![](bg8.png)
NCuts segmentation in the learned space. As can be
seen in the figures, simple used added equivalence
constraints can improve the segmentations.
5 SUMMARY AND
CONCLUSIONS
In this paper we proposed a semi-supervised distance
metric learning method, which uses pairwise equiv-
alence constraints to discover the desired groups in
high-dimensional data. The method works in both
the input and kernel induced-feature space and it
can learn nonsquare projection matrices that yield
low rank distance metrics. The optimization proce-
dure involves minimizing two terms defined based on
sigmoids. The first term encourages pulling simi-
lar sample pairs closer while the second term max-
imizes the local margin. The solution is found by
a gradient descent procedure that involves an eigen-
decomposition.
Experimental results show that the proposed
method increases performance of subsequent cluster-
ing and classification algorithms. Moreover, it yields
better results than methods applying unsupervised di-
mensionality reduction followed by full rank metric
learning.
ACKNOWLEDGEMENTS
Roberto Paredes is supported by the grant from Span-
ish project TIN2008-04571.
REFERENCES
Bar-Hillel, A., Hertz, T., Shental, N., and Weinshall, D.
(2003). Learning distance functions using equivalence
relations. In International Conference on Machine
Learning.
Basu, S., Banerjee, A., and Mooney, R. J. (2004). Active
semi-supervision for pairwise constrained clustering.
In the SIAM International Conference on Data Min-
ing.
Bilenko, M., Basu, S., and Mooney, R. J. (2004). Integrat-
ing constraints and metric learning in semi-supervised
clustering. In the 21st International Conference on
Machine Learning.
Cevikalp, H., Verbeek, J., Jurie, F., and Klaser, A. (2008).
Semi-supervised dimensionality reduction using pair-
wise equivalence constraints. In Computer Vision The-
ory and Applications.
Chen, H. T., Liu, T. L., and Fuh, C. S. (2005). Learning ef-
fective image metrics from few pairwise examples. In
IEEE International Conference on Computer Vision.
Globerson, A. and Roweis, S. (2005). Metric learning by
collapsing classes. In Advances in Neural Information
Processing Systems.
Goldberger, J., Roweis, S., Hinton, G., and Salakhutdinov,
R. (2004). Neighbourhood component analysis. In
Advances in Neural Information Processing Systems.
Hadsell, R., Chopra, S., and LeCun, Y. (2006). Dimension-
ality reduction by learning and invariant mapping. In
IEEE Computer Society Conference on Computer Vi-
sion and Pattern Recognition.
He, X. and Niyogi, P. (2003). Locality preserving direc-
tions. In Advances in Neural Information Processing
Systems.
Hertz, T., Shental, N., Bar-Hillel, A., and Weinshall, D.
(2003). Enhancing image and video retrieval: Learn-
ing via equivalence constraints. In IEEE Computer
Society Conference on Computer Vision and Pattern
Recognition.
Kwok, J. T. and Tsang, I. W. (2003). Learning with ideal-
ized kernels. In International Conference on Machine
Learning.
Lazebnik, S., Schmid, C., and Ponce, J. (2005). A max-
imum entropy framework for part-based texture and
objcect recognition. In International Conference on
Computer Vision (ICCV).
Scholkopf, B., Smola, A. J., and Muller, K. R. (1998). Non-
linear component analysis as a kernel eigenvalue prob-
lem. Neural Computation, 10:1299–1319.
Shalev-Shwartz, S., Singer, Y., and Ng, A. Y. (2004). Online
and batch learning of pseudo metrics. In International
Conference on Machine Learning.
Shi, J. and Malik, J. (2000). Normalized cuts and image
segmentation. IEEE Transactions on PAMI, 22:885–
905.
Torresani, L. and Lee, K. C. (2006). Large margin com-
ponent analysis. In Advances in Neural Information
Processing Systems.
Turk, M. and Pentland, A. P. (1991). Eigenfaces for recog-
nition. Journal of Cognitive Neuroscience, 3:71–86.
van de Weijer, J. and Schmid, C. (2006). Coloring local fea-
ture extraction. In European Conference on Computer
Vision (ECCV).
Villegas, M. and Paredes, R. (2008). Simultaneous learn-
ing of a discriminative projection and prototype for
nearest-neighbor classification. In IEEE Computer
Society Conference on Computer Vision and Pattern
Recognition.
Weinberger, K. Q., Blitzer, J., and Saul, L. K. (2005). Dis-
tance metric learning for large margin nearest neigh-
bor classification. In Advances in Neural Information
Processing Systems.
Xing, E. P., Ng, A. Y., Jordan, M. I., and Russell, S. (2003).
Distance metric learning with application to clustering
with side-information. In Advances in Neural Infor-
mation Processing Systems.
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