Abstract

This paper presents a deep-learning (DL) based approach to the design of multi-colored visible light communication (VLC) systems where RGB light-emitting diode (LED) lamps accomplish multi-dimensional color modulation under color and illuminance requirements. It is aimed to identify a pair of multi-color modulation transmitter and receiver leading to efficient symbol recovery performance. To this end, an autoencoder (AE), an unsupervised deep learning technique, is adopted to train the end-to-end symbol recovery process that includes the VLC transceiver pair and a channel layer characterizing the optical channel along with additional LED intensity control features. As a result, the VLC transmitter and receiver are jointly designed and optimized. Intensive numerical results demonstrate that the learned VLC system outperforms existing techniques in terms of the average symbol error probability. This framework sheds light on the viability of DL techniques in the optical communication system design.

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

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References

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    [Crossref]
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    [Crossref]
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2018 (1)

S. Dorner, S. Cammer, J. Hoydis, and S. Brink, “Deep learning based communication over the air,” IEEE J. Sel. Topics Signal Process. 12(1), 132–143 (2018).
[Crossref]

2017 (4)

J. Dong, Y. Zhang, and Y. Zhu, “Convex relaxation for illumination control of multi-color multiple-input-multiple-output visible light communications with linear minimum mean square error detection,” Appl. Opt. 56(23), 6587–6595 (2017).
[Crossref] [PubMed]

Q. Gao, C. Gong, and Z. Xu, “Joint transceiver and offset design for visible light communications with input-dependent shot noise,” IEEE Trans. Wireless Commun. 16(5), 2736–2747 (2017).
[Crossref]

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

T. O’Shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Trans. Cogn. Commun. Netw. 3(4), 563–575 (2017).
[Crossref]

2015 (2)

S. H. Lee, S.-Y. Jung, and J. K. Kwon, “Modulation and coding for dimmable visible light communication,” IEEE Commun. Mag. 53(2), 136–143 (2015).
[Crossref]

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

2014 (2)

R. Singh, T. O’Farrell, and J. P. R. David, “An enhanced color shift keying modulation scheme for high-speed wireless visible light communications,” J. Lightw. Technol. 32(14), 2582–2592 (2014).
[Crossref]

E. Monteiro and S. Hranilovic, “Design and implementation of color-shift keying for visible light communications,” J. Lightw. Technol. 32(10), 2053–2060 (2014).
[Crossref]

2012 (2)

P. Das, B.-Y. Kim, Y. Park, and K.-D. Kim, “Color-independent VLC based on a color space without sending target color information,” Optics Communications 286(1), 69–73 (2012)
[Crossref]

K.-I. Ahn and J. K. Kwon, “Color intensity modulation for multicolored visible light communications,” IEEE Photon. Technol. Lett. 24(24), 2254–2257 (2012).
[Crossref]

2010 (1)

K. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun. 9(12), 3671–3681 (2010).
[Crossref]

2009 (1)

H. Sung, S.-R. Lee, and I. Lee, “Generalized channel inversion methods for multiuser MIMO systems,” IEEE Trans. Commun. 57(11), 3489–3499 (2009).
[Crossref]

2006 (1)

H. Lee, B. Lee, and I. Lee, “Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems,” IEEE J. Sel. Areas Commun. 24(3), 504–513 (2006).
[Crossref]

2004 (1)

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

Ahn, K.-I.

K.-I. Ahn and J. K. Kwon, “Color intensity modulation for multicolored visible light communications,” IEEE Photon. Technol. Lett. 24(24), 2254–2257 (2012).
[Crossref]

Bae, J. S.

Z. Chen, J. S. Bae, S.-K. Chung, J.-W. Koh, and S. G. Kang, “Multi-envelope 3-dimensional constellations for polarization shift keying modulation,” in Proceedings of Information and Communication Technology Convergence (KICS, 2010), pp. 173–174.

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in Proceedings of International Conference on Artificial Intelligence and Statistics (PMLR, 2010), pp. 249–256.

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

Brink, S.

S. Dorner, S. Cammer, J. Hoydis, and S. Brink, “Deep learning based communication over the air,” IEEE J. Sel. Topics Signal Process. 12(1), 132–143 (2018).
[Crossref]

Cammer, S.

