Abstract

For the benefit of designing scalable, fault resistant optical neural networks (ONNs), we investigate the effects architectural designs have on the ONNs’ robustness to imprecise components. We train two ONNs – one with a more tunable design (GridNet) and one with better fault tolerance (FFTNet) – to classify handwritten digits. When simulated without any imperfections, GridNet yields a better accuracy (98%) than FFTNet (95%). However, under a small amount of error in their photonic components, the more fault tolerant FFTNet overtakes GridNet. We further provide thorough quantitative and qualitative analyses of ONNs’ sensitivity to varying levels and types of imprecisions. Our results offer guidelines for the principled design of fault-tolerant ONNs as well as a foundation for further research.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. N. H. Farhat, D. Psaltis, A. Prata, and E. Paek, “Optical implementation of the hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
    [Crossref] [PubMed]
  2. Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Reports 2, 287 (2012).
    [Crossref]
  3. 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] [PubMed]
  4. A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
    [Crossref]
  5. A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
    [Crossref]
  6. J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
    [Crossref]
  7. Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, and M. S. Englund, Dirk, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441 (2017).
    [Crossref]
  8. I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, vol. 1 (MIT Cambridge, 2016).
  9. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
    [Crossref] [PubMed]
  10. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
    [Crossref]
  11. R. Barak and Y. Ben-Aryeh, “Quantum fast fourier transform and quantum computation by linear optics,” JOSA B 24, 231–240 (2007).
    [Crossref]
  12. J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
    [Crossref] [PubMed]
  13. N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
    [Crossref]
  14. S. Pai, B. Bartlett, O. Solgaard, and D. A. Miller, “Matrix optimization on universal unitary photonic devices,” arXiv preprint arXiv:1808.00458 (2018).
  15. N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
    [Crossref]
  16. R. Burgwal, W. R. Clements, D. H. Smith, J. C. Gates, W. S. Kolthammer, J. J. Renema, and I. A. Walmsley, “Using an imperfect photonic network to implement random unitaries,” Opt. Express 25, 28236–28245 (2017).
    [Crossref]
  17. D. A. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
    [Crossref]
  18. C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 db high-extinction auto-configured mach–zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
    [Crossref] [PubMed]
  19. M. Y.-S. Fang, “Imprecise optical neural networks,” https://github.com/mike-fang/imprecise_optical_neural_network (2019).
  20. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comput. 19, 297–301 (1965).
    [Crossref]
  21. Y. Ma, Y. Zhang, S. Yang, A. Novack, R.A. Ding, E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for soi optical interconnect,” Opt. Express 21, 29374–29382 (2013).
    [Crossref]
  22. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219 (2008).
    [Crossref]
  23. G. Panusa, Y. Pu, J. Wang, C. Moser, and D. Psaltis, “Photoinitiator-free multi-photon fabrication of compact optical waveguides in polydimethylsiloxane,” Opt. Mater. Express 9, 128–138 (2019).
    [Crossref]
  24. M. J. Connelly, Semiconductor Optical Amplifiers(Springer Science & Business Media, 2007).
  25. D. A. Miller, “Silicon photonics: Meshing optics with applications,” Nat. Photonics 11, 403 (2017).
    [Crossref]
  26. Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
    [Crossref]
  27. V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in” Proceedings of the 27th international conference on machine learning (ICML-10), (2010), pp. 807–814.
  28. M. Arjovsky, A. Shah, and Y. Bengio, “Unitary evolution recurrent neural networks,” in International Conference on Machine Learning, (2016), pp. 1120–1128.
  29. Q. Xu and M. Lipson, “Optical bistability based on the carrier dispersion effect in soi ring resonators,” in Integrated Photonics Research and Applications, (Optical Society of America, 2006), p. IMD2.
    [Crossref]
  30. Y. Jiang, P. T. DeVore, and B. Jalali, “Analog optical computing primitives in silicon photonics,” Opt. Lett. 41, 1273–1276 (2016).
    [Crossref] [PubMed]
  31. M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
    [Crossref] [PubMed]
  32. Y. LeCun, “The mnist database of handwritten digits,” http://yann.lecun.com/exdb/mnist/ .
  33. A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).
  34. C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.
  35. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) (Wiley-Interscience, New York, NY, USA, 2006).
  36. L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.
  37. C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).
  38. H. Robbins and S. Monro, “A stochastic approximation method,” in Herbert Robbins Selected Papers, (Springer, 1985), pp. 102–109.
    [Crossref]
  39. P. Y. Simard, D. Steinkraus, and J. C. Platt, “Best practices for convolutional neural networks applied to visual document analysis,” in null, (IEEE, 2003), p. 958.
  40. N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).
  41. F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
    [Crossref]
  42. F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
    [Crossref]
  43. D. F. Walls and G. J. Milburn, Quantum optics(Springer Science & Business Media, 2007).
  44. A. Fawzi, S.-M. Moosavi-Dezfooli, and P. Frossard, “Robustness of classifiers: from adversarial to random noise,” in Advances in Neural Information Processing Systems, (2016), pp. 1632–1640.
  45. L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.
  46. K. Kikuchi, “Characterization of semiconductor-laser phase noise and estimation of bit-error rate performance with low-speed offline digital coherent receivers,” Opt. Express 20, 5291–5302 (2012).
    [Crossref] [PubMed]
  47. M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.
  48. A. Selden, “Pulse transmission through a saturable absorber,” Br. J. Appl. Phys. 18, 743 (1967).
    [Crossref]
  49. I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).
  50. M. Rastegari, V. Ordonez, J. Redmon, and A. Farhadi, “Xnor-net: Imagenet classification using binary convolutional neural networks,” in European Conference on Computer Vision, (Springer, 2016), pp. 525–542.

