Abstract
Many problems require a notion of distance between a set of points in a metric space, e.g., clustering data points in an N-dimensional space, object retrieval in pattern recognition, and image segmentation. However, these applications often require that the distances must be a metric, meaning that they must satisfy a set of conditions, with triangle inequality being the focus of this paper. Given an \(N \times N\) dissimilarity matrix with triangle inequality violations, the metric nearness problem requires to find a closest distance matrix, which satisfies the triangle inequality condition. This paper introduces a new deep learning approach for approximating a nearest matrix with more efficient runtime complexity than existing algorithms. We have experimented with several deep learning architectures, and our experimental results demonstrate that deep neural networks can learn to construct a close-distance matrix efficiently by removing most of the triangular inequality violations.
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References
Baraty, S., Simovici, D.A., Zara, C.: The impact of triangular inequality violations on medoid-based clustering. In: Kryszkiewicz, M., Rybinski, H., Skowron, A., Raś, Z.W. (eds.) ISMIS 2011. LNCS (LNAI), vol. 6804, pp. 280–289. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21916-0_31
Bronstein, A.M., Bronstein, M.M., Bruckstein, A.M., Kimmel, R.: Partial similarity of objects, or how to compare a centaur to a horse. Int. J. Comput. Vision 84(2), 163 (2008)
Demirci, M.F.: Efficient shape retrieval under partial matching. In: 20th International Conference on Pattern Recognition, pp. 3057–3060, August 2010
Fleming, C.L., Griffis, S.E., Bell, J.E.: The effects of triangle inequality on the vehicle routing problem. Eur. J. Oper. Res. 224(1), 1–7 (2013)
Gilbert, A.C., Jain, L.: If it ain’t broke, don’t fix it: sparse metric repair. In: 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 612–619, October 2017
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. CoRR abs/1512.03385 (2015)
Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. Commun. ACM 60(6), 84–90 (2017)
Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. CoRR abs/1409.1556 (2014)
Solomonik, E., Bulu, A., Demmel, J.: Minimizing communication in all-pairs shortest paths. In: IEEE 27th International Symposium on Parallel and Distributed Processing, pp. 548–559, May 2013
Sra, S., Tropp, J., Dhillon, I.S.: Triangle fixing algorithms for the metric nearness problem. In: Saul, L.K., Weiss, Y., Bottou, L. (eds.) Advances in Neural Information Processing Systems, vol. 17, pp. 361–368. MIT Press, Cambridge (2005)
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Gabidolla, M., Iskakov, A., Demirci, M.F., Yazici, A. (2019). On Approximating Metric Nearness Through Deep Learning. In: Rutkowski, L., Scherer, R., Korytkowski, M., Pedrycz, W., Tadeusiewicz, R., Zurada, J. (eds) Artificial Intelligence and Soft Computing. ICAISC 2019. Lecture Notes in Computer Science(), vol 11508. Springer, Cham. https://doi.org/10.1007/978-3-030-20912-4_6
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DOI: https://doi.org/10.1007/978-3-030-20912-4_6
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