Abstract

Recently, integrated optics has gained interest as a hardware platform for implementing machine learning algorithms. Of particular interest are artificial neural networks, since matrix-vector multiplications, which are used heavily in artificial neural networks, can be done efficiently in photonic circuits. The training of an artificial neural network is a crucial step in its application. However, currently on the integrated photonics platform there is no efficient protocol for the training of these networks. In this work, we introduce a method that enables highly efficient, in situ training of a photonic neural network. We use adjoint variable methods to derive the photonic analogue of the backpropagation algorithm, which is the standard method for computing gradients of conventional neural networks. We further show how these gradients may be obtained exactly by performing intensity measurements within the device. As an application, we demonstrate the training of a numerically simulated photonic artificial neural network. Beyond the training of photonic machine learning implementations, our method may also be of broad interest to experimental sensitivity analysis of photonic systems and the optimization of reconfigurable optics platforms.

© 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)

J. M. Shainline, S. M. Buckley, R. P. Mirin, and S. W. Nam, “Superconducting optoelectronic circuits for neuromorphic computing,” Phys. Rev. Appl. 7, 1–27 (2017).
[Crossref]

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]

T. Hughes, G. Veronis, K. P. Wootton, R. J. England, and S. Fan, “Method for computationally efficient design of dielectric laser accelerator structures,” Opt. Express 25, 15414–15427 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light-automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

D. A. B. Miller, “Setting up meshes of interferometers; reversed local light interference method,” Opt. Express 25, 29233–29248 (2017).
[Crossref]

2016 (2)

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walsmley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
[Crossref]

A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using a neural network with dynamic external memory,” Nature 538, 471–476 (2016).
[Crossref]

2015 (5)

M. Hermans, M. Burm, T. Van Vaerenbergh, J. Dambre, and P. Bienstman, “Trainable hardware for dynamical computing using error backpropagation through physical media,” Nat. Commun. 6, 6729 (2015).
[Crossref]

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

M. Prezioso, F. Merrikh-Bayat, B. D. Hoskins, G. C. Adam, K. K. Likharev, and D. B. Strukov, “Training and operation of an integrated neuromorphic network based on metal-oxide memristors,” Nature 521, 61–64 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

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

2014 (4)

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

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[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. Lightwave Technol. 32, 4029–4041 (2014).
[Crossref]

K. Vandoorne, P. Mechet, T. Van Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 3541 (2014).
[Crossref]

2013 (5)

F. Alibart, E. Zamanidoost, and D. B. Strukov, “Pattern classification by memristive crossbar circuits using ex situ and in situ training,” Nat. Commun. 4, 2072 (2013).
[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, 1364–1367 (2013).
[Crossref]

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

D. A. B. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
[Crossref]

D. A. B. Miller, “Self-configuring universal linear optical component,” Photon. Res. 1, 1–15 (2013).
[Crossref]

2012 (1)

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain maxwell’s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
[Crossref]

2009 (1)

2004 (1)

2002 (1)

N. Georgieva, S. Glavic, M. Bakr, and J. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microw. Theory Tech. 50, 2751–2758 (2002).
[Crossref]

1996 (1)

S. Jutamulia, “Overview of hybrid optical neural networks,” Science 28, 59–72 (1996).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

1988 (1)

1987 (2)

K. Wagner and D. Psaltis, “Multilayer optical learning networks,” Appl. Opt. 26, 5061–5076 (1987).
[Crossref]

Y. S. Abu-Mostafa and D. Pslatis, “Optical neural computers,” Sci. Am. 256, 88–95 (1987).
[Crossref]

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa and D. Pslatis, “Optical neural computers,” Sci. Am. 256, 88–95 (1987).
[Crossref]

Adam, G. C.

M. Prezioso, F. Merrikh-Bayat, B. D. Hoskins, G. C. Adam, K. K. Likharev, and D. B. Strukov, “Training and operation of an integrated neuromorphic network based on metal-oxide memristors,” Nature 521, 61–64 (2015).
[Crossref]

Agapiou, J.

