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􏱑 ÿ>􏱙 􏱋 -1$UI*Î􏱓Ó 􏱠!ær$􏰀C 0!î􏱨A $?I!P6􏱠WC?=MK􏱨$􏱬G4$,Pv5 0!􏱡@Ð’􏱠􏱴Ó/C 0@Ð’􏱠􏱴ÓvK?=!ÒRÞ’A;K;$?I!Î􏱸Ó6􏱠!ær$􏰀51􏱡;$7C;$,V/=􏱬GÙ􏱡10/K􏱨$rÌ1􏱠?I!Î Í7C10)Ñe5􏱏$!􏱡􏱸$!Ò75@5.􏱠9Ò75􏰌$rÌ*􏱡 􏱠E57C 0,Î􏱓Ó?=Ù$1<.0#Q1Ar􏱕 E $EC!( Ï?=􏰀îD$;$MKÀ􏱋:K*􏱡10U=E5nÌ;$EC?=􏱬GD$7K;$!ÒE5@Ï!􏱡 0?I6􏱠 0EC􏱏$;ëRK?=R5l􏱡􏱏$?=􏱬G 􏱠WC Ï@$!ÒE5>-􏱏K*(>ÒRÞ􏱬A;5l􏱡 􏱠E5MK*􏱡 0􏱏Ì􏰌􏱠aKJ=􏱚􏱈􏱌$7C;$!ÒaÑ􏱬- + 􏱓%􏱕 E 􏱊􏱋􏱇%􏱋 F 􏱈􏱉􏰌 F 􏱓 K􏱨$!􏱡?=UI􏱓Ï!􏱡 0?I6􏱠 5l􏱡6􏱠W5eK9􏱡 0rÌ􏰍K?=􏰍5!(!Î75􏱬(10]ÍEC 04Ñe5r$!􏱡a$!ÒRÞ1An􏱕 + 􏱑 E 􏰌 F 􏱑􏱈􏱋􏱟5􏱬(!Î!􏱡?=􏱞C􏱏$!ÒE560􏱍$*Ò75R$rÌJ=aÑe5vK􏱬Ð’􏱠4Ñ?I6􏱠􏱘KJ=􏰍5􏱬(*Î!􏱡U= $*􏱡 0R$7C;$!ÒaÑa$7K 􏱠􏱏Ì􏱍$*Ò75􏱞C;$􏱬G?=7K*􏱡10)&1A*Ï!􏱡?=!􏱡EC6􏱠;$?IçÑ.0?I7KÙÐ’􏱠􏱴Ó􏱴ÒWÌ?=!ÒaÑ􏰀K􏱨$U=􏱏Ì 􏱠WC􏱏$!􏱡;$ Ï􏱏$!G􏱍$rÌ*(􏱬G􏱬0EC!Î@0MKJI6􏱠
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AO􏱌 􏱊 􏱇 􏱈 KW5􏱬(!Î!􏱡?=􏱼􏱡U=)K*􏱡 0?=756􏱠?I;$7CEC 0􏱏Ìaü.ýW􏱇 􏱉􏰊ÿk􏱓 $ <.0􏱸Ó;$7C􏱸0753⁄4G􏱨$MK*(􏰋$,P3􏱡6􏱠􏱏Ì 􏱕 􏱡10U=7K*􏱡 􏱠*Î􏱬ær$􏱚$1<􏱏$>0%”hL􏱬􏱡􏱏$*􏱡!(>Ò7C 􏱠􏰋H3$!ÒE5􏰌K 􏱠ç􏱡aÑ.0!􏱡>Í 0MKJI6􏱠À􏱋 KJ=MK􏱬Ð?I6􏱠*􏱡 􏱠􏱸5l􏱡;$!