The requirement for suitable ways to measure the distance or similarity between data is omnipresent in machine learning, pattern recognition and data mining, but extracting such good metrics for particular problems is in general challenging. This has led to the emergence of metric learning ideas, which intend to automatically learn a distance function tuned to a specific task. In many tasks and data types, there are natural transformations to which the classification result should be invariant or insensitive. This demand and its implications are essential in many machine learning applications, and insensitivity to image transformations was in the first place achieved by using invariant feature vectors. In this paper, a new representation model on Grassmann manifolds for data points and a novel method for learning a Mahalanobis metric which uses the geodesic distance on Grassmann manifolds are proposed. In fact, we use an appropriate geodesic distance metric on the Grassmann manifolds, called projection metric, for measuring primary similarities between the new representations of the data points. This makes learning of the Mahalanobis metric invariant to similarity transforms and intensity changes, and therefore improve the performance. Experiments on face and handwritten digit datasets demonstrate that our proposed method yields performance improvements in a state-of-the-art metric learning algorithm.
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Absil P-A, Mahony R, Sepulchre R (2009) Optimization algorithms on matrix manifolds. Princeton University Press, Princeton
Bar-Hillel A, Hertz T, Shental N, Weinshall D (2005) Learning a Mahalanobis metric from equivalence constraints. J Mach Learn Res 6:937–965
Bi Y, Fan B, Wu F (2015) Beyond Mahalanobis metric: Cayley-Klein metric learning. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2339–2347
Candemir S, Borovikov E, Santosh KC, Antani S, Thoma G (2015) Rsilc: rotation-and scale-invariant, line-based color-aware descriptor. Image Vis Comput 42:1–12
Crammer K, Singer Y (2001) On the algorithmic implementation of multiclass kernel-based vector machines. J Mach Learn Res 2(Dec):265–292
Davis JV, Kulis B, Jain P, Sra S, Dhillon IS (2007) Information-theoretic metric learning. In: Proceedings of the 24th international conference on machine learning, ACM, pp 209–216.
Domeniconi C, Gunopulos D (2002) Adaptive nearest neighbor classification using support vector machines. In: Advances in neural information processing systems, pp 665–672
Duda RO, Hart PE, Stork DG (2012) Pattern classification. Wiley, New York
Edelman A, Arias TA, Smith ST (1998) The geometry of algorithms with orthogonality constraints. SIAM J Matrix Anal Appl 20(2):303–353
Goldberger J, Hinton GE, Roweis ST, Salakhutdinov RR (2005) Neighbourhood components analysis. In: Advances in neural information processing systems, pp 513–520
Guillaumin M, Verbeek J, Schmid C (2009) Is that you? Metric learning approaches for face identification. In: 2009 IEEE 12th international conference on computer Vision, pp 498–505. IEEE
Halmos PR (2012) A Hilbert space problem book, vol 19. Springer Science & Business Media, New York
Hamm J, Lee DD (2008) Grassmann discriminant analysis: a unifying view on subspace-based learning. In: Proceedings of the 25th international conference on machine learning, pp 376–383. ACM
Harandi M, Sanderson C, Shen C, Lovell BC (2013) Dictionary learning and sparse coding on Grassmann manifolds: An extrinsic solution. In: Proceedings of the IEEE international conference on computer vision, pp 3120–3127
Harandi MT, Salzmann M, Jayasumana S, Hartley R, Li H (2014) Expanding the family of Grassmannian kernels: an embedding perspective. In: European conference on computer vision, Springer, pp 408–423.
