A kernel view of the dimensionality reduction of manifolds

Authors:
Jihun Ham;Daniel D. Lee;Sebastian Mika;Bernhard Schölkopf
Affiliations:
University of Pennsylvania, Philadelphia, PA;University of Pennsylvania, Philadelphia, PA;Fraunhofer FIRST.IDA, Berlin, Germany;Max Planck Institute for Biological Cybernetics, Tübingen, Germany
Venue:
ICML '04 Proceedings of the twenty-first international conference on Machine learning
Year:
2004

Citing 5
Cited 64

Nonlinear component analysis as a kernel eigenvalue problem

Neural Computation
Discrete Green's functions

Journal of Combinatorial Theory Series A
Diffusion Kernels on Graphs and Other Discrete Input Spaces

ICML '02 Proceedings of the Nineteenth International Conference on Machine Learning
Laplacian Eigenmaps for dimensionality reduction and data representation

Neural Computation
Segmentation Using Eigenvectors: A Unifying View

ICCV '99 Proceedings of the International Conference on Computer Vision-Volume 2 - Volume 2

Learning a kernel matrix for nonlinear dimensionality reduction

ICML '04 Proceedings of the twenty-first international conference on Machine learning
A duality view of spectral methods for dimensionality reduction

ICML '06 Proceedings of the 23rd international conference on Machine learning
Graph Embedding and Extensions: A General Framework for Dimensionality Reduction

IEEE Transactions on Pattern Analysis and Machine Intelligence
Robust kernel Isomap

Pattern Recognition
Dimensionality Reduction of Multimodal Labeled Data by Local Fisher Discriminant Analysis

The Journal of Machine Learning Research
A transductive framework of distance metric learning by spectral dimensionality reduction

Proceedings of the 24th international conference on Machine learning
Regression on manifolds using kernel dimension reduction

Proceedings of the 24th international conference on Machine learning
Robust path-based spectral clustering

Pattern Recognition
A two-stage head pose estimation framework and evaluation

Pattern Recognition
A circuit framework for robust manifold learning

Neurocomputing
Comparison of visualization methods for an atlas of gene expression data sets

Information Visualization
Quasi-isometric parameterization for texture mapping

Pattern Recognition
Quasi-isometric parameterization for texture mapping

Pattern Recognition
Locally adaptive subspace and similarity metric learning for visual data clustering and retrieval

Computer Vision and Image Understanding
A family of dissimilarity measures between nodes generalizing both the shortest-path and the commute-time distances

Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining
Development of an Agreement Metric Based Upon the RAND Index for the Evaluation of Dimensionality Reduction Techniques, with Applications to Mapping Customer Data

MLDM '07 Proceedings of the 5th international conference on Machine Learning and Data Mining in Pattern Recognition
Texture Classification with Generalized Fourier Descriptors in Dimensionality Reduction Context: An Overview Exploration

ANNPR '08 Proceedings of the 3rd IAPR workshop on Artificial Neural Networks in Pattern Recognition
Nonlinear Principal Manifolds --- Adaptive Hybrid Learning Approaches

HAIS '08 Proceedings of the 3rd international workshop on Hybrid Artificial Intelligence Systems
Graph nodes clustering with the sigmoid commute-time kernel: A comparative study

Data & Knowledge Engineering
Nonlinear dimensionality reduction with relative distance comparison

Neurocomputing
Local linear transformation embedding

Neurocomputing
Nonlinear process monitoring based on maximum variance unfolding projections

Expert Systems with Applications: An International Journal
Information Maximization in a Linear Manifold Topographic Map

Neural Processing Letters
Sparsity preserving projections with applications to face recognition

Pattern Recognition
Manifold integration with Markov random walks

AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 1
Nonlinear embedding preserving multiple local-linearities

Pattern Recognition
Kernel extrapolation

Neurocomputing
Adaptive nonlinear manifolds and their applications to pattern recognition

Information Sciences: an International Journal
Marginal discriminant projections: An adaptable margin discriminant approach to feature reduction and extraction

Pattern Recognition Letters
Spectral sampling of manifolds

ACM SIGGRAPH Asia 2010 papers
Incremental embedding and learning in the local discriminant subspace with application to face recognition

IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews
Learning an affine transformation for non-linear dimensionality reduction

ECML PKDD'10 Proceedings of the 2010 European conference on Machine learning and knowledge discovery in databases: Part II
An incremental manifold learning algorithm based on the small world model