S. Dorner, S. Cammer, J. Hoydis, and S. Brink, “Deep learning based communication over the air,” IEEE J. Sel. Topics Signal Process. 12(1), 132–143 (2018).
[Crossref]

Chen, Y.

M. Li, T. Zhang, Y. Chen, and A. J. Smola, “Efficient mini-batch training for stochastic optimization,” in Proceedings of International Conference on Knowledge Discovery and Data mining (ACM, 2014), pp. 661–670.

Chen, Z.

Z. Chen, J. S. Bae, S.-K. Chung, J.-W. Koh, and S. G. Kang, “Multi-envelope 3-dimensional constellations for polarization shift keying modulation,” in Proceedings of Information and Communication Technology Convergence (KICS, 2010), pp. 173–174.

Chung, S.-K.

Z. Chen, J. S. Bae, S.-K. Chung, J.-W. Koh, and S. G. Kang, “Multi-envelope 3-dimensional constellations for polarization shift keying modulation,” in Proceedings of Information and Communication Technology Convergence (KICS, 2010), pp. 173–174.

Das, P.

P. Das, B.-Y. Kim, Y. Park, and K.-D. Kim, “Color-independent VLC based on a color space without sending target color information,” Optics Communications 286(1), 69–73 (2012)
[Crossref]

David, J. P. R.

R. Singh, T. O’Farrell, and J. P. R. David, “An enhanced color shift keying modulation scheme for high-speed wireless visible light communications,” J. Lightw. Technol. 32(14), 2582–2592 (2014).
[Crossref]

Ding, Z.

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

Dong, J.

Dorner, S.

S. Dorner, S. Cammer, J. Hoydis, and S. Brink, “Deep learning based communication over the air,” IEEE J. Sel. Topics Signal Process. 12(1), 132–143 (2018).
[Crossref]

Drost, R. J.

R. J. Drost and B. M. Sadler, “Constellation design for color-shift keying using billiards algorithms,” in Proceedings of GLOBECOM Workshops (IEEE, 2010).

Gao, Q.

Q. Gao, C. Gong, and Z. Xu, “Joint transceiver and offset design for visible light communications with input-dependent shot noise,” IEEE Trans. Wireless Commun. 16(5), 2736–2747 (2017).
[Crossref]

Glorot, X.

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in Proceedings of International Conference on Artificial Intelligence and Statistics (PMLR, 2010), pp. 249–256.

Gong, C.

Q. Gao, C. Gong, and Z. Xu, “Joint transceiver and offset design for visible light communications with input-dependent shot noise,” IEEE Trans. Wireless Commun. 16(5), 2736–2747 (2017).
[Crossref]

Hinton, G.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

V. Nair and G. Hinton, “Rectified linear units improve restricted boltzmann machines,” in Proceedings of International Conference on Machine Learning (IMLS, 2010), pp. 807–814.

Hoydis, J.

S. Dorner, S. Cammer, J. Hoydis, and S. Brink, “Deep learning based communication over the air,” IEEE J. Sel. Topics Signal Process. 12(1), 132–143 (2018).
[Crossref]

T. O’Shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Trans. Cogn. Commun. Netw. 3(4), 563–575 (2017).
[Crossref]

Hranilovic, S.

E. Monteiro and S. Hranilovic, “Design and implementation of color-shift keying for visible light communications,” J. Lightw. Technol. 32(10), 2053–2060 (2014).
[Crossref]

Jiang, M.

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

Jung, S.-Y.

S. H. Lee, S.-Y. Jung, and J. K. Kwon, “Modulation and coding for dimmable visible light communication,” IEEE Commun. Mag. 53(2), 136–143 (2015).
[Crossref]

Kang, S. G.

Z. Chen, J. S. Bae, S.-K. Chung, J.-W. Koh, and S. G. Kang, “Multi-envelope 3-dimensional constellations for polarization shift keying modulation,” in Proceedings of Information and Communication Technology Convergence (KICS, 2010), pp. 173–174.

Kim, B.-Y.

P. Das, B.-Y. Kim, Y. Park, and K.-D. Kim, “Color-independent VLC based on a color space without sending target color information,” Optics Communications 286(1), 69–73 (2012)
[Crossref]

Kim, K.-D.

