Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic

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sigma-summation Underscript k equals 1 Overscript n Endscripts e 1 left-parenthesis normal x Subscript k Baseline right-parenthesis StartFraction partial-differential f Subscript upper F upper M Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis Over partial-differential sigma Subscript j Superscript negative 2 Baseline EndFraction minus alpha sigma-summation Underscript k equals 1 Overscript n Endscripts e 2 left-parenthesis normal x Subscript k Baseline right-parenthesis StartFraction partial-differential f Subscript upper F upper M Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis Over partial-differential sigma Subscript j Superscript negative 2 Baseline EndFraction"/>

      where

      (4.73)StartLayout 1st Row StartFraction partial-differential f Subscript upper F upper M Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis Over partial-differential sigma Subscript j Superscript negative 2 Baseline EndFraction equals minus sigma-summation Underscript i equals 1 Overscript c Superscript f Baseline Endscripts left-parenthesis x Subscript italic k j Baseline minus z Subscript italic i j Baseline right-parenthesis squared lamda Subscript i Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis theta Subscript i Baseline plus left-parenthesis sigma-summation Underscript i equals 1 Overscript c Superscript f Baseline Endscripts left-parenthesis x Subscript italic k j Baseline minus z Subscript italic i j Baseline right-parenthesis squared normal phi Subscript i Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis right-parenthesis left-parenthesis sigma-summation Underscript i equals 1 Overscript normal c prime Endscripts normal phi Subscript i Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis theta Subscript 0 i Baseline right-parenthesis 2nd Row minus left-parenthesis sigma-summation Underscript i equals 1 Overscript c prime Endscripts left-parenthesis x Subscript k normal j Baseline minus z Subscript italic i j Baseline right-parenthesis squared normal phi Subscript i Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis theta Subscript 0 i Baseline right-parenthesis left-parenthesis sigma-summation Underscript i equals 1 Overscript c prime Endscripts normal phi Subscript i Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis right-parenthesis 3rd Row normal lamda Subscript i Baseline left-parenthesis normal x Subscript k Baseline right-parenthesis equals product Underscript j equals 1 Overscript d Endscripts exp left-parenthesis minus left-parenthesis StartFraction x Subscript italic k j Baseline minus z Subscript italic iota j Baseline Over italic sigma j EndFraction right-parenthesis squared right-parenthesis period EndLayout

      and σj are updated as

      (4.74)sigma Subscript j Superscript negative 2 Baseline equals sigma Subscript j Superscript negative 2 Baseline minus eta StartFraction partial-differential upper E Over partial-differential sigma Subscript j Superscript negative 2 Baseline EndFraction

      where η is the constant step size.

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