Optimal Cluster Preserving Embedding of Nonmetric Proximity Data

@article{Roth2003OptimalCP,
  title={Optimal Cluster Preserving Embedding of Nonmetric Proximity Data},
  author={Volker Roth and Julian Laub and Motoaki Kawanabe and Joachim M. Buhmann},
  journal={IEEE Trans. Pattern Anal. Mach. Intell.},
  year={2003},
  volume={25},
  pages={1540-1551}
}
For several major applications of data analysis, objects are often not represented as feature vectors in a vector space, but rather by a matrix gathering pairwise proximities. Such pairwise data often violates metricity and, therefore, cannot be naturally embedded in a vector space. Concerning the problem of unsupervised structure detection or clustering, in this paper, a new embedding method for pairwise data into Euclidean vector spaces is introduced. We show that all clustering methods… 

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References

SHOWING 1-10 OF 27 REFERENCES
A theory of proximity based clustering: structure detection by optimization
Pairwise Data Clustering by Deterministic Annealing
TLDR
A deterministic annealing approach to pairwise clustering is described which shares the robustness properties of maximum entropy inference and the resulting Gibbs probability distributions are estimated by mean-field approximation.
Clustering in large graphs and matrices
TLDR
It is argued that in fact the relaxation provides a generalized clustering which is useful in its own right and can be applied to problems of very large size which typically arise in modern applications.
Data clustering: a review
TLDR
An overview of pattern clustering methods from a statistical pattern recognition perspective is presented, with a goal of providing useful advice and references to fundamental concepts accessible to the broad community of clustering practitioners.
Nonlinear dimensionality reduction by locally linear embedding.
TLDR
Locally linear embedding (LLE) is introduced, an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs that learns the global structure of nonlinear manifolds.
Classification with Nonmetric Distances: Image Retrieval and Class Representation
TLDR
It is shown that in nonmetric spaces, boundary points are less significant for capturing the structure of a class than in Euclidean spaces, and it is suggested that atypical points may be more important in describing classes.
A global geometric framework for nonlinear dimensionality reduction.
TLDR
An approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set and efficiently computes a globally optimal solution, and is guaranteed to converge asymptotically to the true structure.
Kernel PCA and De-Noising in Feature Spaces
TLDR
This work presents ideas for finding approximate pre-images, focusing on Gaussian kernels, and shows experimental results using these pre- images in data reconstruction and de-noising on toy examples as well as on real world data.
Normalized cuts and image segmentation
  • Jianbo Shi, J. Malik
  • Computer Science
    Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition
  • 1997
TLDR
This work treats image segmentation as a graph partitioning problem and proposes a novel global criterion, the normalized cut, for segmenting the graph, which measures both the total dissimilarity between the different groups as well as the total similarity within the groups.
Investigation of measures for grouping by graph partitioning
  • P. Soundararajan, Sudeep Sarkar
  • Computer Science
    Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001
  • 2001
TLDR
Using probabilistic analysis and a rigorous empirical evaluation, it is shown that the minimization of the average cut and the normalized cut measure, using recursive bi-partitioning will, on an average, result in the correct segmentation.
...
1
2
3
...