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S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Statist. 22
1951
Earlier work this paper cites.
J J Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proceedings of the National Academy of Sciences 79
1982
Earlier work this paper cites.
J. Pearl, Bayesian Networks: A Model of Self-activated Memory for Evidential Reasoning , Report (UCLA, Computer Science Department, 1985)
1985
Earlier work this paper cites.
Geoffrey E Hinton and Terrence J Sejnowski, “Learning and relearning in boltzmann machines,” in Parallel distributed processing: Explorations in the microstructure of cognition , Vol. 1 (1986) pp. 282–317
1986
Earlier work this paper cites.
Xuedong D Huang, Yasuo Ariki, and Mervyn A Jack, “Hidden markov models for speech recognition,” (1990)
1990
Earlier work this paper cites.
F. Verstraete and J. I. Cirac, “Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions,” eprint arXiv:cond-mat/0407066 (2004), cond-mat/0407066
2004
Earlier work this paper cites.
D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix Product State Representations,” eprint arXiv:quant-ph/0608197 (2006), quant-ph/0608197
2006
Earlier work this paper cites.
Y.-Y. Shi, L.-M. Duan, and G. Vidal, “Classical simulation of quantum many-body systems with a tree tensor network,” Phys. Rev. A 74
2006
Earlier work this paper cites.
F. R. Kschischang, B. J. Frey, and H. A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theor. 47
2006
Earlier work this paper cites.
G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett. 99
2007
Earlier work this paper cites.
Michael Levin and Cody P. Nave, “Tensor renormalization group approach to two-dimensional classical lattice models,” Phys. Rev. Lett. 99
2007
Earlier work this paper cites.
G. Vidal, “Entanglement Renormalization,” Physical Review Letters 99
2007
Earlier work this paper cites.
N Le Roux and Y Bengio, “Representational power of restricted Boltzmann machines and deep belief networks,” Neural computation 20
2008
Earlier work this paper cites.
Ruslan Salakhutdinov and Iain Murray, “On the quantitative analysis of deep belief networks,” in Proceedings of the 25th Annual International Conference on Machine Learning (ICML 2008) , edited by Andrew McCallum and Sam Roweis (Omnipress, 2008) pp. 872–879
2008
Earlier work this paper cites.

Similar

Geoffrey E Hinton, “Deep belief networks,” Scholarpedia 4
2009
Cited alongside, same era.
L. Tagliacozzo, G. Evenbly, and G. Vidal, “Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law,” Phys. Rev. B 80
Original
2009
Cited alongside, same era.
J. Eisert, M. Cramer, and M. B. Plenio, “ Colloquium
2010
Cited alongside, same era.
D. P Kingma and M. Welling, “Auto-Encoding Variational Bayes,” ArXiv e-prints (2013), arXiv:1312.6114 [stat.ML]
Original
2013
Cited alongside, same era.
C. Bény, “Deep learning and the renormalization group,” ArXiv e-prints (2013), arXiv:1301.3124 [quant-ph]
Original
2013
Cited alongside, same era.

Then

E. Miles Stoudenmire and D. J. Schwab, “Supervised Learning with Quantum-Inspired Tensor Networks,” Advances in Neural Information Processing Systems 29, 4799 (2016) (2016), arXiv:1605.05775 [stat.ML]
Original
2016
Later among the works it cites.
Yoav Levine, David Yakira, Nadav Cohen, and Amnon Shashua, “Deep learning and quantum entanglement: Fundamental connections with implications to network design,” (2017) , arXiv:1704.01552 [cs.LG]
Original
2017
Later among the works it cites.
Xun Gao, Zhengyu Zhang, and Luming Duan, “An efficient quantum algorithm for generative machine learning,” arXiv:1711.02038 [quant-ph, stat] (2017) , arXiv: 1711.02038
Original
2017
Later among the works it cites.
H. W. Lin, M. Tegmark, and D. Rolnick, “Why Does Deep and Cheap Learning Work So Well?” Journal of Statistical Physics 168
Original
2017
Later among the works it cites.

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Reading the bibliography…

2019

Tree Tensor Networks for Generative Modeling

Cheng, Song, Wang, Lei, Xiang, Tao et al.

Understand

Matrix product states (MPS), a tensor network designed for one-dimensional quantum systems, has been recently proposed for generative modeling of natural data (such as images) in terms of `Born machine'.

However, the exponential decay of correlation in MPS restricts its representation power heavily for modeling complex data such as natural images.
In this work, we push forward the effort of applying tensor networks to machine learning by employing the Tree Tensor Network (TTN) which exhibits balanced performance in expressibility and efficient training and sampling.
We design the tree tensor network to utilize the 2-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing Graphical Processing Units (GPU).