2019 (1)

2018 (2)

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

2017 (8)

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

D. A. Miller, “Silicon photonics: Meshing optics with applications,” Nat. Photonics 11, 403 (2017).
[Crossref]

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

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

R. Burgwal, W. R. Clements, D. H. Smith, J. C. Gates, W. S. Kolthammer, J. J. Renema, and I. A. Walmsley, “Using an imperfect photonic network to implement random unitaries,” Opt. Express 25, 28236–28245 (2017).
[Crossref]

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

2016 (3)

2015 (3)

D. A. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

2014 (2)

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
[Crossref]

2013 (1)

2012 (2)

K. Kikuchi, “Characterization of semiconductor-laser phase noise and estimation of bit-error rate performance with low-speed offline digital coherent receivers,” Opt. Express 20, 5291–5302 (2012).
[Crossref] [PubMed]

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

2011 (2)

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] [PubMed]

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

2008 (1)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219 (2008).
[Crossref]

2007 (1)

R. Barak and Y. Ben-Aryeh, “Quantum fast fourier transform and quantum computation by linear optics,” JOSA B 24, 231–240 (2007).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

1985 (1)

1967 (1)

A. Selden, “Pulse transmission through a saturable absorber,” Br. J. Appl. Phys. 18, 743 (1967).
[Crossref]

1965 (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Allen, T. G.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Antiga, L.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Appeltant, L.

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] [PubMed]

Arjovsky, M.

M. Arjovsky, A. Shah, and Y. Bengio, “Unitary evolution recurrent neural networks,” in International Conference on Machine Learning, (2016), pp. 1120–1128.

Babaeian, M.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Baehr-Jones, T.

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

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Y. Ma, Y. Zhang, S. Yang, A. Novack, R.A. Ding, E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for soi optical interconnect,” Opt. Express 21, 29374–29382 (2013).
[Crossref]

Bao, Q.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Barak, R.

R. Barak and Y. Ben-Aryeh, “Quantum fast fourier transform and quantum computation by linear optics,” JOSA B 24, 231–240 (2007).
[Crossref]

Bartlett, B.

S. Pai, B. Bartlett, O. Solgaard, and D. A. Miller, “Matrix optimization on universal unitary photonic devices,” arXiv preprint arXiv:1808.00458 (2018).

Bélisle, F.

C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.

Ben-Aryeh, Y.