P. Das, B.-Y. Kim, Y. Park, and K.-D. Kim, “Color-independent VLC based on a color space without sending target color information,” Optics Communications 286(1), 69–73 (2012)
[Crossref]

Koh, J.-W.

Z. Chen, J. S. Bae, S.-K. Chung, J.-W. Koh, and S. G. Kang, “Multi-envelope 3-dimensional constellations for polarization shift keying modulation,” in Proceedings of Information and Communication Technology Convergence (KICS, 2010), pp. 173–174.

Komine, T.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

Kwon, J. K.

S. H. Lee, S.-Y. Jung, and J. K. Kwon, “Modulation and coding for dimmable visible light communication,” IEEE Commun. Mag. 53(2), 136–143 (2015).
[Crossref]

K.-I. Ahn and J. K. Kwon, “Color intensity modulation for multicolored visible light communications,” IEEE Photon. Technol. Lett. 24(24), 2254–2257 (2012).
[Crossref]

LeCun, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

Lee, B.

H. Lee, B. Lee, and I. Lee, “Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems,” IEEE J. Sel. Areas Commun. 24(3), 504–513 (2006).
[Crossref]

Lee, H.

H. Lee, B. Lee, and I. Lee, “Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems,” IEEE J. Sel. Areas Commun. 24(3), 504–513 (2006).
[Crossref]

Lee, I.

K. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun. 9(12), 3671–3681 (2010).
[Crossref]

H. Sung, S.-R. Lee, and I. Lee, “Generalized channel inversion methods for multiuser MIMO systems,” IEEE Trans. Commun. 57(11), 3489–3499 (2009).
[Crossref]

H. Lee, B. Lee, and I. Lee, “Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems,” IEEE J. Sel. Areas Commun. 24(3), 504–513 (2006).
[Crossref]

Lee, K.

K. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun. 9(12), 3671–3681 (2010).
[Crossref]

Lee, S. H.

S. H. Lee, S.-Y. Jung, and J. K. Kwon, “Modulation and coding for dimmable visible light communication,” IEEE Commun. Mag. 53(2), 136–143 (2015).
[Crossref]

Lee, S.-R.

H. Sung, S.-R. Lee, and I. Lee, “Generalized channel inversion methods for multiuser MIMO systems,” IEEE Trans. Commun. 57(11), 3489–3499 (2009).
[Crossref]

Li, M.

M. Li, T. Zhang, Y. Chen, and A. J. Smola, “Efficient mini-batch training for stochastic optimization,” in Proceedings of International Conference on Knowledge Discovery and Data mining (ACM, 2014), pp. 661–670.

Liang, X.

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

Monteiro, E.

E. Monteiro and S. Hranilovic, “Design and implementation of color-shift keying for visible light communications,” J. Lightw. Technol. 32(10), 2053–2060 (2014).
[Crossref]

Nair, V.

V. Nair and G. Hinton, “Rectified linear units improve restricted boltzmann machines,” in Proceedings of International Conference on Machine Learning (IMLS, 2010), pp. 807–814.

Nakagawa, M.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

O’Farrell, T.

R. Singh, T. O’Farrell, and J. P. R. David, “An enhanced color shift keying modulation scheme for high-speed wireless visible light communications,” J. Lightw. Technol. 32(14), 2582–2592 (2014).
[Crossref]

O’Shea, T.

T. O’Shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Trans. Cogn. Commun. Netw. 3(4), 563–575 (2017).
[Crossref]

Park, E.

K. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun. 9(12), 3671–3681 (2010).
[Crossref]

Park, Y.

P. Das, B.-Y. Kim, Y. Park, and K.-D. Kim, “Color-independent VLC based on a color space without sending target color information,” Optics Communications 286(1), 69–73 (2012)
[Crossref]

Sadler, B. M.

R. J. Drost and B. M. Sadler, “Constellation design for color-shift keying using billiards algorithms,” in Proceedings of GLOBECOM Workshops (IEEE, 2010).

Singh, R.

R. Singh, T. O’Farrell, and J. P. R. David, “An enhanced color shift keying modulation scheme for high-speed wireless visible light communications,” J. Lightw. Technol. 32(14), 2582–2592 (2014).
[Crossref]

Smola, A. J.

M. Li, T. Zhang, Y. Chen, and A. J. Smola, “Efficient mini-batch training for stochastic optimization,” in Proceedings of International Conference on Knowledge Discovery and Data mining (ACM, 2014), pp. 661–670.

