Learning distance metric for regression by semidefinite programming with application to human age estimation

ABSTRACT

A good distance metric for the input data is crucial in many pattern recognition and machine learning applications. Past studies have demonstrated that learning a metric from labeled samples can significantly improve the performance of classification and clustering algorithms. In this paper, we investigate the problem of learning a distance metric that measures the semantic similarity of input data for regression problems. The particular application we consider is human age estimation. Our guiding principle for learning the distance metric is to preserve the local neighborhoods based on a specially designed distance as well as to maximize the distances between data that are not in the same neighborhood in the semantic space.Without any assumption about the structure and the distribution of the input data, we show that this can be done by using semidefinite programming. Furthermore, the low-level feature space can be mapped to the high-level semantic space by a linear transformation with very low computational cost. Experimental results on the publicly available FG-NET database show that 1) the learned metric correctly discovers the semantic structure of the data even when the amount of training data is small and 2) significant improvement over the traditional Euclidean metric for regression can be obtained using the learned metric. Most importantly, simple regression methods such as k nearest neighbors (kNN), combined with our learned metric, become quite competitive (and sometimes even superior) in terms of accuracy when compared with the state-of-the-art human age estimation approaches.

REFERENCES

• 1: The FG-NET Aging Database [Online], http://www.fgnet.rsunit.com/.
• 2: V. N. Balasubramanian, J. Ye, and S. Panchanathan. Biased manifold embedding: A framework for person-independent head pose estimation. In IEEE Conf. CVPR, pages 1--7, 2007.
• 3: M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionally reduction and data representation. Neural Computation, 15(6):1373--1396, 2003.
• 4: M. Belkin, P. Niyogi, and V. Sindhwani. Manifold regularization: a geometric framework for learing from labeled and unlabeled examples. Journal of Machine Learning Research, 7:2399--2434, 2006.
• 5: D. B. Borchers. Csdp, a c library for semidefinite programming. Optimization Methods and Software, 11(1):613--623, 1999.
• 6: D. Cai, X. He, J. Han, and H. J. Zhang. Orthogonal laplacianfaces for face recognition. IEEE Trans. Image Processing, 15:3608--3614, Nov. 2006.
• 7: S. Chopra, R. Hadsell, and Y. LeCun. Learning a similarity metric discriminatively, with application to face verification. IEEE Conf. on Computer Vision and Pattern Recognition, 1:539--546, 2005.
• 8: T. Cootes, G. Edwards, and C. Taylar. Active appearance models. IEEE Trans. Pattern Analysis and Machine Intelligence, 23(6):681--685, 2001.
• 9: T. Cox and M. Cox. Multidimensional Scaling. Chapman & Hall, Lodon, 1994.
• 10: D. L. Donoho and C. E. Grimes. When does geodesic distance recover the true hidden parametrization of families of articulated images? Proceedings European Symposium on Artificial Neural Networks, 2002.
• 11: A. L. C. Draganova and C. Christodoulou. Comparing different classifiers for automatic age estimation. IEEE Trans. Systems, Man, and Cybernetics, 34(1):621--628, 2004.
• 12: X. Geng, K. Smith-Miles, and Z.-Z. Zhou. Facial age estimation by nonlinear aging pattern subspace. Proc. ACM Conf. Multimedia, 2008.
• 13: X. Geng, Z.-H. Zhou, Y. Zhang, G. Li, and H. Dai. Learning from facial aging patterns for automatic age estimation. Proc. ACM Conf. Multimedia, pages 307--316, 2006.