R. Barak and Y. Ben-Aryeh, “Quantum fast fourier transform and quantum computation by linear optics,” JOSA B 24, 231–240 (2007).
[Crossref]

Bengio, Y.

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

M. Arjovsky, A. Shah, and Y. Bengio, “Unitary evolution recurrent neural networks,” in International Conference on Machine Learning, (2016), pp. 1120–1128.

I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, vol. 1 (MIT Cambridge, 2016).

I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, vol. 1 (MIT Cambridge, 2016).

C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Bhardwaj, A.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Bilaniuk, O.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Blanche, P.-A.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Bunandar, D.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Burgwal, R.

Carolan, J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Chakhmakhchyan, L.

N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

Chan, K.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Chanan, G.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Chang, J.

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

Chen, C.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Chen, V. W.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Chi, S.-H.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Chintala, S.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Chiu, E.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Clements, W. R.

Connelly, M. J.

M. J. Connelly, Semiconductor Optical Amplifiers(Springer Science & Business Media, 2007).

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Courbariaux, M.

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

Courville, A.

I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, vol. 1 (MIT Cambridge, 2016).

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) (Wiley-Interscience, New York, NY, USA, 2006).

Crespi, A.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Cun, Y. Le

L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.

D’ambrosio, V.

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Dambre, J.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Reports 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] [PubMed]

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] [PubMed]

Desmaison, A.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

DeVito, Z.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

DeVore, P. T.

Ding, R.A.

Dubcek, T.

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Dugas, C.

C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.

Dun, X.

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

Duport, F.

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

El-Yaniv, R.

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

Englund, D.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Englund, M. S.

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

Farhadi, A.

M. Rastegari, V. Ordonez, J. Redmon, and A. Farhadi, “Xnor-net: Imagenet classification using binary convolutional neural networks,” in European Conference on Computer Vision, (Springer, 2016), pp. 525–542.

Farhat, N. H.

Fawzi, A.

A. Fawzi, S.-M. Moosavi-Dezfooli, and P. Frossard, “Robustness of classifiers: from adversarial to random noise,” in Advances in Neural Information Processing Systems, (2016), pp. 1632–1640.

Feng, Y.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Fergus, R.

L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.

Fischer, I.

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] [PubMed]

Flamini, F.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Frossard, P.

A. Fawzi, S.-M. Moosavi-Dezfooli, and P. Frossard, “Robustness of classifiers: from adversarial to random noise,” in Advances in Neural Information Processing Systems, (2016), pp. 1632–1640.

Garcia, R.

C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.

Gates, J. C.

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219 (2008).
[Crossref]

Goodfellow, I.

I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, vol. 1 (MIT Cambridge, 2016).

Gross, S.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Haelterman, M.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Reports 2, 287 (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, and M. S. Englund, Dirk, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Harrold, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Heidrich, W.

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

Hinton, G.

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

Hinton, G. E.

V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in” Proceedings of the 27th international conference on machine learning (ICML-10), (2010), pp. 807–814.

Hochberg, M.

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

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Y. Ma, Y. Zhang, S. Yang, A. Novack, R.A. Ding, E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for soi optical interconnect,” Opt. Express 21, 29374–29382 (2013).
[Crossref]

Huang, X.-d.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Hubara, I.

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

Humphreys, P. C.

Jalali, B.

Jiang, Y.

Jing, L.

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Kaplas, T.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Keiffer, P.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Kikuchi, K.

Koh, P.-C.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Kolthammer, W. S.

Krizhevsky, A.

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

Lahini, Y.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Laing, A.

N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

Larochelle, H.

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

Larson, M.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

LeCun, Y.

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Lerer, A.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Lim, E.-J.

Lima, T. F.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

Lin, Z.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Lipson, M.

Q. Xu and M. Lipson, “Optical bistability based on the carrier dispersion effect in soi ring resonators,” in Integrated Photonics Research and Applications, (Optical Society of America, 2006), p. IMD2.
[Crossref]

Lloyd, S.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Lo, G.-Q.

Loh, K. P.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Ma, Y.

Magrini, L.

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Marshall, G. D.

Martín-López, E.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Massar, S.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Reports 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] [PubMed]

Mataloni, P.

