LECTURE #2 C: ompel x Gradients Vectors · 3 . Lecture #2: Complex Gradient Vectors. EE210B:...

Proc. IEEE, vol. 102, no. 4, pp. 460-497, April 2014. Foundations and Trends in Machine Learning, vol. 7, no. 4-5, pp. 311-801, July 2014.

Course EE210B Spring Quarter 2015

1

Professor Ali H. Sayed UCLA Electrical Engineering

INFERENCE OVER NETWORKS

LECTURE #2: Complex Gradients Vectors

Proc. IEEE, vol. 102, no. 4, pp. 460-497, April 2014. Foundations and Trends in Machine Learning, vol. 7, no. 4-5, pp. 311-801, July 2014.

Course EE210B Spring Quarter 2015

Part I: Background Material

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 3

Reference

Appendix A (Complex Gradient Vectors, pp. 712-719):

A. H. Sayed, ``Adaptation, learning, and optimization over networks,'' Foundations and Trends in Machine Learning, vol. 7, issue 4-5, pp. 311-801, NOW Publishers, 2014.

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 4

Setting

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Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 5

Cauchy-Riemann Conditions

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 6

Cauchy-Riemann Conditions

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 7

Cauchy-Riemann Conditions

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 8

Cauchy-Riemann Conditions

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 9

Cauchy-Riemann Conditions

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 10

Example #A.1 10 10

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 11

Example #A.1 11 11

Proc. IEEE, vol. 102, no. 4, pp. 460-497, April 2014. Foundations and Trends in Machine Learning, vol. 7, no. 4-5, pp. 311-801, July 2014.

Course EE210B Spring Quarter 2015

Scalar Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 13

Scalar Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 14

Scalar Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 15

Scalar Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 16

Scalar Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 17

Scalar Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 18

Example #A.2 18 18

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 19

Example #A.2 19 19

Proc. IEEE, vol. 102, no. 4, pp. 460-497, April 2014. Foundations and Trends in Machine Learning, vol. 7, no. 4-5, pp. 311-801, July 2014.

Course EE210B Spring Quarter 2015

Vector Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 21

Vector Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 22

Vector Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 23

Vector Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 24

Vector Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 25

Vector Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 26

Example #A.3 26 26

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 27

Example #A.3 27 27

Proc. IEEE, vol. 102, no. 4, pp. 460-497, April 2014. Foundations and Trends in Machine Learning, vol. 7, no. 4-5, pp. 311-801, July 2014.

Course EE210B Spring Quarter 2015

Real Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 29

Real Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 30

Real Arguments

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 31

Example #A.4 31 31

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 32

Example #A.4 32 32

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 33

Example #A.4 33 33

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 34

Example #A.4 34 34

Lecture #2: Complex Gradient Vectors EE210B: Inference over Networks (A. H. Sayed) 35

Example #A.5 35 35

Proc. IEEE, vol. 102, no. 4, pp. 460-497, April 2014. Foundations and Trends in Machine Learning, vol. 7, no. 4-5, pp. 311-801, July 2014.

Course EE210B Spring Quarter 2015

End of Lecture