A feasible k-means kernel trick under non-Euclidean feature space

Ackerman, M., Ben-David, S. and Loker, D. (2010). Towards property-based classification of clustering paradigms, in J. Lafferty et al. (Eds), Advances in Neural Information Processing Systems 23, Curran Associates, Red Hook, NY, pp. 10–18.Search in Google Scholar

Balaji, R. and Bapat, R. (2007). On Euclidean distance matrices, Linear Algebra and Its Applications424(1): 108–117.10.1016/j.laa.2006.05.013Search in Google Scholar

Bao, T. and Kadobayashi, Y. (2008). On tighter inequalities for efficient similarity search in metric spaces, IAENG International Journal of Computer Science35(3): IJCS_35_3_17.Search in Google Scholar

Bradley, P.S., Mangasarian, O.L. and Street, W.N. (1996). Clustering via concave minimization, Proceedings of the 9th International Conference on Neural Information Processing Systems, NIPS’96, Denver, CO, USA, pp. 368–374.Search in Google Scholar

Cailliez, F. (1983). The analytical solution of the additive constant problem, Psychometrika48(2): 305–308.10.1007/BF02294026Search in Google Scholar

Chitta, R., Jin, R., Havens, T. and Jain, A. (2011). Approximate kernel k-means: Solution to large scale kernel clustering, Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’11, San Diego, CA, USA, pp. 895–903.Search in Google Scholar

Choi, H. and Choi, S. (2005). Kernel Isomap on noisy manifold, 4th International Conference on Development and Learning, Osaka, Japan, pp. 208 – 213.Search in Google Scholar

Cox, T.F. and Cox, M.A.A. (2001). Multidimensional Scaling, 2nd Edn, Chapman & Hall, London.10.1201/9780367801700Search in Google Scholar

Demontis, A., Melis, M., Biggio, B., Fumera, G. and Roli, F. (2017). Super-sparse learning in similarity spaces, CoRR: abs/1712.06131.Search in Google Scholar

Dokmanic, I., Parhizkar, R., Ranieri, J. and Vetterli, M. (2015). Euclidean distance matrices: A short walk through theory, algorithms and applications, CoRR: abs/1502.07541.Search in Google Scholar

Du, L., Zhou, P., Shi, L., Wang, H., Fan, M., Wang, W. and Shen, Y. (2015). Robust multiple kernel k-means using l₂₁-norm, Proceedings of the 24th International Conference on Artificial Intelligence, IJCAI’15, Buenos Aries, Argentina, pp. 3476–3482.Search in Google Scholar

Gisbrecht, A. and Schleif, F.-M. (2015). Metric and non-metric proximity transformations at linear costs, Neurocomputing167(1): 643–657.10.1016/j.neucom.2015.04.017Search in Google Scholar

Gonzalez, J. and Munoz, A. (2010). Representing functional data in reproducing kernel Hilbert spaces with applications to clustering and classification, Statistics and Econometrics Series013, Working paper 10-27.Search in Google Scholar

Gower, J.C. (1982). Euclidean distance geometry, The Mathematical Scientist7: 1–14.Search in Google Scholar

Gower, J.C. (1985). Properties of Euclidean and non-Euclidean distance matrices, Linear Algebra and Its Applications67: 81–97.10.1016/0024-3795(85)90187-9Search in Google Scholar

Gower, J. and Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients, Journal of Classification3(1): 5–48.10.1007/BF01896809Search in Google Scholar

Handhayania, T. and Hiryantob, L. (2015). Intelligent kernel k-means for clustering gene expression, International Conference on Computer Science and Computational Intelligence (ICCSCI 2015), Jakarta, Indonesia, Vol. 59, pp. 171–177.Search in Google Scholar

Higham, N.J. (1988). Computing a nearest symmetric positive semidefinite matrix, Linear Algebra and Its Applications103: 103–118.10.1016/0024-3795(88)90223-6Search in Google Scholar

Hofmann, T., Schölkopf, B. and Smola, A.J. (2008). Kernel methods in machine learning, Annals of Statistics36(3): 1171–1220.10.1214/009053607000000677Search in Google Scholar

Jacobs, D., Weinshall, D. and Gdalyahu, Y. (2000). Classification with nonmetric distances: Image retrieval and class representation, IEEE Transactions on Pattern Analysis and Machine Intelligence22(6): 583–600.10.1109/34.862197Search in Google Scholar

Jain, A. and Zongker, D. (1998). Representation and recognition of handwritten digits using deformable templates, IEEE Transactions on Pattern Analysis and Machine Intelligence19(12): 1386–1390.10.1109/34.643899Search in Google Scholar

Jaworski, M. (2018). Regression function and noise variance tracking methods for data streams with concept drift, International Journal of Applied Mathematics and Computer Science28(3): 559–567, DOI: 10.2478/amcs-2018-0043.10.2478/amcs-2018-0043Search in Google Scholar

Kashima, H., Hu, J., Ray, B. and Singh, M. (2008). K-means clustering of proportional data using l1 distance, Proceedings of the 19th International Conference on Pattern Recognition, Tampa, FL, USA.10.1109/ICPR.2008.4760982Search in Google Scholar