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Matsuda, N.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219 (2008).
[Crossref]

Mehri, S.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Metcalf, B. J.

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum optics(Springer Science & Business Media, 2007).

Miller, D. A.

D. A. Miller, “Silicon photonics: Meshing optics with applications,” Nat. Photonics 11, 403 (2017).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 db high-extinction auto-configured mach–zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref] [PubMed]

D. A. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
[Crossref]

S. Pai, B. Bartlett, O. Solgaard, and D. A. Miller, “Matrix optimization on universal unitary photonic devices,” arXiv preprint arXiv:1808.00458 (2018).

Mirasso, C. R.

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] [PubMed]

Moewe, M.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Monro, S.

H. Robbins and S. Monro, “A stochastic approximation method,” in Herbert Robbins Selected Papers, (Springer, 1985), pp. 102–109.
[Crossref]

Moosavi-Dezfooli, S.-M.

A. Fawzi, S.-M. Moosavi-Dezfooli, and P. Frossard, “Robustness of classifiers: from adversarial to random noise,” in Advances in Neural Information Processing Systems, (2016), pp. 1632–1640.

Moser, C.

Mower, J.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Nadeau, C.

C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.

Nahmias, M. A.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
[Crossref]

Nair, V.

V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in” Proceedings of the 27th international conference on machine learning (ICML-10), (2010), pp. 807–814.

Ni, Z.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Norwood, R. A.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Novack, A.

O’Brien, J. L.

Oguma, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Ordonez, V.

M. Rastegari, V. Ordonez, J. Redmon, and A. Farhadi, “Xnor-net: Imagenet classification using binary convolutional neural networks,” in European Conference on Computer Vision, (Springer, 2016), pp. 525–542.

Osellame, R.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Paek, E.

Paesani, S.

Pai, S.

S. Pai, B. Bartlett, O. Solgaard, and D. A. Miller, “Matrix optimization on universal unitary photonic devices,” arXiv preprint arXiv:1808.00458 (2018).

Pal, C. J.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Panusa, G.

Paquot, Y.

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

Paszke, A.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Patwardhan, A.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Perry, J. W.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Peurifoy, J.

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Platt, J. C.

P. Y. Simard, D. Steinkraus, and J. C. Platt, “Best practices for convolutional neural networks applied to visual document analysis,” in null, (IEEE, 2003), p. 958.

Polavarapu, L.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Prabhu, M.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

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

Prata, A.

Prucnal, P. R.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
[Crossref]

Psaltis, D.

Pu, Y.

Qiang, X.

Rab, A. S.

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Ramponi, R.

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Rastegari, M.

M. Rastegari, V. Ordonez, J. Redmon, and A. Farhadi, “Xnor-net: Imagenet classification using binary convolutional neural networks,” in European Conference on Computer Vision, (Springer, 2016), pp. 525–542.

Reck, M.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Redmon, J.

M. Rastegari, V. Ordonez, J. Redmon, and A. Farhadi, “Xnor-net: Imagenet classification using binary convolutional neural networks,” in European Conference on Computer Vision, (Springer, 2016), pp. 525–542.

Renema, J. J.

Robbins, H.

H. Robbins and S. Monro, “A stochastic approximation method,” in Herbert Robbins Selected Papers, (Springer, 1985), pp. 102–109.
[Crossref]

Rostamzadeh, N.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Russell, N. J.

N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Salakhutdinov, R.

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

Santagati, R.

Santos, J. F.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Schrauwen, B.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Reports 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] [PubMed]

Sciarrino, F.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Selden, A.

A. Selden, “Pulse transmission through a saturable absorber,” Br. J. Appl. Phys. 18, 743 (1967).
[Crossref]

Semakov, A.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

Serdyuk, D.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Shadbolt, P. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Shah, A.

M. Arjovsky, A. Shah, and Y. Bengio, “Unitary evolution recurrent neural networks,” in International Conference on Machine Learning, (2016), pp. 1120–1128.

Shastri, B. J.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
[Crossref]

Shen, Y.

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

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Shen, Z.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Silverstone, J. W.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Simard, P. Y.