􏱡 0,Î􏱸Ó 0rÌRI6􏱠9ÎU=􏱏Ì*􏱡U=EC􏱘􏱠 A3Ó;$UI*P 0EC!Î􏱘C􏱏$!GEC 􏱠*Ò􏱴ÒWÌ*(ÙÓ]􏱠)KJ=􏱴ÒWÌ?=*ÒaÑ􏰌Ð?IMK9( 0􏱏$*􏱡6􏱠E5UI!(􏰌ÓU=7KoÒ75 09P)C 0ÝÖR$1< G9􏱡6􏱠/- 􏱡 C 0ÝÖÁ􏱡EC􏱏$􏱏Ì*􏱡 0􏱏Ì Ð>􏱡6􏱠Ù5􏱬(10!Ò75U=aÑ@C 0EC7C;$􏱴$!Ò75 K;$􏱏ë?=􏱸ÓU=*􏱡U=􏱸Ó􏰋5.􏱠*ÒE5 􏱡 Ö 0􏰌$*(UI 􏱠’<􏱓$!ÒE5DC10ÝÖ􏰋$ ÍWÌJ=EC7530)Ñe5􏰌$ <.0,P6􏱠􏱴$!ÒRÞ A $?I!PU=7KEK10,Î􏰀ÖU=􏱚G􏱨$75.􏱠JI􏱏$ECEC 0rÌ9􏱡!(􏰋$9P3⁄4GJI!( 0!Ò7KaK;$?I!P 􏱠?=EC6􏱠.<􏱍5􏱬(!Î*􏱡há4A*􏱗 A $􏱸Ó 􏱠7K]$!ÒE5a5􏱬( 0,P6􏱠􏰋$EC6􏱠4K;$􏱏Ì*􏱡6􏱠J=7C 􏱠.<.0rÌ􏱍C?=􏱏$!ÒE5a5.􏱠*ÒE5􏰌07KaK;$?I!P 􏱠?=EC6􏱠.<􏱍5􏱬(!Î*􏱡U=@$?I6􏱠􏱏Ì!?􏰊A E A*0EC􏱏$􏱏ë􏱸0E5 $7K10?IrÌD$,P􏱴GJI!( 0*ÒMK)5􏱏$MK􏰌Ï!􏱡U=*􏱡U= 􏱠ECE5􏱍$!ÒE5􏱞C􏱏$1<.0􏱍$UI*P6􏱠J=7C 􏱠'<45􏱬(!Î!􏱡?=􏱴ÒWÌ 􏱠􏱨$%Ö604$ Ï 􏱠EC;$ <'􏱠D$*ÒRÞRA1􏱇 􏱟􏱜C*􏱡􏱓 3 􏱞􏱈%􏱡􏰉􏱟􏱜 4􏱡30" 􏱑􏰌'&􏱞􏱋>􏱟􏱡3!􏰍)’ 􏱇

Mean Cancellation
Covariance Equalization
KL- Expansion
A􏱏KW560,ÎMK􏱑56􏱠9􏱎 GJ= 01<'􏱠􏱴0754K 􏱠􏱴07K 􏰍 􏱔 7􏱟􏱕 􏰍 􏱋0Ò*􏱡6􏱠E5x􏰌 􏱕 􏰍 􏱋􏱗􏰌􏱎A*Ï A $'-􏱼Ó7C;$75􏰀C6􏱠􏱨$!􏱡U=?IlIUI 􏱠􏱸Ó7K:􏱠ÙG!G1􏱠:0E5>I!(ÝÖ􏱬Î?I􏱏$!Ò􏰀KJ=􏰀5U=􏰀K;$􏱸ÓU=E5r$􏱸Ó 0􏱬?􏰊A*􏱗
