Abstract
Semi Non-negative Matrix Factorization (SNMF) is a machine learning algorithm that is used to decompose large data matrices where the data matrix is unconstrained (i.e., it may have mixed signs). We develop the quantum version of SNMF using quantum gradient descent, and we show that the quantum version of SNMF provides an exponential speedup compared to its classical version. Our approach is well adapted for high-dimensional problems since in the context of using a quantum approach only a few number of iterations is enough to obtain a close approximation of the factorization. From the algorithmic analysis point of view, the proposed algorithm for Quantum Semi Non-negative Matrix Factorization (QSNMF) yields a performance \(T\mathcal {O}( polylog K (M+N)),\) while the classical version of Semi Non-negative Matrix Factorization (SNMF) yields a performance \(T\mathcal {O}(poly (K M N))\) where M and N are the dimensions of the matrix, K is the number of clusters and T is the maximum number of iterations to prepare the quantum states.
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References
Benlamine, K., Grozavu, N., Bennani, Y., Matei, B.: Collaborative non-negative matrix factorization. In: International Conference on Artificial Neural Networks, pp. 655–666. Springer (2019)
Cichocki, A., Phan, A.-H.: Fast local algorithms for large scale nonnegative matrix and tensor factorizations. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 92(3), 708–721 (2009)
Ding, C., He, X., Simon, H.D.: On the equivalence of nonnegative matrix factorization and spectral clustering. In: Proceedings of the 2005 SIAM International Conference on Data Mining, pp. 606–610. SIAM (2005)
Ding, C.H.Q., Li, T., Jordan, M.I.: Convex and semi-nonnegative matrix factorizations. IEEE Trans. Pattern Anal. Mach. Intell. 32(1), 45–55 (2008)
Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 150502 (2009)
Iten, R., Colbeck, R., Kukuljan, I., Home, J., Christandl, M.: Quantum circuits for isometries. Phys. Rev. A, 93, 032318 (2016)
Kim, J., Park, H.: Sparse nonnegative matrix factorization for clustering. Technical report, Georgia Institute of Technology (2008)
Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788 (1999)
Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. In: Advances in Neural Information Processing Systems, pp. 556–562 (2001)
Lloyd, S., Mohseni, M., Rebentrost, P.: Quantum algorithms for supervised and unsupervised machine learning. arXiv preprint arXiv:1307.0411 (2013)
Lloyd, S., Mohseni, M., Rebentrost, P.: Quantum principal component analysis. Nat. Phys. 10(9), 631 (2014)
Paatero, P., Tapper, U.: Positive matrix factorization: a non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5(2), 111–126 (1994)
Rebentrost, P., Mohseni, M., Lloyd, S.: Quantum support vector machine for big feature and big data classification. arXiv preprint arXiv:1307.0471 (2013)
Rebentrost, P., Schuld, M., Wossnig, L., Petruccione, F., Lloyd, S.: Quantum gradient descent and newton’s method for constrained polynomial optimization. New J. Phys. 21(7) (2019)
Shahnaz, F., Berry, M.W., Pauca, V.P., Plemmons, R.J.: Document clustering using nonnegative matrix factorization. Inf. Process. Manage. 42(2), 373–386 (2006)
Xie, Y.-L., Hopke, P.K., Paatero, P.: Positive matrix factorization applied to a curve resolution problem. J. Chemom. 12(6), 357–364 (1999)
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Benlamine, K., Bennani, Y., Matei, B., Grozavu, N. (2021). Quantum Semi Non-negative Matrix Factorization. In: Abraham, A., et al. Proceedings of the 12th International Conference on Soft Computing and Pattern Recognition (SoCPaR 2020). SoCPaR 2020. Advances in Intelligent Systems and Computing, vol 1383. Springer, Cham. https://doi.org/10.1007/978-3-030-73689-7_14
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DOI: https://doi.org/10.1007/978-3-030-73689-7_14
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