P. Y. Simard, D. Steinkraus, and J. C. Platt, “Best practices for convolutional neural networks applied to visual document analysis,” in null, (IEEE, 2003), p. 958.

Sitzmann, V.

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

Skirlo, S.

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

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Smerieri, A.

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

Smith, D. H.

Solgaard, O.

S. Pai, B. Bartlett, O. Solgaard, and D. A. Miller, “Matrix optimization on universal unitary photonic devices,” arXiv preprint arXiv:1808.00458 (2018).

Soljacic, M.

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Soriano, M. C.

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] [PubMed]

Soudry, D.

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

Spagnolo, N.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Sparrow, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Srivastava, N.

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

Steinbrecher, G. R.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Steinkraus, D.

P. Y. Simard, D. Steinkraus, and J. C. Platt, “Best practices for convolutional neural networks applied to visual document analysis,” in null, (IEEE, 2003), p. 958.

Subramanian, S.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Sun, X.

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

Sutskever, I.

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

Svirko, Y.

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

Tait, A. N.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
[Crossref]

Tang, D.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Tegmark, M.

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) (Wiley-Interscience, New York, NY, USA, 2006).

Thompson, M. G.

Trabelsi, C.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Tukey, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Van der Sande, G.

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] [PubMed]

Viggianiello, N.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum optics(Springer Science & Business Media, 2007).

Walmsley, I. A.

Wan, L.

L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.

Wang, J.

Wang, Y.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Wetzstein, G.

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

Wilkes, C. M.

Wong, F. N.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

Wu, A. X.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

Xu, Q.

Q. Xu and M. Lipson, “Optical bistability based on the carrier dispersion effect in soi ring resonators,” in Integrated Photonics Research and Applications, (Optical Society of America, 2006), p. IMD2.
[Crossref]

Xu, Q.-H.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Yang, E.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

Yang, S.

Zandrini, T.

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Zeiler, M.

L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Zhang, H.

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Zhang, S.

L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.

Zhang, Y.

Y. Ma, Y. Zhang, S. Yang, A. Novack, R.A. Ding, E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for soi optical interconnect,” Opt. Express 21, 29374–29382 (2013).
[Crossref]

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

Zhao, S.

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

Zhou, E.

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

Zhou, X.

Appl. Opt. (1)

Br. J. Appl. Phys. (1)

A. Selden, “Pulse transmission through a saturable absorber,” Br. J. Appl. Phys. 18, 743 (1967).
[Crossref]

J. Light. Technol. (1)

A. N. Tait, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Broadcast and weight: an integrated network for scalable photonic spike processing,” J. Light. Technol. 32, 3427–3439 (2014).
[Crossref]

JOSA B (1)

R. Barak and Y. Ben-Aryeh, “Quantum fast fourier transform and quantum computation by linear optics,” JOSA B 24, 231–240 (2007).
[Crossref]

Light. Sci. & Appl. (1)

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light. Sci. & Appl. 4, e354 (2015).
[Crossref]

Math. Comput. (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Nano Res. (1)

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

Nat. Commun. (2)

M. Babaeian, P.-A. Blanche, R. A. Norwood, T. Kaplas, P. Keiffer, Y. Svirko, T. G. Allen, V. W. Chen, S.-H. Chi, and J. W. Perry, “Nonlinear optical components for all-optical probabilistic graphical model,” Nat. Commun. 9,2128 (2018).
[Crossref] [PubMed]

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] [PubMed]

Nat. Photonics (4)

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

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447 (2017).
[Crossref]

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219 (2008).
[Crossref]

D. A. Miller, “Silicon photonics: Meshing optics with applications,” Nat. Photonics 11, 403 (2017).
[Crossref]

New J. Phys. (1)

N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. Mater. Express (1)

Optica (2)

Phys. Rev. Lett. (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Sci. Reports (4)

J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Reports 8, 12324 (2018).
[Crossref]

A. N. Tait, T. F. Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic networks using silicon photonic weight banks,” Sci. Reports 7, 7430 (2017).
[Crossref]

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

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Reports 7, 15133 (2017).
[Crossref]

Science (1)

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, and G. D. M. M. G. T. J. C. F. M. T. H. J. L. O. A. L. Itoh, Mikitaka, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

The J. Mach. Learn. Res. (2)

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The J. Mach. Learn. Res. 15, 1929–1958 (2014).