A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using a neural network with dynamic external memory,” Nature 538, 471–476 (2016).
[Crossref]

Akopyan, F.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Alibart, F.

F. Alibart, E. Zamanidoost, and D. B. Strukov, “Pattern classification by memristive crossbar circuits using ex situ and in situ training,” Nat. Commun. 4, 2072 (2013).
[Crossref]

Alvarez-Icaza, R.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Amir, A.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Annoni, A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light-automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

Appuswamy, R.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Arthur, J. V.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Badia, A. P.

A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using a neural network with dynamic external memory,” Nature 538, 471–476 (2016).
[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]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Bakr, M.

N. Georgieva, S. Glavic, M. Bakr, and J. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microw. Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Bandler, J.

N. Georgieva, S. Glavic, M. Bakr, and J. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microw. Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Bengio, Y.

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

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bienstman, P.

M. Hermans, M. Burm, T. Van Vaerenbergh, J. Dambre, and P. Bienstman, “Trainable hardware for dynamical computing using error backpropagation through physical media,” Nat. Commun. 6, 6729 (2015).
[Crossref]

K. Vandoorne, P. Mechet, T. Van Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 3541 (2014).
[Crossref]

Black, D. S.

T. W. Hughes, S. Tan, Z. Zhao, N. V. Sapra, Y. J. Lee, K. J. Leedle, H. Deng, Y. Miao, D. S. Black, M. Qi, O. Solgaard, J. S. Harris, J. Vuckovic, R. L. Byer, and S. Fan, “On-chip laser power delivery system for dielectric laser accelerators,” Phys. Rev. Appl. (to be published).

Blunsom, P.

A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using a neural network with dynamic external memory,” Nature 538, 471–476 (2016).
[Crossref]

Brady, D.

Brezzo, B.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Brunner, D.

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, 1364–1367 (2013).
[Crossref]

Buckley, S. M.

J. M. Shainline, S. M. Buckley, R. P. Mirin, and S. W. Nam, “Superconducting optoelectronic circuits for neuromorphic computing,” Phys. Rev. Appl. 7, 1–27 (2017).
[Crossref]

Bunandar, D.

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Burm, M.

M. Hermans, M. Burm, T. Van Vaerenbergh, J. Dambre, and P. Bienstman, “Trainable hardware for dynamical computing using error backpropagation through physical media,” Nat. Commun. 6, 6729 (2015).
[Crossref]

Byer, R. L.

T. W. Hughes, S. Tan, Z. Zhao, N. V. Sapra, Y. J. Lee, K. J. Leedle, H. Deng, Y. Miao, D. S. Black, M. Qi, O. Solgaard, J. S. Harris, J. Vuckovic, R. L. Byer, and S. Fan, “On-chip laser power delivery system for dielectric laser accelerators,” Phys. Rev. Appl. (to be published).

Cain, A.

A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using a neural network with dynamic external memory,” Nature 538, 471–476 (2016).
[Crossref]

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A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light-automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
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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).
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Walsmley, I. A.

Watts, M. R.