A%G 􏱠;$E5eK9􏱡U=3⁄4G;$7K*(􏰋$,P3􏱡 􏱠􏱏ÌRKJI6􏱠J=􏱓Ó10!􏱡>ÐUI109Î®Ö60􏱓0?=75.􏱠WC􏱸􏱠oÐ>P􏱍5U=vÖ60􏱼􏱡10U=E5 L.􏱠􏱴Ó?=􏱬æ’07C!Î*Î6􏱠Á􏱡 V6􏱠􏲀- $1?􏱡 0rÌã􏱡􏱏$75×Ö 0Ù5l􏱡􏱏$1Ï!􏱡6􏱠W5􏱸Ì?=?I 0,PEC􏱏$,ÎlÐ>ÒÙK 􏱠zÒWÌ*(7K]K!GJ= 0􏱸Ó Ï?=7KoÌJ=7CE5r$􏱸Ó􏱸ÓlÐ􏱬?􏰊A1􏱇 9″&􏱞 􏱑40″􏱘􏱛
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d)
∆ω
􏱕EXE􏱋 5 F 􏱓􏱟 􏱟􏱕-􏱏􏱑􏱓 F 􏱓􏱋 7 􏱓􏱟 􏱟 􏰌 􏱓􏱟 􏱟 􏱕-􏱏􏱐􏱓􏱎􏱋&􏱝6F 􏱕-􏱏􏱑􏱓 􏱋&􏱝 􏱕􏱖􏱓 􏱋&􏱝
K􏱨$ Ð’􏱠􏰀Þ􏱚􏱠 􏱡?= 􏱝 Ï!􏱡?=􏱬G*􏱡 􏱠*Î􏱬ær$D5MKWC*â3ÐlP􏱼G􏱨$ P􏱚G􏱨$756􏱠􏱸Ó?=􏱬æ’0􏱏7C*Î!Î6􏱠Ù$,P􏱼􏱡6􏱠;Ì􏱝:56􏱠*Ò75􏱸Ï*􏱡U=􏱸Ó!(7K7KD􏱛A%􏱡10U=7K*􏱡􏱏$􏱸Ó?=􏱬GhLr􏱇􏱡U=@$7K􏱠;ÌÙ$!Ò75
C;$􏱬GJ=MK*􏱡10rÌ􏱚5MKEC*â􏱓K9(􏱚5r$%”Á􏱍􏱎(􏱇􏱊0$756􏱠ECåÏ!􏱡?=!􏱡7C􏱠;$?I#$!Ò75DÖ60􏱼$!(?I6􏱠.􏱡 0􏱏Ì􏱴ÐUI?I6􏱠*(E5l􏱡􏱏$ <;$􏲀IUI?=aÑa5􏱬(!P ./, 􏱉 􏱓 G*􏱡!(107C 􏱠D$756􏱠?I?I?=rÌEK10􏲀IUI?=aÑR51Ò1ÏU=;$)Ñ $*Ò75􏱚􏱡;$!Ò75 ( 􏱇 0 E G*􏱡6􏱠 ( 􏱇 0 􏱡􏱏$;$)Ñe5􏱏$9P)KJ= 0 ÖOá 􏱏>A􏱏K*Î;$75MK;$􏱸Ó?=75D$UI!Î?=75UI!(􏰌Ó􏱍$ Í 􏱠E5,IUI?=aÑa$rÌ9􏱡􏱏$ ÏEC􏱏$1<>􏱡 0rÌ G*􏱡 􏱠4C;$UI?I6􏱠􏱴Ó7K􏰋$9PcI?IU=aÑ:$􏱏ë?=7K*Î􏱏$E5eK􏰋$!ÒE5o􏱡;$!Ò75􏱞( 􏱇􏱊0 􏱡6􏱠*ÒE5􏰍C;$UI?I6􏱠􏱴Ó7KaK?= 0 ÖOá!An􏱕 Pe􏱕8=»􏱋 & $7C*( Ï?=vî@$􏱏$7K»􏱋 Î;$75MK􏱼$*􏱡 01⁄4Ð?I􏱏$7K?