I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Quantized neural networks: Training neural networks with low precision weights and activations,” The J. Mach. Learn. Res. 18, 6869–6898 (2017).

Other (20)

M. Rastegari, V. Ordonez, J. Redmon, and A. Farhadi, “Xnor-net: Imagenet classification using binary convolutional neural networks,” in European Conference on Computer Vision, (Springer, 2016), pp. 525–542.

M. Larson, Y. Feng, P.-C. Koh, X.-d. Huang, M. Moewe, A. Semakov, A. Patwardhan, E. Chiu, A. Bhardwaj, K. Chan, and et al., “Narrow linewidth high power thermally tuned sampled-grating distributed bragg reflector laser,” in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), (IEEE, 2013), pp. 1–3.

M. J. Connelly, Semiconductor Optical Amplifiers(Springer Science & Business Media, 2007).

D. F. Walls and G. J. Milburn, Quantum optics(Springer Science & Business Media, 2007).

A. Fawzi, S.-M. Moosavi-Dezfooli, and P. Frossard, “Robustness of classifiers: from adversarial to random noise,” in Advances in Neural Information Processing Systems, (2016), pp. 1632–1640.

L. Wan, M. Zeiler, S. Zhang, Y. Le Cun, and R. Fergus, “Regularization of neural networks using dropconnect,” in International Conference on Machine Learning, (2013), pp. 1058–1066.

S. Pai, B. Bartlett, O. Solgaard, and D. A. Miller, “Matrix optimization on universal unitary photonic devices,” arXiv preprint arXiv:1808.00458 (2018).

I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, vol. 1 (MIT Cambridge, 2016).

M. Y.-S. Fang, “Imprecise optical neural networks,” https://github.com/mike-fang/imprecise_optical_neural_network (2019).

V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in” Proceedings of the 27th international conference on machine learning (ICML-10), (2010), pp. 807–814.

M. Arjovsky, A. Shah, and Y. Bengio, “Unitary evolution recurrent neural networks,” in International Conference on Machine Learning, (2016), pp. 1120–1128.

Q. Xu and M. Lipson, “Optical bistability based on the carrier dispersion effect in soi ring resonators,” in Integrated Photonics Research and Applications, (Optical Society of America, 2006), p. IMD2.
[Crossref]

Y. LeCun, “The mnist database of handwritten digits,” http://yann.lecun.com/exdb/mnist/ .

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in NIPS-Workshop, (2017).

C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, “Incorporating second-order functional knowledge for better option pricing,” in Advances in neural information processing systems, (2001), pp. 472–478.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) (Wiley-Interscience, New York, NY, USA, 2006).

L. Jing, Y. Shen, T. Dubcek, J. Peurifoy, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (eunn) and their application to rnns,” in Proceedings of the 34th International Conference on Machine Learning-Volume 70, ( JMLR.org , 2017), pp. 1733–1741.

C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” arXiv preprint arXiv:1705.09792 (2017).

H. Robbins and S. Monro, “A stochastic approximation method,” in Herbert Robbins Selected Papers, (Springer, 1985), pp. 102–109.
[Crossref]

P. Y. Simard, D. Steinkraus, and J. C. Platt, “Best practices for convolutional neural networks applied to visual document analysis,” in null, (IEEE, 2003), p. 958.