Sung, H.

K. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun. 9(12), 3671–3681 (2010).
[Crossref]

H. Sung, S.-R. Lee, and I. Lee, “Generalized channel inversion methods for multiuser MIMO systems,” IEEE Trans. Commun. 57(11), 3489–3499 (2009).
[Crossref]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

Wang, J.

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

Xu, Z.

Q. Gao, C. Gong, and Z. Xu, “Joint transceiver and offset design for visible light communications with input-dependent shot noise,” IEEE Trans. Wireless Commun. 16(5), 2736–2747 (2017).
[Crossref]

Yuan, M.

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

Zhang, T.

M. Li, T. Zhang, Y. Chen, and A. J. Smola, “Efficient mini-batch training for stochastic optimization,” in Proceedings of International Conference on Knowledge Discovery and Data mining (ACM, 2014), pp. 661–670.

Zhang, Y.

Zhao, C.

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

Zhu, Y.

Appl. Opt. (1)

IEEE Commun. Mag. (1)

S. H. Lee, S.-Y. Jung, and J. K. Kwon, “Modulation and coding for dimmable visible light communication,” IEEE Commun. Mag. 53(2), 136–143 (2015).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

H. Lee, B. Lee, and I. Lee, “Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems,” IEEE J. Sel. Areas Commun. 24(3), 504–513 (2006).
[Crossref]

IEEE J. Sel. Topics Signal Process. (1)

S. Dorner, S. Cammer, J. Hoydis, and S. Brink, “Deep learning based communication over the air,” IEEE J. Sel. Topics Signal Process. 12(1), 132–143 (2018).
[Crossref]

IEEE Photon. Technol. Lett. (1)

K.-I. Ahn and J. K. Kwon, “Color intensity modulation for multicolored visible light communications,” IEEE Photon. Technol. Lett. 24(24), 2254–2257 (2012).
[Crossref]

IEEE Trans. Cogn. Commun. Netw. (1)

T. O’Shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Trans. Cogn. Commun. Netw. 3(4), 563–575 (2017).
[Crossref]

IEEE Trans. Commun. (1)

H. Sung, S.-R. Lee, and I. Lee, “Generalized channel inversion methods for multiuser MIMO systems,” IEEE Trans. Commun. 57(11), 3489–3499 (2009).
[Crossref]

IEEE Trans. Consum. Electron. (1)

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

IEEE Trans. Wireless Commun. (2)

K. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun. 9(12), 3671–3681 (2010).
[Crossref]

Q. Gao, C. Gong, and Z. Xu, “Joint transceiver and offset design for visible light communications with input-dependent shot noise,” IEEE Trans. Wireless Commun. 16(5), 2736–2747 (2017).
[Crossref]

J. Lightw. Tech. (1)

X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying modulation of quadrichromatic LEDs in visible light communications,” J. Lightw. Tech. 35(17), 3650–3663 (2017).
[Crossref]

J. Lightw. Technol. (2)

R. Singh, T. O’Farrell, and J. P. R. David, “An enhanced color shift keying modulation scheme for high-speed wireless visible light communications,” J. Lightw. Technol. 32(14), 2582–2592 (2014).
[Crossref]

E. Monteiro and S. Hranilovic, “Design and implementation of color-shift keying for visible light communications,” J. Lightw. Technol. 32(10), 2053–2060 (2014).
[Crossref]

Nature (1)

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

Optics Communications (1)

P. Das, B.-Y. Kim, Y. Park, and K.-D. Kim, “Color-independent VLC based on a color space without sending target color information,” Optics Communications 286(1), 69–73 (2012)
[Crossref]

Other (9)

R. J. Drost and B. M. Sadler, “Constellation design for color-shift keying using billiards algorithms,” in Proceedings of GLOBECOM Workshops (IEEE, 2010).