Helmke U, Hüper K, Trumpf J (2007) Newton’s method on Grassmann manifolds. arXiv preprint arXiv:0709.2205
Hochuli AG, Oliveira LS, Britto AS Jr, Sabourin R (2018) Handwritten digit segmentation: is it still necessary? Pattern Recognit 78:1–11
Hoi SCH, Liu W, Chang S-F (2010) Semi-supervised distance metric learning for collaborative image retrieval and clustering. ACM Trans Multimed Comput Commun Appl (TOMM) 6(3):18
Huang Z, Wang R, Shan S, Chen X (2015) Projection metric learning on Grassmann manifold with application to video based face recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 140–149
Koestinger M, Hirzer M, Wohlhart P, Roth PM, Bischof H (2012) Large scale metric learning from equivalence constraints. In: 2012 IEEE conference on computer vision and pattern recognition (CVPR) , pp 2288–2295. IEEE
Le H (1991) On geodesics in Euclidean shape spaces. J Lond Math Soc 2(2):360–372
Lim D, Lanckriet G (2014) Efficient learning of Mahalanobis metrics for ranking. In: Proceedings of the 31st international conference on machine learning (ICML-14), pp 1980–1988
Patil S, Talbar S (2012) Content based image retrieval using various distance metrics. In: Data engineering and management, Springer, pp 154–161
Peng J, Heisterkamp DR, Dai HK (2002) Adaptive kernel metric nearest neighbor classification. In: Proceedings of 16th international conference on pattern recognition, 2002, vol 3, pp 33–36. IEEE
Santosh KC (2011) Character recognition based on dtw-radon. In: 2011 international conference on document analysis and recognition (ICDAR), pp 264–268. IEEE
Santosh KC, Aafaque A, Antani S, Thoma GR (2017) Line segment-based stitched multipanel figure separation for effective biomedical cbir. Int J Pattern Recognit Artif Intell 31(06):1757003
Santosh KC, Lamiroy B, Wendling L (2014) Integrating vocabulary clustering with spatial relations for symbol recognition. Int J Doc Anal Recognit (IJDAR) 17(1):61–78
Santosh KC, Roy PP (2018) Arrow detection in biomedical images using sequential classifier. Int J Mach Learn Cybern 9(6):993–1006
Santosh KC, Wendling L, Antani S, Thoma GR (2016) Overlaid arrow detection for labeling regions of interest in biomedical images. IEEE Intell Syst 31(3):66–75
Sethi S, Rohila VK, Agarwal P (2018) Hand written and natural scene character recognition. Hand 5(05):908–911
Shen C, Kim J, Wang L, Hengel AVD (2012) Positive semidefinite metric learning using boosting-like algorithms. J Mach Learn Res 13:1007–1036
Srivastava A, Klassen E (2004) Bayesian and geometric subspace tracking. Adv Appl Probab 36(01):43–56
Vajda S, Santosh KC (2016) A fast k-nearest neighbor classifier using unsupervised clustering. In: International conference on recent trends in image processing and pattern recognition, Springer, pp 185–193.
Wang D, Tan X (2018) Robust distance metric learning via bayesian inference. IEEE Trans Image Process 27(3):1542–1553
Wang J, Kalousis A, Woznica A (2012) Parametric local metric learning for nearest neighbor classification. In: Advances in neural information processing systems, pp 1601–1609
Weinberger KQ, Saul LK (2009) Distance metric learning for large margin nearest neighbor classification. J Mach Learn Res 10(Feb):207–244
Xiang S, Nie F, Zhang C (2008) Learning a Mahalanobis distance metric for data clustering and classification. Pattern Recognit 41(12):3600–3612
Ying Y, Li P (2012) Distance metric learning with eigenvalue optimization. J Mach Learn Res 13:1–26
Zadeh P, Hosseini R, Sra S (2016) Geometric mean metric learning. In: International conference on machine learning, pp 2464–2471
Zuo W, Wang F, Zhang D, Lin L, Huang Y, Meng D, Zhang L (2015) Iterated support vector machines for distance metric learning. arXiv preprint arXiv:1502.00363
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Goudarzi, Z., Adibi, P., Grigat, R. et al. Making metric learning algorithms invariant to transformations using a projection metric on Grassmann manifolds. Int. J. Mach. Learn. & Cyber. 10, 3407–3416 (2019). https://doi.org/10.1007/s13042-019-00927-4
- Metric learning
- Grassmann manifold
- Mahalanobis metric