Supplementary Material (1)

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Figures (16)

Fig. 1
Fig. 1 a) A schematic of a universal 8×4 optical linear multiplier with two unitary multipliers (red) consisting of MZIs in a grid-like layout and a diagonal layer (yellow). The MZIs of GridUnitary multipliers are indexed according to their layer depth (l) and dimension (d). Symbols at the top represent the mathematical operations performed by the various modules. Inset: A MZI with two 50:50 beamsplitters and two tunable phaseshifters b) An FFT-like, non-universal multiplier with FFTUnitary multipliers (blue).
Fig. 2
Fig. 2 Network design used for the MNIST classification task. GridNet used universal unitary multipliers while FFTNet used FFT-Unitary multipliers. See Fig. 1 for details of physical implementation of the three linear layers.
Fig. 3
Fig. 3 Visualizing the degradation of ONN outputs, FFTNet is seen to be much more robust than GridNet. Identical input is fed through GridNet (a, b) and FFTNet (c, d), simulated with ideal components (a, c) and imprecise components (b, d) with σBS = 0.01 and σPS = 0.01 rad. Imprecise networks are simulated 100 times and their mean output is represented by bar plots. Error bars represent the 20th to 80th percentile range.
Fig. 4
Fig. 4 The decrease in classification accuracy is visualized for GridNet and FFTNet. (a,b) The two networks were tested with simulated noise of various levels for 20 runs. The mean accuracy is plotted as a function of σPS and σBS. Note the difference in color map ranges between the two plots. (c) The accuracies of GridNet and FFTNet are compared along the σPS = σBS cutline.
Fig. 5
Fig. 5 The architecture of a) StackedFFT and b) TruncGrid shown with FFTUnitary and GridUnitary from which they were derived. For clarity, the dimension, here, is N = 24 = 16 so FFTUnitary was stacked four times and GridUnitary was truncated at the fourth layer. In the experiments described in this section, the dimension was taken to be N = 28 = 256.
Fig. 6
Fig. 6 With the same layer depth, multipliers with FFT-like architectures are shown to be more robust. The fidelity between the error-free and imprecise transfer matrices is plotted as a function of increasing error. Two sets of comparisons between unitary multipliers of the same depth are made. a) Both StackedFFT and GridUnitary have N = 256 layers of MZIs. b) TruncGrid and FFTNet have log N = 8 layers.
Fig. 7
Fig. 7 Change in accuracy due to localized imprecision in layer 2 of GridNet with randomized singular values. A large amount of imprecision (σPS = 0.1 rad) is introduced to 8×8 blocks of MZIs in an otherwise error-free GridNet. The resulting change in accuracy of the network is plotted as a function of the position of the MZI block in GridUnitary multipliers V 2 and U2 (coordinates defined as in Fig. 1(a)). The transmissivity of each waveguide through the diagonal layer Σ2 is also plotted (center panel).
Fig. 8
Fig. 8 Effects of localized imprecision in layer 2 of GridNet with ordered singular values. Similar to Fig. 7, except GridNet has its singular values ordered. Therefore, the transmissivity is also ordered (center panel).
Fig. 9
Fig. 9 The degradation of accuracies with increasing σPS = σBS compared between two GridNets one with ordered and another with randomized (but fixed) singular values.
Fig. 10
Fig. 10 The saturable absorption response curve compared to the corresponding Softplus approximation with various values of T.
Fig. 11
Fig. 11 The degradation of ONN outputs visualized through confusion matrices. Each confusion matrix shows how often each target class (row) is predicted as each of the ten possible classs (column). Both networks, GridNet (a, b, c) and FFTNet (d, e, f) are evaluated. First in the ideal case (a, d) then, with increasing errors (b, e and c ,f). Note the logarithmic scaling.
Fig. 12
Fig. 12 The effects of quantization is shown for both GridNet and FFTNet. 10 instances of GridNet (blue) and FFTNet (red) were trained then quantized to varying levels. The mean classification accuracy at each level is shown by bar plots. The 20-80%quantiles are shown with error bars. The dotted horizontal line denotes the full precision accuracy.
Fig. 13
Fig. 13 The central MZIs of GridNet has lower variance in internal phase shifts (θ). a) The spatial distribution of internal phase shift (θd,l) of MZIs in U2 of GridNet. Reference Fig. 1(a) for coordinates and Fig. 2 for location of U2 in context of network architecture. b) Histogram of phase shifts near the center (red), edge (green), and corner (blue) of the GridUnitary multiplier. These phases are obtained from multiple instances of trained GridNets with random initialization.
Fig. 14
Fig. 14 The variance of internal phase shifts of FFTNet is uniform spatially (a) Spatial distribution of phase shifts for a FFTUnitary multiplier. The MZIs are ordered as shown in Fig. 1(b). (b) Histogram of phase shifts of FFTUnitary near the center (red) and top (green). These phases are obtained from mulitple trained FFTNets with random initialization.
Fig. 15
Fig. 15 a) A schematic of BlockFFTUnitary. Blocks of MZIs in dashed, blue boxes are similar to GridNet. The crossing waveguide, similar to those in FFTNet are between the blocks. b) The distribution of phases after being trained. The dashed white lines denote the locations of the crossing waveguides.
Fig. 16
Fig. 16 No improvement in robustness to imprecision is seen with BlockFFTNet over GridNet. In fact, there is a significant decrease.