Kleinberg, J. (2002). An impossibility theorem for clustering, Proceedings of the 16th Neural Information Processing Systems Conference, NIPS 2002, Vancouver, BC, Canada, pp. 446–453.Search in Google Scholar

Kłopotek, M. (2019). On the existence of kernel function for kernel-trick of k-means in the light of Gower theorem, Fundamenta Informaticae168(1): 25–43.10.3233/FI-2019-1822Search in Google Scholar

Legendre, P. and Legendre, L. (1998). Numerical Ecology, 2nd Edn, Elsevier, Amsterdam.Search in Google Scholar

Li, C., Georgiopoulos, M. and Anagnostopoulos, G.C. (2013). Kernel-based distance metric learning in the output space, International Joint Conference on Neural Networks (IJCNN), Dallas, TX, USA, pp. 1–8.Search in Google Scholar

Lingoes, J. (1971). Some boundary conditions for a monotone analysis of symmetric matrices, Psychometrika36(2): 195–203.10.1007/BF02291398Search in Google Scholar

Loosli, G., Canu, S. and Ong, C. (2016). Learning SVM in Kreĭn spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence38(6): 1204–1216.10.1109/TPAMI.2015.247783026372207Search in Google Scholar

Marin, D., Tang, M., Ayed, I.B. and Boykov, Y. (2019). Kernel clustering: Density biases and solutions, IEEE Transactions on Pattern Analysis and Machine Intelligence41(1): 136–147.10.1109/TPAMI.2017.278016629990278Search in Google Scholar

Marteau, P.-F. (2019). Times series averaging and denoising from a probabilistic perspective on time-elastic kernels, International Journal of Applied Mathematics and Computer Science29(2): 375–392, DOI: 10.2478/amcs-2019-0028.10.2478/amcs-2019-0028Search in Google Scholar

Pan, V.Y. and Chen, Z.Q. (1999). The complexity of the matrix eigenproblem, Proceedings of the 31st Annual ACM Symposium on Theory of Computing, STOC’99, Atlanta, GA, USA, pp. 507–516.Search in Google Scholar

Pękalska, E. and Duin, R. (2005). The Dissimilarity Representation for Pattern Recognition: Foundations and Applications (Machine Perception and Artificial Intelligence), World Scientific, River Edge, NJ.10.1142/5965Search in Google Scholar

Pękalska, E., Paclík, P. and Duin, R. (2002). A generalized kernel approach to dissimilarity-based classification, Journal of Machine Learning Research2: 175–211.Search in Google Scholar

Qi, H. (2016). A convex matrix optimization for constant problem in multidimensional scaling with application to LLE, SIAM Journal on Optimization26(4): 2564–2590.10.1137/15M1043133Search in Google Scholar

Richter, R., Kyprianidis, J., Springborn, B. and Alexa, M. (2017). Constrained modelling of 3-valent meshes using a hyperbolic deformation metric, Computer Graphics Forum36(6): 62–75.10.1111/cgf.12805Search in Google Scholar

Roth, V., Laub, J., Kawanabe, M. and Buhmann, J. (2003). Optimal cluster preserving embedding of nonmetric proximity data, IEEE Transactions on Pattern Analysis and Machine Intelligence25(12): 1540–1551.10.1109/TPAMI.2003.1251147Search in Google Scholar

Sarma, T., Vishwanath, P. and Reddy, B. (2013). Single pass kernel k-means clustering method, Sadhana38(3): 407–419.10.1007/s12046-013-0143-3Search in Google Scholar

Schleif, F. and Tino, P. (2015). Indefinite proximity learning: A review, Neural Computation27(10): 2039–2096.10.1162/NECO_a_0077026313601Search in Google Scholar

Schoenberg, I.J. (1938). Metric spaces and positive definite functions, Transactions of the American Mathematical Society44(3): 522–536.10.1090/S0002-9947-1938-1501980-0Search in Google Scholar

Shawe-Taylor, J. and Cristianini, N. (2004). Kernel Methods for Pattern Analysis, Cambridge University Press, Cambridge.10.1017/CBO9780511809682Search in Google Scholar

Szalkai, B. (2013). An implementation of the relational k-means algorithm, CoRR: abs/1304.6899.Search in Google Scholar

Szekely, G. and Rizzo, M. (2014). Partial distance correlation with methods for dissimilarities, CoRR: abs/1310.2926.Search in Google Scholar

Tzortzis, G. and Likas, A.C. (2009). The global kernel k-means algorithm for clustering in feature space, IEEE Transactions on Neural Networks7(20): 1181–94.10.1109/TNN.2009.201972219493848Search in Google Scholar

Villmann, T., Kaden, M., Nebel, D. and Bohnsack, A. (2016). Similarities, dissimilarities and types of inner products for data analysis in the context of machine learning, in L. Rutkowski et al. (Eds), Artificial Intelligence and Soft Computing: 15th International Conference, ICAISC 2016, Springer, Cham, pp. 125–133.10.1007/978-3-319-39384-1_11Search in Google Scholar

Wierzchoń, S. and Kłopotek, M. (2018). Modern Clustering Algorithms, Springer, Cham.Search in Google Scholar

Yao, Y. and Chen, H. (2018). Multiple kernel k-means clustering by selecting representative kernels, CoRR: abs/1811.00264.Search in Google Scholar