Partial order embedding with multiple kernels

We consider the problem of embedding arbitrary objects (e.g., images, audio, documents) into Euclidean space subject to a partial order over pair-wise distances. Partial order constraints arise naturally when modeling human perception of similarity. Our partial order framework enables the use of graph-theoretic tools to more efficiently produce the embedding, and exploit global structure within the constraint set. We present an embedding algorithm based on semidefinite programming, which can be parameterized by multiple kernels to yield a unified space from heterogeneous features.

[1]  R. Varga.  Geršgorin And His Circles , 2004 .

[2]  Amos Storkey,et al.  Advances in Neural Information Processing Systems 20 , 2007 .

[3]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[4]  Daniel P. W. Ellis,et al.  The Quest for Ground Truth in Musical Artist Similarity , 2002, ISMIR.

[5]  Nello Cristianini,et al.  Learning the Kernel Matrix with Semidefinite Programming , 2002, J. Mach. Learn. Res..

[6]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[7]  Arnold W. M. Smeulders,et al.  The Amsterdam Library of Object Images , 2004, International Journal of Computer Vision.

[8]  Amir Globerson,et al.  Visualizing pairwise similarity via semidefinite programming , 2007, AISTATS.

[9]  David J. Kriegman,et al.  Generalized Non-metric Multidimensional Scaling , 2007, AISTATS.

[10]  Carla E. Brodley,et al.  Proceedings of the twenty-first international conference on Machine learning , 2004, International Conference on Machine Learning.

[11]  Kilian Q. Weinberger,et al.  Learning a kernel matrix for nonlinear dimensionality reduction , 2004, ICML.

[12]  Ivor W. Tsang,et al.  Distance metric learning with kernels , 2003 .

[13]  Joachim M. Buhmann,et al.  Going Metric: Denoising Pairwise Data , 2002, NIPS.

[14]  Jaroslav Opatrny,et al.  Total Ordering Problem , 1979, SIAM J. Comput..

[15]  Claire Cardie,et al.  Constrained K-means Clustering with Background Knowledge , 2001, ICML.

[16]  Anne Lohrli.  Chapman and Hall , 1985 .

[17]  Bernhard Schölkopf,et al.  A Generalized Representer Theorem , 2001, COLT/EuroCOLT.

[18]  Thorsten Joachims,et al.  Learning a Distance Metric from Relative Comparisons , 2003, NIPS.

[19]  Yann LeCun,et al.  Dimensionality Reduction by Learning an Invariant Mapping , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[20]  Michael I. Jordan,et al.  Distance Metric Learning with Application to Clustering with Side-Information , 2002, NIPS.

[21]  Kilian Q. Weinberger,et al.  Distance Metric Learning for Large Margin Nearest Neighbor Classification , 2005, NIPS.

[22]  Le Song,et al.  Colored Maximum Variance Unfolding , 2007, NIPS.

[23]  François E. Cellier,et al.  Computer Aided Control Systems Design , 1983, ESC.

[24]  Alfred V. Aho,et al.  The Transitive Reduction of a Directed Graph , 1972, SIAM J. Comput..

[25]  Raymond J. Mooney,et al.  Integrating constraints and metric learning in semi-supervised clustering , 2004, ICML.