Equations (39)

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U M Z ( θ , ϕ ) = U B S U P S ( θ ) U B S U P S ( ϕ ) = i e i θ / 2 ( e i ϕ sin  θ 2 cos  θ 2 e i ϕ cos  θ 2 sin  θ 2 ) .
M = β U Σ V .
U Σ V = ( U Π 1 ) ( ΠΣΠ 1 ) ( Π V ) .
F ( U 0 , U ) = | Tr ( U U 0 ) N | 2 .
U B S ( r ) = ( r i t i t r )
U P S ( θ ) = ( e i θ 0 0 1 ) .
U M Z I ( θ , ϕ ; r , r ) = U B S ( r ) U P S ( θ ) U B S ( r ) U P S ( ϕ )
= ( r i t i t r ) ( e i θ 0 0 1 ) ( r i t i t r ) ( e i ϕ 0 0 1 )
= ( e i ϕ ( e i θ r r t t ) i ( t ρ + e i θ r t ) i e i ϕ ( e i θ t r + r t ) r ρ e i θ t t )
U B S U B S ( π / 2 ) = 1 2 ( 1 i i 1 )
U M Z I ( θ , ϕ ) = i e i θ / 2 ( e i ϕ sin  θ 2 cos  θ 2 e i ϕ cos  θ 2 sin  θ 2 )
T = | cos  θ / 2 | 2  and  R = | sin  θ / 2 | 2
H = 1 2 ( 1 1 1 1 ) .
U B S = ( 1 0 0 i ) H ( 1 0 0 i ) = U P S ( π / 2 ) H U P S ( π / 2 )
U M Z ( θ , ϕ ) = U P S ( π / 2 ) H U P S ( θ π ) H U P S ( ϕ π / 2 ) .
σ ϕ ( τ ) 2 = 2 π δ f τ .
σ ϕ 2 = 10 4 = 2 π δ f τ
6 3 × 10 13 s δ f
δ f 5 × 10 7 Hz
= 50 MHz .
u 0 = 1 2 log  ( T / T 0 ) 1 T
u = 1 2 W ( 2 T 0 u 0 e 2 u 0 ) f ( u 0 )
σ ( u ) = β 1 log  ( 1 + e β ( u u 0 ) 1 + e β u 0 ) .
σ ( 0 ) = e β u 0 1 + e β u 0
= ( 1 + e β u 0 ) 1 .
u 0 = β 1 log  ( T 0 1 1 ) .
σ ( u ) ( u u 0 ) β 1 log  ( 1 + e β u 0 ) .
u 0 + β 1 log  ( 1 + e β u 0 ) = 1 2 log  T 0
β u 0 + log  ( 1 + 1 T 0 1 1 ) = 1 2 β log  T 0
log  ( T 0 ) = 1 2 β log  T 0
β = 2 .
u 0 = 1 2 log  ( T 0 1 1 ) .
θ V 2 θ = 2 π ( V V 2 π ) 2 2 π u 2 θ 2 π = u
r d , l Beta ( 1 , β d , l ) .
β N 2 max  ( | d N / 2 | , | l N / 2 | )
= N 2 | | ( d , l ) ( N / 2 , N / 2 ) | | .
X k = 1 N m = 0 N 1 x n e 2 π i N n k .
X k = 1 2 ( E k + e 2 π i N k O k )
X k + N / 2 = 1 2 ( E k e 2 π i N k O k ) .

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