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A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using a neural network with dynamic external memory,” Nature 538, 471–476 (2016).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic of the ANN architecture demonstrated in Ref. [11]. The boxed regions correspond to OIUs that perform a linear operation represented by the matrix W ^ l . Integrated phase shifters (blue) are used to control the OIU and train the network. The red regions correspond to nonlinear activations f l ( · ) . (b) Illustration of operation and gradient computation in an ANN. The top and bottom rows correspond to the forward and backward propagation steps, respectively. Propagation through a square cell corresponds to matrix multiplication. Propagation through a rounded region corresponds to activation. is element-wise vector multiplication.
Fig. 2.
Fig. 2. Schematic illustration of our proposed method for experimental measurement of gradient information. The box region represents the OIU. The colored ovals represent tunable phase shifters, and we illustrate computing the gradient with respect to the red and the yellow phase shifters, labeled 1 and 2, respectively. (a) We send the original set of amplitudes X l 1 and measure the constant intensity terms at each phase shifter. (b) We send the adjoint mode amplitudes, given by δ l , through the output side of our device, recording X TR * from the opposite side, as well as | e aj | 2 in each phase shifter. (c) We send in X l 1 + X TR , interfering e og and e aj * inside the device and recovering the gradient information for all phase shifters simultaneously.
Fig. 3.
Fig. 3. Numerical demonstration of the time-reversal procedure of Section 4. (a) Relative permittivity distribution for three MZIs arranged to perform a 3 × 3 linear operation. Blue boxes represent where phase shifters would be placed in this system. As an example, we compute the gradient information for a layer with X l 1 = [ 0 0 1 ] T and δ l = [ 0 1 0 ] T , corresponding to the bottom left and middle right ports, respectively. (b) Real part of the simulated electric field E z corresponding to injection from the bottom left port. (c) Real part of the adjoint E z , corresponding to injection from the middle right port. (d) Time-reversed adjoint field as constructed by our method, fed in through all three ports on the left. (e) Gradient information d L d ε l ( x , y ) as obtained directly by the adjoint method, normalized by its maximum absolute value. (f) Gradient information as obtained by the method introduced in this work, normalized by its maximum absolute value. Namely, the field pattern from (b) is interfered with the time-reversed adjoint field of (d), and the constant intensity terms are subtracted from the resulting intensity pattern. Panels (e) and (f) match with high precision.
Fig. 4.
Fig. 4. Numerical demonstration of a photonic ANN implementing an XOR gate using the backpropagation algorithm and adjoint method described in this work. (a) Architecture of the ANN. Two layers of 3 × 3 OIUs with z 2 activations. (b) Mean-squared error (MSE) between the predictions and targets as a function of training iterations. (c) Absolute value of the network predictions (blue circles) and targets (black crosses) before training. (d) Absolute value of the network predictions after training, showing that the network has successfully learned the XOR function.

Equations (26)

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W ^ X in = Z out ,
( X out Z out ) = ( 0 W ^ T W ^ 0 ) ( X in Z in ) .
X l = f l ( W ^ l X l 1 ) f l ( Z l ) .
L = 1 2 ( X L T ) ( X L T ) ,
d L d ε L = R { ( X L T ) d X L d ε L } ,
= R { ( Γ L f L ( Z L ) ) T d W ^ L d ε L X L 1 } ,
R { δ L T d W ^ L d ε L X L 1 } ,
Γ l = W ^ l + 1 T δ l + 1 ,
δ l = Γ l f l ( Z l ) ,
d L d ε l = R { δ l T d W ^ l d ε l X l 1 } .
[ ^ × ^ × k 0 2 ε ^ r ] e = i ω μ 0 j ,
A ^ ( ε r ) e = b .
X in , i = b i T e ,
X in P ^ in e .
Z out , i = b i + N T e ,
Z out P ^ out e ,
W ^ P ^ in e = P ^ out e .
d L d ε l = R { δ l T P ^ out A ^ 1 d A ^ d ε l A ^ 1 b x , l 1 } .
A ^ e og = b x , l 1 ,
A ^ e aj = P ^ out T δ ,
d L d ε l = R { e aj T d A ^ d ε l e og } .
d A ^ d ε l = k 0 2 r r ϕ δ ^ r , r ,
d L d ε l = k 0 2 R { r r ϕ e aj ( r ) e og ( r ) } .
I = | e og | 2 + | e aj | 2 + 2 R { e og e aj } ,
X TR * = W ^ l T δ l .
[ 0 0 ] T 0 , [ 0 1 ] T 1 , [ 1 0 ] T 1 , [ 1 1 ] T 0 .

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