LSMS/ICSEE'10 Proceedings of the 2010 international conference on Life system modeling and and intelligent computing, and 2010 international conference on Intelligent computing for sustainable energy and environment: Part I
Robust Positive semidefinite L-Isomap Ensemble

Pattern Recognition Letters
Supervised principal component analysis: Visualization, classification and regression on subspaces and submanifolds

Pattern Recognition
Guided Locally Linear Embedding

Pattern Recognition Letters
Letters: Kernel-view based discriminant approach for embedded feature extraction in high-dimensional space

Neurocomputing
Locally Defined Principal Curves and Surfaces

The Journal of Machine Learning Research
A RANSAC-based ISOMAP for Filiform manifolds in nonlinear dynamical systems: an application to chaos in a dripping faucet

ICANN'11 Proceedings of the 21st international conference on Artificial neural networks - Volume Part II
PERTURBO: a new classification algorithm based on the spectrum perturbations of the Laplace-Beltrami operator

ECML PKDD'11 Proceedings of the 2011 European conference on Machine learning and knowledge discovery in databases - Volume Part I
High dimensional correspondences from low dimensional manifolds: an empirical comparison of graph-based dimensionality reduction algorithms

ACCV'10 Proceedings of the 2010 international conference on Computer vision - Volume part II
Segmentation via graph-spectral methods and riemannian geometry

CAIP'05 Proceedings of the 11th international conference on Computer Analysis of Images and Patterns
An integrated two-stage framework for robust head pose estimation

AMFG'05 Proceedings of the Second international conference on Analysis and Modelling of Faces and Gestures
Nonlinear dimensionality reduction using circuit models

SCIA'05 Proceedings of the 14th Scandinavian conference on Image Analysis
Neighborhood preserving projections (NPP): a novel linear dimension reduction method

ICIC'05 Proceedings of the 2005 international conference on Advances in Intelligent Computing - Volume Part I
Kernel laplacian eigenmaps for visualization of non-vectorial data

AI'06 Proceedings of the 19th Australian joint conference on Artificial Intelligence: advances in Artificial Intelligence
The connections between principal component analysis and dimensionality reduction methods of manifolds

ICIC'11 Proceedings of the 7th international conference on Advanced Intelligent Computing Theories and Applications: with aspects of artificial intelligence
An experimental investigation of kernels on graphs for collaborative recommendation and semisupervised classification

Neural Networks
A unifying probabilistic perspective for spectral dimensionality reduction: insights and new models

The Journal of Machine Learning Research
Orthogonal projection analysis

IScIDE'11 Proceedings of the Second Sino-foreign-interchange conference on Intelligent Science and Intelligent Data Engineering
Modeling and monitoring of multimodes process

ISNN'12 Proceedings of the 9th international conference on Advances in Neural Networks - Volume Part I
Data-based modeling and monitoring for multimode processes using local tangent space alignment

ISNN'12 Proceedings of the 9th international conference on Advances in Neural Networks - Volume Part I
Predicting expression-related features of chromosomal domain organization with network-structured analysis of gene expression and chromosomal location

Proceedings of the ACM Conference on Bioinformatics, Computational Biology and Biomedicine
Commute times of random walks on trees

Discrete Applied Mathematics
Joint discriminative dimensionality reduction and dictionary learning for face recognition

Pattern Recognition
Learning with boundary conditions

Neural Computation
Editor's choice article: Homeomorphic Manifold Analysis (HMA): Generalized separation of style and content on manifolds

Image and Vision Computing
Image classification with manifold learning for out-of-sample data

Signal Processing
Type 1 and 2 mixtures of Kullback-Leibler divergences as cost functions in dimensionality reduction based on similarity preservation

Neurocomputing
Beam search algorithms for multilabel learning

Machine Learning
Isometric sliced inverse regression for nonlinear manifold learning

Statistics and Computing
Parallel vector field embedding

The Journal of Machine Learning Research
Large-scale SVD and manifold learning

The Journal of Machine Learning Research
Unsupervised categorization of human motion sequences

Intelligent Data Analysis

Quantified Score

Hi-index	0.00

Visualization

Abstract

We interpret several well-known algorithms for dimensionality reduction of manifolds as kernel methods. Isomap, graph Laplacian eigenmap, and locally linear embedding (LLE) all utilize local neighborhood information to construct a global embedding of the manifold. We show how all three algorithms can be described as kernel PCA on specially constructed Gram matrices, and illustrate the similarities and differences between the algorithms with representative examples.