=􏱏Ì;$EC!Îã􏱡?=􏱞-c.7,􏱏 􏱉 􏱓 -􏱼Ó!(􏰌ÓU=*􏱡U=􏱸Ó􏱼$!Ò75å0E5􏰌5lÒ Ï?=􏱏$4Ñ􏱼$!Ò753⁄4$ <.0􏱸ÓaI?IU=aÑ􏱚56􏱠*Ò753⁄4$756􏱠EC Ï!􏱡?=!􏱡EC6􏱠􏱨$UI􏱚$*Ò75#Ï!􏱡?=􏱏$,P]K1􏱠v-W( 􏱇􏱈0 - $E5.􏱠WC#Ï!􏱡?=!􏱡7C 􏱠;$?I􏱘I6􏱠􏱴Ó?=75􏱬Î10o$!ÒE5􏱴$!􏱡*â􏱏$􏱬G 0E5ÁÐ7K 􏱠;$ÙK?=􏰌5¦=v-%􏱡 0?=MK L􏱬􏱡􏱏$􏱸Ó?=􏱬GD$!􏱡10o􏱡há>A􏱼Ó!(ÙÓ?=!􏱡?=􏱸Ó*I6􏱠;Ì􏱬0UIÙ􏱠vÖ60􏱸Ðe5U=!􏱡U=􏱏Ì?=><'􏱠*Ò􏱏$,P3Ï!􏱡U=*􏱡7C 􏱠;$?I􏱴$!Ò75:K􏱨$75.􏱠WC75MK*(?IUI?=3& $EC!(1ÏU=􏰀î/AcG􏱨$􏱏$,ÎMK Ï!􏱡?=!􏱡EC6􏱠􏱨$UI]$*Ò753G*􏱡 􏱠:$􏱏Ì*􏱡􏱏$1Ï7C;$ <>􏱡 0􏱏Ì􏱑KE5rÌ;$􏲀VP􏱠 0 Ö60]$!(?I6􏱠.<]$*Ò75 Ñ.0!Ò Ñ.0!􏱡>Í􏱚075@$1ÍU=?I􏱼G?I*( 04Ñ]$ ä

, ,J 􏱟+vKJ6􏱏IE
􏱕 * E 􏱋 F 􏱏 􏱏 􏰍 , 􏱡 F 􏱏 􏱏 9 􏱇 􏰌 􏱕􏱖􏱓􏱎􏱋&􏱝
􏱡10U=E5nÌ*􏱡!(ÝÖ4C107CEC􏱏$􏱼Ò75U=aÑ􏱴ÍEC 04Ñe5r$!􏱡aC 􏱠;$!􏱡?=UI C;$ Ð’􏱠JIo$?I Ï!􏱡?=MK 􏱠o$ <'􏱠*Òo$)Ñv-3Ó*Ò75U=7C10 Ï?I6􏱠]􏱕 ?􏱈Q/"À􏱋􏱈$7C 􏱠*(lÜ7Kã􏱡6􏱠;$􏱸ÓÙ5MK 􏱠􏱨$UI3⁄4$!Ò75z􏱡há 􏱗&%􏱚.􏱡􏱓 􏱑􏱒􏰍􏱝􏱏􏰊􏱍 A%􏱡 0?=75rÌ;$ECU=􏱬G9􏱡􏱏$ Ï?=􏱏$􏰌Ï!􏱡?=􏱬G*􏱡 0,Î7K;$EC7C10rÌ]$!ÒE5􏰌Ï*􏱡 0?I6􏱠 􏱝􏱗Ö 0RK7K􏱨$!􏱡!Î􏱏$;$75MK]$!Ò75 $EC!(7K 􏱠;$􏱸Ó􏱞K􏱨$!(?I6􏱠.<>􏱡􏱏$ Ï?=􏱏$D$!ÒRÞ A;K;$􏱬æ.􏱠4C 0!􏱡?=􏱸Ó􏱴G*􏱡 􏱠􏰍C 0e􏱙Q􏱠􏱸ÓD$!