Built on

S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Statist. 22
1951
Earlier work this paper cites.
J J Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proceedings of the National Academy of Sciences 79
1982
Earlier work this paper cites.
J. Pearl, Bayesian Networks: A Model of Self-activated Memory for Evidential Reasoning , Report (UCLA, Computer Science Department, 1985)
1985
Earlier work this paper cites.
Geoffrey E Hinton and Terrence J Sejnowski, “Learning and relearning in boltzmann machines,” in Parallel distributed processing: Explorations in the microstructure of cognition , Vol. 1 (1986) pp. 282–317
1986
Earlier work this paper cites.
Xuedong D Huang, Yasuo Ariki, and Mervyn A Jack, “Hidden markov models for speech recognition,” (1990)
1990
Earlier work this paper cites.
F. Verstraete and J. I. Cirac, “Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions,” eprint arXiv:cond-mat/0407066 (2004), cond-mat/0407066
2004
Earlier work this paper cites.
D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix Product State Representations,” eprint arXiv:quant-ph/0608197 (2006), quant-ph/0608197
2006
Earlier work this paper cites.
Y.-Y. Shi, L.-M. Duan, and G. Vidal, “Classical simulation of quantum many-body systems with a tree tensor network,” Phys. Rev. A 74
2006
Earlier work this paper cites.
F. R. Kschischang, B. J. Frey, and H. A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theor. 47
2006
Earlier work this paper cites.
G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett. 99
2007
Earlier work this paper cites.
Michael Levin and Cody P. Nave, “Tensor renormalization group approach to two-dimensional classical lattice models,” Phys. Rev. Lett. 99
2007
Earlier work this paper cites.
G. Vidal, “Entanglement Renormalization,” Physical Review Letters 99
2007
Earlier work this paper cites.
N Le Roux and Y Bengio, “Representational power of restricted Boltzmann machines and deep belief networks,” Neural computation 20
2008
Earlier work this paper cites.
Ruslan Salakhutdinov and Iain Murray, “On the quantitative analysis of deep belief networks,” in Proceedings of the 25th Annual International Conference on Machine Learning (ICML 2008) , edited by Andrew McCallum and Sam Roweis (Omnipress, 2008) pp. 872–879
2008
Earlier work this paper cites.

Similar

Geoffrey E Hinton, “Deep belief networks,” Scholarpedia 4
2009
Cited alongside, same era.
L. Tagliacozzo, G. Evenbly, and G. Vidal, “Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law,” Phys. Rev. B 80
Original
2009
Cited alongside, same era.
J. Eisert, M. Cramer, and M. B. Plenio, “ Colloquium
2010
Cited alongside, same era.
D. P Kingma and M. Welling, “Auto-Encoding Variational Bayes,” ArXiv e-prints (2013), arXiv:1312.6114 [stat.ML]
Original
2013
Cited alongside, same era.
C. Bény, “Deep learning and the renormalization group,” ArXiv e-prints (2013), arXiv:1301.3124 [quant-ph]
Original
2013
Cited alongside, same era.

Then

E. Miles Stoudenmire and D. J. Schwab, “Supervised Learning with Quantum-Inspired Tensor Networks,” Advances in Neural Information Processing Systems 29, 4799 (2016) (2016), arXiv:1605.05775 [stat.ML]
Original
2016
Later among the works it cites.
Yoav Levine, David Yakira, Nadav Cohen, and Amnon Shashua, “Deep learning and quantum entanglement: Fundamental connections with implications to network design,” (2017) , arXiv:1704.01552 [cs.LG]
Original
2017
Later among the works it cites.
Xun Gao, Zhengyu Zhang, and Luming Duan, “An efficient quantum algorithm for generative machine learning,” arXiv:1711.02038 [quant-ph, stat] (2017) , arXiv: 1711.02038
Original
2017
Later among the works it cites.
H. W. Lin, M. Tegmark, and D. Rolnick, “Why Does Deep and Cheap Learning Work So Well?” Journal of Statistical Physics 168
Original
2017
Later among the works it cites.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio, “Generative adversarial nets,” in Advances in Neural Information Processing Systems 27 , edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (Curran Associates, Inc., 2014) pp. 2672–2680

2014

Cited alongside, same era.

R. Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349

2014

Cited alongside, same era.

Roman Orus, “Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography,” The European Physical Journal B 87

2014

Cited alongside, same era.

P. Mehta and D. J. Schwab, “An exact mapping between the Variational Renormalization Group and Deep Learning,” ArXiv e-prints (2014), arXiv:1410.3831 [stat.ML]

Original

2014

Cited alongside, same era.

Ruslan Salakhutdinov, “Learning deep generative models,” Annual Review of Statistics and Its Application 2

2015

Cited alongside, same era.

Yann LeCun, Yoshua Bengio, and Geoffrey Hinton, “Deep learning,” Nature 521

2015

Cited alongside, same era.

Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov, “Importance Weighted Autoencoders,” arXiv:1509.00519 [cs, stat] (2015) , arXiv: 1509.00519

Original

2015

Cited alongside, same era.

V. Pestun and Y. Vlassopoulos, “Tensor network language model,” ArXiv e-prints (2017), arXiv:1710.10248 [cs.CL]

Original

2017