Melonic dominance and the largest eigenvalue of a large random tensor. (English) Zbl 1509.15004
Summary: We consider a Gaussian rotationally invariant ensemble of random real totally symmetric tensors with independent normally distributed entries, and estimate the largest eigenvalue of a typical tensor in this ensemble by examining the rate of growth of a random initial vector under successive applications of a nonlinear map defined by the random tensor. In the limit of a large number of dimensions, we observe that a simple form of melonic dominance holds, and the quantity we study is effectively determined by a single Feynman diagram arising from the Gaussian average over the tensor components. This computation suggests that the largest tensor eigenvalue in our ensemble in the limit of a large number of dimensions is proportional to the square root of the number of dimensions, as it is for random real symmetric matrices.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A69 | Multilinear algebra, tensor calculus |
| 15B52 | Random matrices (algebraic aspects) |
| 60B20 | Random matrices (probabilistic aspects) |
References:
| [1] | Mehta, ML, Random Matrices (2004), USA: Academic Press, USA · Zbl 1107.15019 |
| [2] | Gurau, R., Random Tensors (2017), Oxford: Oxford University Press, Oxford · Zbl 1371.81007 |
| [3] | Qi, L., Eigenvalues of a real supersymmetric tensor, J. Symb. Comp., 40, 1302 (2005) · Zbl 1125.15014 · doi:10.1016/j.jsc.2005.05.007 |
| [4] | Lim, L.-H.: Singular values and eigenvalues of tensors: a variational approach. In Proceedings of CAMSAP ’05, vol. 1, p. 129. (2005) |
| [5] | Qi, L.: The spectral theory of tensors (rough version). (2012). arXiv:1201.3424 [math.SP] |
| [6] | Qi, L.; Chen, H.; Chen, Y., Tensor Eigenvalues and Their Applications (2018), New York: Springer, New York · Zbl 1398.15001 · doi:10.1007/978-981-10-8058-6 |
| [7] | Dolotin, V., Morozov, A.: Introduction to non-linear algebra (World Scientific, 2007). arXiv:hep-th/0609022 · Zbl 1134.15001 |
| [8] | Cartwright, D.; Sturmfels, B., The number of eigenvalues of a tensor, Linear Algebra Appl., 438, 942 (2013) · Zbl 1277.15007 · doi:10.1016/j.laa.2011.05.040 |
| [9] | Morozov, A.; Shakirov, SH, Analogue of the identity Log Det = Trace Log for resultants, J. Geom. Phys., 16, 708 (2011) · Zbl 1214.15005 · doi:10.1016/j.geomphys.2010.12.001 |
| [10] | Kolda, TG; Bader, BW, Tensor decompositions and applications, SIAM Rev., 51, 455 (2009) · Zbl 1173.65029 · doi:10.1137/07070111X |
| [11] | Comon, P.; Golub, G.; Lim, L-H; Mourrain, B., Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl., 30, 1254 (2008) · Zbl 1181.15014 · doi:10.1137/060661569 |
| [12] | Robeva, E., Orthogonal decomposition of symmetric tensors, SIAM J. Matrix Anal. Appl., 37, 86 (2016) · doi:10.1137/140989340 |
| [13] | Auffinger, A.; Ben Arous, G.; Černý, J., Random matrices and complexity of spin glasses, Commun. Pure Appl. Math., 66, 165 (2013) · Zbl 1269.82066 · doi:10.1002/cpa.21422 |
| [14] | Cooper, J., Adjacency spectra of random and complete hypergraphs, Linear Algebra Appl., 596, 184 (2020) · Zbl 1435.05150 · doi:10.1016/j.laa.2020.03.013 |
| [15] | Bonzom, V.; Gurau, R.; Riello, A.; Rivasseau, V., Critical behavior of colored tensor models in the large N limit, Nucl. Phys. B, 853, 174 (2011) · Zbl 1229.81222 · doi:10.1016/j.nuclphysb.2011.07.022 |
| [16] | Gurau, R.; Ryan, JP, Colored tensor models-a review, SIGMA, 8, 020 (2012) · Zbl 1242.05094 |
| [17] | Li, G.; Qi, L.; Yu, G., The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Num. Linear Algebra Appl., 20, 1001 (2013) · Zbl 1313.65081 · doi:10.1002/nla.1877 |
| [18] | M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl. 31 (2009) 1090 · Zbl 1197.65036 |
| [19] | Montanari, A., Richard, E.: A statistical model for tensor PCA. In: Advances in Neural Information Processing Systems 27 (NIPS), (2014). arXiv:1411.1076 |
| [20] | Rivasseau, V.: Melonic non-linear flows and the spiked tensor model, talk at the mini-symposium Holographic Tensors. OIST, Okinawa (2018). (https://groups.oist.jp/sites/default/files/imce/u1390/RivasseauOIST2018.pdf) |
| [21] | Dartois, S.; Evnin, O.; Lionni, L.; Rivasseau, V.; Valette, G., Melonic turbulence, Commun. Math. Phys., 374, 1179 (2020) · Zbl 1436.81095 · doi:10.1007/s00220-020-03683-7 |
| [22] | May, RM, Will a large complex system be stable?, Nature, 238, 413 (1972) · doi:10.1038/238413a0 |
| [23] | Ros, V.; Ben Arous, G.; Biroli, G.; Cammarota, C., Complex energy landscapes in spiked-tensor and simple glassy models, Phys. Rev. X, 9, 00110003 (2019) |
| [24] | Cvitanović, P., Group Theory: Birdtracks, Lie’s, and Exceptional Groups (2008), New Jersey: Princeton, New Jersey · Zbl 1152.22001 · doi:10.1515/9781400837670 |
| [25] | Vidal, G.: Pedagogical introduction to tensor networks: MPS, PEPS and MERA, talk at the conference Tensor networks for quantum field theories. (2011). https://www.perimeterinstitute.ca/videos/pedagogical-introduction-tensor-networks-mps-peps-and-mera |
| [26] | Xu, C., Zhang, Z.: Random tensors and their normal distributions. arXiv:1908.01131 [math.ST] |
| [27] | Zinn-Justin, J.: Gaussian integrals. In: Path Integrals in Quantum Mechanics. Oxford University Press, Oxford (2004) |
| [28] | Makeenko, Y., Methods of Contemporary Gauge Theory (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1088.81004 · doi:10.1017/CBO9780511535147 |
| [29] | Tao, T.; Vu, V., Random matrices: the four moment theorem for Wigner ensembles, Random Matrix Theory, Interacting Particle Systems and Integrable Systems (2014), Cambridge: Cambridge University Press, Cambridge · Zbl 1327.15075 |
| [30] | Gurau, R., Universality for random tensors, Ann. H. Poincaré Probab. Stat., 50, 1474 (2014) · Zbl 1318.60010 |
| [31] | Brézin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, (2000) 111 arXiv:math-ph/9910005; Characteristic polynomials of random matrices at edge singularities. Phys. Rev. E 62, (2000) 3558 arXiv:math-ph/0004018 · Zbl 1042.82017 |
| [32] | Gurau, R.: On the generalization of the Wigner semicircle law to real symmetric tensors. arXiv:2004.02660 |
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