernel Approach
Neural Computation
Unsupervised Learning of Image Manifolds by Semidefinite Programming
International Journal of Computer Vision
Fast nonparametric clustering with Gaussian blurring mean-shift
ICML '06 Proceedings of the 23rd international conference on Machine learning
Journal of VLSI Signal Processing Systems
A high resolution time frequency representation with significantly reduced cross-terms
ICASSP '00 Proceedings of the Acoustics, Speech, and Signal Processing, 2000. on IEEE International Conference - Volume 02
Decision-directed recursive least squares MIMO channels tracking
EURASIP Journal on Wireless Communications and Networking
An Algorithm for Finding Intrinsic Dimensionality of Data
IEEE Transactions on Computers
IEEE Transactions on Signal Processing
Semi-Supervised Learning
Improving the readability of time-frequency and time-scalerepresentations by the reassignment method
IEEE Transactions on Signal Processing
Adaptive MIMO decision feedback equalization for receivers with time-varying channels
IEEE Transactions on Signal Processing
Principal curves with bounded turn
IEEE Transactions on Information Theory
IEEE Journal on Selected Areas in Communications
IEEE Transactions on Neural Networks
Journal of Visual Communication and Image Representation
Contour-based shape representation using principal curves
Pattern Recognition
A generative model and a generalized trust region Newton method for noise reduction
Computational Optimization and Applications
Hi-index | 0.00 |
Principal curves are defined as self-consistent smooth curves passing through the middle of the data, and they have been used in many applications of machine learning as a generalization, dimensionality reduction and a feature extraction tool. We redefine principal curves and surfaces in terms of the gradient and the Hessian of the probability density estimate. This provides a geometric understanding of the principal curves and surfaces, as well as a unifying view for clustering, principal curve fitting and manifold learning by regarding those as principal manifolds of different intrinsic dimensionalities. The theory does not impose any particular density estimation method can be used with any density estimator that gives continuous first and second derivatives. Therefore, we first present our principal curve/surface definition without assuming any particular density estimation method. Afterwards, we develop practical algorithms for the commonly used kernel density estimation (KDE) and Gaussian mixture models (GMM). Results of these algorithms are presented in notional data sets as well as real applications with comparisons to other approaches in the principal curve literature. All in all, we present a novel theoretical understanding of principal curves and surfaces, practical algorithms as general purpose machine learning tools, and applications of these algorithms to several practical problems.