Nonlinear component analysis as a kernel eigenvalue problem
Neural Computation
Advances in kernel methods: support vector learning
Advances in kernel methods: support vector learning
The Effect of the Input Density Distribution on Kernel-based Classifiers
ICML '00 Proceedings of the Seventeenth International Conference on Machine Learning
Normalized Cuts and Image Segmentation
CVPR '97 Proceedings of the 1997 Conference on Computer Vision and Pattern Recognition (CVPR '97)
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
Spectral partitioning works: planar graphs and finite element meshes
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Think globally, fit locally: unsupervised learning of low dimensional manifolds
The Journal of Machine Learning Research
Learning generative models of similarity matrices
UAI'03 Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence
Principal Surfaces from Unsupervised Kernel Regression
IEEE Transactions on Pattern Analysis and Machine Intelligence
Nonlocal Estimation of Manifold Structure
Neural Computation
Robust non-linear dimensionality reduction using successive 1-dimensional Laplacian Eigenmaps
Proceedings of the 24th international conference on Machine learning
A transductive framework of distance metric learning by spectral dimensionality reduction
Proceedings of the 24th international conference on Machine learning
A survey of kernel and spectral methods for clustering
Pattern Recognition
A tutorial on spectral clustering
Statistics and Computing
Identifying biologically relevant genes via multiple heterogeneous data sources
Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining
Learning a Kernel Matrix for Time Series Data from DTW Distances
Neural Information Processing
Functional Learning of Kernels for Information Fusion Purposes
CIARP '08 Proceedings of the 13th Iberoamerican congress on Pattern Recognition: Progress in Pattern Recognition, Image Analysis and Applications
Representing Functional Data Using Support Vector Machines
CIARP '08 Proceedings of the 13th Iberoamerican congress on Pattern Recognition: Progress in Pattern Recognition, Image Analysis and Applications
Large-Scale Clustering through Functional Embedding
ECML PKDD '08 Proceedings of the European conference on Machine Learning and Knowledge Discovery in Databases - Part II
Fuzzy connectivity clustering with radial basis kernel functions
Fuzzy Sets and Systems
Geometry-aware metric learning
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
AAAI'06 Proceedings of the 21st national conference on Artificial intelligence - Volume 1
Combining Functional Data Projections for Time Series Classification
CIARP '09 Proceedings of the 14th Iberoamerican Conference on Pattern Recognition: Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications
Unsupervised Object Discovery: A Comparison
International Journal of Computer Vision
Representing functional data using support vector machines
Pattern Recognition Letters
Similarity preserving principal curve: an optimal 1-D feature extractor for data representation
IEEE Transactions on Neural Networks
Non-local spatial spectral clustering for image segmentation
Neurocomputing
Composite hashing with multiple information sources
Proceedings of the 34th international ACM SIGIR conference on Research and development in Information Retrieval
Supervised Discriminant Projection with Its Application to Face Recognition
Neural Processing Letters
Hypergraph spectral hashing for similarity search of social image
MM '11 Proceedings of the 19th ACM international conference on Multimedia
Kernel principal components are maximum entropy projections
ICA'06 Proceedings of the 6th international conference on Independent Component Analysis and Blind Signal Separation
Metric and kernel learning using a linear transformation
The Journal of Machine Learning Research
Diffusion maps for the description of meteorological data
HAIS'12 Proceedings of the 7th international conference on Hybrid Artificial Intelligent Systems - Volume Part I
A unifying probabilistic perspective for spectral dimensionality reduction: insights and new models
The Journal of Machine Learning Research
Diffusion maps and local models for wind power prediction
ICANN'12 Proceedings of the 22nd international conference on Artificial Neural Networks and Machine Learning - Volume Part II
PCM'12 Proceedings of the 13th Pacific-Rim conference on Advances in Multimedia Information Processing
Clustering and outlier detection using isoperimetric number of trees
Pattern Recognition
Nonlocal estimation of manifold structure
Neural Computation
| Hi-index | 0.00 |
In this letter, we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem: learning the principal eigenfunctions of an operator defined from a kernel and the unknown data-generating density. Whereas spectral embedding methods provided only coordinates for the training points, the analysis justifies a simple extension to out-of-sample examples (the Nyström formula) for multidimensional scaling (MDS), spectral clustering, Laplacian eigenmaps, locally linear embedding (LLE), and Isomap. The analysis provides, for all such spectral embedding methods, the definition of a loss function, whose empirical average is minimized by the traditional algorithms. The asymptotic expected value of that loss defines a generalization performance and clarifies what these algorithms are trying to learn. Experiments with LLE, Isomap, spectral clustering, and MDS show that this out-of-sample embedding formula generalizes well, with a level of error comparable to the effect of small perturbations of the training set on the embedding.