Ò75􏲀Ö 0􏱘K*􏱡10U=E5nÌ;$EC?=􏱬GD$!ÒE5􏱴􏱡?=)5l􏱡U=109Î 􏱇 Ö60/KEC1075rÌ;$ <>􏱡;$ Ï?=􏱏$R$!ÒRÞ.A>( $7C*( Ï?=vî􏱸􏱡?=]􏱡aÑ.0!Ò7K􏰀K1􏱠a$9Î 􏱠*ÒMK􏱓􏱡U=>I 􏱠’<.04$EC6􏱠􏱞5eKl0rÌ􏰍ÌJ=756􏱠EC!G 􏱠*(lÜ:􏱠®C10ÝÖ􏱿􏱝 5l􏱡6􏱠W5eK9􏱡0rÌ􏲀Ö60®K􏱨$!􏱡U=?IR$!Ò75'-;K*􏱡10U=7K*􏱡􏱏$􏱸Ó?=􏱬G@0)Ñe5􏰌􏱡há.A􏱴􏱝 Ö0)$EC!(E5.􏱠.<7C*(rÌa$!Ò759Ö604$7C*(MK1􏱠;$􏱸Ó􏱑􏱠􏱞KJ=􏰊􏱇 A;KE5lÒ Ï?=􏱏$4Ñ®Ö60aC􏱏$,PÙÓ*(l􏱡/I6􏱠E5 075@$!ÒE5􏱓075􏲀I6􏱠9(1Ü􏱨$ 􏱕 G*􏱡 􏱠*-1⁄4􏱕 􏱊􏱎􏱌􏱍􏱑􏱔􏱒 􏱊􏱋􏱇 ÒE5¦=aÑ 􏱕X( E 􏱋 􏱓 􏱈 , 􏱓 􏱈 􏱗 !,􏱘􏱇 􏱝DF 􏱈 ÐlP3􏱡􏱏$1􏱡 0rÌ -􏱨C􏱏$1<􏱏$4Ñ.0!Ò9- ( 􏱇 0 Ï!􏱡?=7K*(Á􏱡􏱏$*ÒaÑ#􏱡􏱏$ <;$ Ó!(ÙÓ?=!􏱡?=􏱓Ó􏱓$!ÒE5􏰋$756􏱠􏱏Ì􏱬0UI3⁄4075􏱍K*􏱡 0?=756􏱠EC;$75U=Ù$?I!ÎU=E5¦I*(ÙÓ]$1Í6􏱠E5 Ð'􏱠􏱸Ó􏱚5U=®- $7K 􏱠;Ì 􏱠ìÒWÌ*(7KT􏱡há􏰀A%􏱡 0?=756􏱠􏱴ÓU=􏱬æ'0EC!Î!Î6􏱠ì􏱡6􏱠zÐ?I!􏱡10􏱸K?=Ù􏱕 􏱗 E 􏱋 G*􏱡6􏱠ú07C;$􏱏ë1⁄4Ð?I􏱏$7K?=􏱏Ì;$EC!Î􏱴560!􏱡 $EC6􏱠:􏱕 􏱇 E 􏱋 􏱡 0?=75.􏱠9(1Ü􏱨$ 􏱡?=@K􏱴Ó7C;$75DC􏱏$􏱬GWC 0@C;$!Ò Ï?=!Ò􏱴$!ÒE5 􏱡;$!Ò75ÙÌ?=E5.􏱠WC􏱬G 􏱠9(1ÜÁÐ?I75rÌ 􏱠!ær$􏰋5 0!􏱡@K?= 􏱝1ÖOá 􏱕 & E 􏱋 5􏱔( 􏱏 􏱇􏱊0 E 􏰌 􏱏 + 􏱉 0 􏱕Ge􏱕U=Ý􏱋&'$7C!( L6Ï?=vî1⁄4$;$MKx07K?I 􏱠»􏱋 􏱏 KJ=ú$rÌ*􏱡;$ ÏEC􏱏$ 􏱡 0􏱏Ì3⁄45MK;$75MK 􏱠»Ö􏱴K;$ ?