• 14: J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov. Neighbourhood components analysis. NIPS, 2005.
• 15: G. H. Golub and C. F. V. Loan. Matrix Computations. Johns Hopkins Univ. Press, 1996.
• 16: G. Guo, Y. Fu, C. Dyer, and T. S. Huang. Image based human age estimation by manifold learning and locally adjusted robust regression. IEEE Trans. on Image Processing, 17:1178--1188, July 2008.
• 17: J. He, M. Li, H. J. Zhang, H. Tong, and C. Zhang. Manifold ranking based image retrieval. Proc. ACM Multimedia, 2004.
• 18: X. He, W. Y. Ma, and H. J. Zhang. Learning an image manifold for retrieval. Proc. ACM Multimedia, 2004.
• 19: X. He, S. Yan, Y. Hu, and H. J. Zhang. Learning a locality perserving subspace for visual recognition. IEEE Conf. on Computer Vision, 1:385--392, 2003.
• 20: I. T. Jolliffe. Principal Component Analysis. Springer Verlag, New York, 1986.
• 21: W. Matusik, H. Pfister, M. Brand, and L. McMillan. A data-driven reflectance model. Proc. of SIGGRAPH, 2003.
• 22: J. Nilsson, F. Sha, and M. I. Jordan. Regression on manifolds using kernel dimension reduction. IEEE Conf. ICML, pages 265--272, 2007.
• 23: S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323--2326, 2000.
• 24: Y. Rui and T. S. Huang. Optimizing learning in image retrieval. IEEE Conf. on CVPR, 1:236--243, 2000.
• 25: S. Shalev-Shwartz, Y. Singer, and A. Y. Ng. Online and batch learning of pseudo-metrics. Proc. of International Conference on Machine Learning, 2004.
• 26: N. Shental, T. Hertz, D. Weinshall, and M. Pavel. Adjustment learning and relevant component analysis. Proc. European Conference on Computer Vision, 4:776--792, 2002.
• 27: J. F. Sturm. Using sedumi 1.02. a matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 11-12:625--653, 1999.
• 28: M. Sugiyama, H. Hachiya, C. Towell, and S. Vijayakumar. Geodesic gaussian kernels for value function approximation. Autonomous Robots, 25:287--304, 2008.
• 29: J. B. Tenebaum, V. de. Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionally reduction. Science, 290(5500):2319--2323, 2000.
• 30: L. Wang, Y. Zhang, and J. Feng. On the euclidean distance of images. IEEE Trans. on Pattern Anal. Mach. Intell., 27:1334--1339, Aug. 2005.
• 31: K. Weinberger, J. Blitzer, and L. Saul. Distance metric learning for large margin nearest neighbor classification. in Proc. NIPS, pages 1475--1482, 2006.
• 32: K. Q. Weinberger and L. K. Saul. Unsupervised learning of image manifolds by semidefinite programming. In IEEE Conf. CVPR, volume 2, pages 988--995, 2004.
• 33: E. Xing, A. Ng, M. I. Jordan, and S. Russell. Distance metric learning with application to clustering with side-information. in Proc. NIPS, 2002.
• 34: S. Yan, H. W. ang T. S. Huang, and X. Tang. Ranking with uncertain labels. IEEE Conf. Mulitimedia and Expo, pages 96--99, 2007.
• 35: S. Yan, H. Wang, X. Tang, and T. S. Huang. Learning auto-structured regressor from uncertain nonnegative labels. IEEE Conf. ICCV, pages 1--8, 2007.
• 36: Q. Yong and J. Yang. Modified kernel functions by geodesic distance. EURASIP Journal on Applied Signal Processing, 16:2515--2521, 2004.
• 37: Z. Zhang and H. Zha. Principal manifolds and nonlinear dimensionality reduction by local tangent space alignment. SIAM Journal of Scientific Computing, 26(10):313--338, 2004.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.9 Applications

Additional Classification:
  H. Information Systems
  H.5 INFORMATION INTERFACES AND PRESENTATION (I.7)
      H.5.2 User Interfaces (D.2.2, H.1.2, I.3.6)

  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
      I.2.6 Learning

General Terms:
Algorithms, Experimentation

Keywords:
human age estimation, kNN regression, metric learning, semidefinite programming