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

S. Boyd and J. Dattorro, “Alternating projections,” http://www.stanford.edu/class/ee392o/alt_proj.pdf (2013).

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Figures (14)

Fig. 1
Fig. 1 Schematic diagrams for AE-based learning networks and multi-color VLC.
Fig. 2
Fig. 2 Proposed AE structure for dimmable VLC.
Fig. 3
Fig. 3 Average SER performance as a function of SNR with M = 8 and ζ = 0.
Fig. 4
Fig. 4 Learned constellation points by the AE with M = 8 and ζ = 0.
Fig. 5
Fig. 5 Euclidean distance between each constellation point and dimming constraint with M = 8 and ζ = 0.
Fig. 6
Fig. 6 MED gain over the baseline scheme with M = 8 and ζ = 0.
Fig. 7
Fig. 7 Average SER performance as a function of SNR with M = 8 and d = [0.8, 0.6, 0.5]T.
Fig. 8
Fig. 8 Transmitted signal of the AE based VLC with M = 8 and d = [0.8, 0.6, 0.5]T.
Fig. 9
Fig. 9 Noiseless received signal of the AE based VLC with M = 8 and d = [0.8, 0.6, 0.5]T.
Fig. 10
Fig. 10 Average SER performance as a function of ζ with M = 8 and SNR = 20 dB.
Fig. 11
Fig. 11 MED gain over the baseline scheme with M = 8 and d1 = 0.3.
Fig. 12
Fig. 12 Average SER performance as a function of SNR with M = 16 and ζ = 0.
Fig. 13
Fig. 13 Learned constellation points by the AE with M = 16 and ζ = 0.
Fig. 14
Fig. 14 Performance comparison with the constellation design method [6].

Tables (1)

Tables Icon

Table 1 Alternating projection algorithm for (P1).

Equations (21)

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h ( i ) = ϕ 1 ( W 1 x ( i ) + b 1 ) , for i = 1 , , m ,
x ^ ( i ) = ϕ 2 ( W 2 h ( i ) + b 2 ) , for i = 1 , , m ,
J ( { W l , b l } ) 1 m i = 1 m J i ( { W l , b l } ) .
W l W l η J ˜ T ( { W l , b l } ) W l and b l b l η J ˜ T ( { W l , b l } ) b l , for l = 1 , 2 ,
y = Hs + n ,
H = [ 1 ζ 2 ζ 0 0 0 ζ 1 2 ζ 2 ζ 0 0 0 2 ζ 1 2 ζ 0 0 0 0 0 1 2 ζ ζ 0 0 0 2 ζ 1 ζ ]
0 [ s ] j [ a ] j , for j = 1 , N ,
E S [ s ] = 1 M b = 1 M s b = d ,
h 1 ( i ) = max { 0 , W 1 e b ( i ) + b 1 } ,
h 2 ( i ) = W 2 h 1 ( i ) + b 2 ,
( P 1 ) : min s ( 1 ) , , s ( m ) 1 2 i = 1 m s ( i ) h 2 ( i ) 2 subject to 0 [ s ( i ) ] j [ a ] j , for j = 1 , , N and i , , m ,
i = 1 m s ( i ) = m d .
( P 2 ) : min s ( 1 ) , , s ( m ) 1 2 i = 1 m s ( i ) h 2 ( i ) 2 subject to 0 [ s ( i ) ] j [ a ] j , for j = 1 , , N and i , , m ,
( P 3 ) : min s ( 1 ) , , s ( m ) 1 2 i = 1 m s ( i ) h 2 ( i ) 2 subject to i = 1 m s ( i ) = m d .
[ s ( i ) ] j = min { max { 0 , [ h 2 ( i ) ] j } , [ a ] j } for j = 1 , , N and i = 1 , , m .
( s ( 1 ) , , s ( m ) , ν ) = 1 2 i = 1 m s ( i ) h 2 ( i ) 2 ν T ( i = 1 m s ( i ) m d ) ,
s ( i ) = d + h 2 ( i ) 1 m k = 1 m h 2 ( k ) , for i = 1 , , m .
[ S ] j i = min { max { 0 , [ H 2 ] j i } , [ a ] j } for j = 1 , , N and i = 1 , , m , S = d 1 m T + H 2 ( I m 1 m 1 m 1 m T ) ,
h 3 ( i ) = max { 0 , W 3 y ( i ) + b 3 } ,
[ p ( i ) ] j = e [ z ( i ) ] j k = 1 M e [ z ( i ) ] k , for j = 1 , , M ,
J ( { W l , b l } ) = 1 m i = 1 m log ( [ p ( i ) ] b ( i ) ) + λ l = 1 4 W l 2 2 ,

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