Abstract

Photonic neural network implementation has been gaining considerable attention as a potentially disruptive future technology. Demonstrating learning in large-scale neural networks is essential to establish photonic machine learning substrates as viable information processing systems. Realizing photonic neural networks with numerous nonlinear nodes in a fully parallel and efficient learning hardware has been lacking so far. We demonstrate a network of up to 2025 diffractively coupled photonic nodes, forming a large-scale recurrent neural network. Using a digital micro mirror device, we realize reinforcement learning. Our scheme is fully parallel, and the passive weights maximize energy efficiency and bandwidth. The computational output efficiently converges, and we achieve very good performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (6)

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

P. Antonik, M. Haelterman, and S. Massar, “Online training for high-performance analogue readout layers in photonic reservoir computers,” Cognit. Comput. 9, 297–306 (2017).
[Crossref]

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7, 8772 (2017).
[Crossref]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

J. Bueno, D. Brunner, M. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25, 2401–2412 (2017).
[Crossref]

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” Optica 4, 1117–1125 (2017).
[Crossref]

2015 (3)

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40, 3854–3857 (2015).
[Crossref]

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

2013 (1)

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref]

2012 (3)

2011 (1)

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

2008 (1)

2006 (1)

2004 (1)

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref]

1987 (1)

Antonik, P.

P. Antonik, M. Haelterman, and S. Massar, “Online training for high-performance analogue readout layers in photonic reservoir computers,” Cognit. Comput. 9, 297–306 (2017).
[Crossref]

Appeltant, L.

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Baehr-Jones, T.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Baets, R.

Barbastathis, G.

Barnett, S. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Bienstman, P.

Brunner, D.

J. Bueno, D. Brunner, M. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25, 2401–2412 (2017).
[Crossref]

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40, 3854–3857 (2015).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref]

Bueno, J.

Dambre, J.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Danckaert, J.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Denz, C.

C. Denz, Optical Neural Networks (Springer Vieweg, 1998).

Dierckx, W.

Duport, F.

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20, 22783–22795 (2012).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

Edgar, M. P.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

Englund, D.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Fischer, I.

J. Bueno, D. Brunner, M. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25, 2401–2412 (2017).
[Crossref]

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40, 3854–3857 (2015).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Gibson, G. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

Gutierrez, J. M.

Gutiérrez, J. M.

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

Haas, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref]

Haelterman, M.

P. Antonik, M. Haelterman, and S. Massar, “Online training for high-performance analogue readout layers in photonic reservoir computers,” Cognit. Comput. 9, 297–306 (2017).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20, 22783–22795 (2012).
[Crossref]

Harris, N. C.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Hinton, G.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Hochberg, M.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Jaeger, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref]

Kim, S.-J.

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7, 8772 (2017).
[Crossref]

Larger, L.

Larochelle, H.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Lasser, T.

LeCun, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Lee, J.

Leitgeb, R. A.

Leutenegger, M.

Li, S.

Massar, S.

P. Antonik, M. Haelterman, and S. Massar, “Online training for high-performance analogue readout layers in photonic reservoir computers,” Cognit. Comput. 9, 297–306 (2017).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20, 22783–22795 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Mirasso, C. R.

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Naruse, M.

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7, 8772 (2017).
[Crossref]

Ortín, M. C.

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

Padgett, M. J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

Paquot, Y.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

Pesquera, L.

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref]

Phillips, D. B.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

Prabhu, M.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Psaltis, D.

Rao, R.

San-Martín, D.

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

Schneider, B.

Schrauwen, B.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Van Campenhout, “Toward optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182–11192 (2008).
[Crossref]

Shen, Y.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Sinha, A.

Skirlo, S.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Smerieri, A.

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20, 22783–22795 (2012).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref]

Soljacic, M.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Soriano, M.

Soriano, M. C.

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
[Crossref]

Soriano, S.

M. C. Ortín, S. Soriano, L. Pesquera, D. Brunner, D. San-Martín, I. Fischer, C. R. Mirasso, and J. M. Gutiérrez, “A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron,” Sci. Rep. 5, 14945 (2015).
[Crossref]

Sun, M.-J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref]

Sun, X.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Taylor, J. M.

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Figures (3)

Fig. 1.
Fig. 1. (a) Schematic illustration of a recurrent neural network. (b) Experimental setup. The NN state encoded on the SLM is imaged onto the camera (CAM) through a polarizing beam splitter (PBS) and a diffractive optical element (DOE). In this way, we realize a recurrently coupled network of Ikeda maps. A digital micromirror device (DMD) creates a spatially modulated image of the SLM’s state. The integrated state, obtained via superimposing the modulated intensities, is detected, creating the photonic RNN’s output. (c) RNN’s coupling matrix established by the DOE, with the inset showing a zoom into a smaller region.
Fig. 2.
Fig. 2. (a) Learning curves for a photonic RNN with two phase offsets ϕi across the network. A fraction of (1μ) nodes have a response function with a negative slope (Θ0=0.17π), and the remaining nodes experience a response function with a positive slope (Θ0=0.43π). (b) An almost symmetric distribution of phase offsets, resulting in positive and negative node responses, benefits computational performance.
Fig. 3.
Fig. 3. (a) Learning performance at optimal parameters (β=0.8, γ=0.4, μ=0.45). (b) Photonic RNN’s predicted output in μW (blue line) can hardly be differentiated from the prediction target (red dots). Prediction error ϵ is given by the yellow dashed data.

Equations (10)

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Ei(n)=Ei0cos(2πκSLM(xiSLM(n)+θi0)),
xi(n+1)=α|Ei0|2cos2(2πκSLM(βx˜iC(n)+θ˜i)).
xi(n+1)=α|Ei0|2cos2[β·α|jNWi,jDOEEj(n)|2+γWiinju(n+1)+θi].
yout(n+1)|iNWi,kDMD(Ei0Ei(n+1))|2.
Wkselect=rand(N)·Wbias,
[lk,Wkselect,max]=max(Wkselect),
Wlk,kDMD=¬(Wlk,k1DMD).
Wbias=1N+Wbias,Wlkbias=0.
r(k)={1if  ϵk<ϵk10if  ϵkϵk1,
Wlk,kDMD=r(k)Wlk,kDMD+(r(k)1)Wlk,k1DMD,

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