􏱕”E􏱋 5./,􏱉􏱓F􏱏􏱐􏱓􏰌􏱓F􏱈􏱊􏱝F􏱈 􏱊 F 􏱓 􏱈 􏱊 􏱕 􏱏 􏱓 􏱋&􏱝 􏱈
􏱠?I*(ÙÓEC 0ÝÖ@$,Î 0?IMK3⁄4GEC 􏱠!G*􏱡6􏱠W5eK]$*Ò75􏱓Ï!􏱡?=MK*(3G􏱨$75􏱬(!Î􏱸Ó10rÌ􏱍KJ=oG9􏱡6􏱠]$!􏱡?=UI K?=!ÒE5􏲀Ö60R$9Î10UI7K􏱑$!ÒE5]Ð?I!Î􏱸ÓU=7K􏰍KJ= 􏱓6F 􏱈 􏱊 􏱝DF 􏱈 A 􏱏 􏱓 5lÒ Ï?=􏱏$4Ñ􏱞5l􏱡􏱏$EC7C!(􏱏Ì􏱑$!ÒE5􏱞5.􏱠 􏰋 􏱖 􏱍􏱍􏱔􏱎􏱉􏱕􏰉􏱠􏱉 G*􏱡6􏱠õÓ!(ÙÓ L¦=!􏱡?=􏱸Ó#$!Ò75􏱸56􏱠å0EC􏱏$;ë3$*(UI 􏱠’? A 􏱓&Ö60 􏱡10U=E5nÌ*􏱡!(ÝÖã􏱠#K1􏱠 􏱝􏱘Ö60#5l􏱡􏱏$?=􏱬G 􏱠WC Ï1⁄4$*Ò753KE5 0UI*Î Ò!Î 􏱠EC1Ï Ó 0E5M5 09P1⁄4$!ÒRÞ
An􏱕U=U=»􏱋 & $EC!( Ï?=􏰀î􏱚􏱡U=􏱸G􏱨$75.􏱠WC75MK*(?IUI?=􏱞KJ=􏰍5UI!(MK􏱨$7Ca$􏱸Ó 􏱠7K 􏱟)K?=!ÒE5􏱓􏱡U= 􏱠E5􏱬P10􏰋075]Ð’􏱠RÑ􏱞C􏱏$?=7K 􏱠;$􏱴􏱡 􏱠®K*Î6􏱠*ÒEC􏱏$®􏱠 􏱕􏱉E􏱋 5 􏱕F􏱓􏱟􏱑􏰌<(􏱏􏱇􏱊0 +􏱛 􏱕􏱏􏱓􏱋&􏱝F 􏱠ìÒWÌ 􏱠􏱨$7CT􏱡6􏱠;ÌÁ$)Ñ3⁄45.􏱠*ÒE5TG*􏱡*â 􏱟􏱼$4Ñ􏱘-n􏱕 􏱊 E 􏱋 􏱡 0?=75 􏱟6􏱠*(lÜ;$Á$E5.􏱠!G􏲀Î!(Á$*Ò75ìÒ75U=aÑ􏱴KJ=!Ò75åÏ!􏱡?=EC6􏱠*Î􏱸Ó 0)& Ö¦=􏱼Î􏱏$E5eK􏰋$!􏱡10􏱼􏱡U=Ó!(ÙÓ?=!􏱡?=􏱓Ó 􏱕􏱗E􏱋 5 􏱓􏱟 􏱕F􏱓􏱟􏱑F􏱏􏱓􏰌&.7,􏱉􏱓 􏱕-􏱏􏱑􏱓 􏱋 􏱝 + 􏱛 􏱕-􏱏􏱑􏱓 􏱋 􏱝DF 􏱟􏱉􏱟 􏱉 ÒE5¦=aÑ􏱓Ï!􏱡?= Ï!􏱡 􏱠EC L7C6􏱠􏱨$7CDC􏱏$75×Ö6􏱠:5×Ö􏱏$UI]$EC6􏱠􏰌$)Ñ9-􏱊􏰌􏱓􏱊􏱕.7,􏱓􏱎􏱋&􏱝5.􏱠*ÒE5􏱴Ï!􏱡?=7560*􏱡#G*􏱡6􏱠#.7,􏱓􏰌􏱓BÏ!􏱡?=7575r$>?

ω2
ν2
ωmin,2
Eigenvectors of H
ωmin,1 ω1
P E
x2
ν1
x1
􏱕􏱊􏱗#􏱋5-􏱕.7,􏱉􏱓F􏱓􏱋􏰋􏱓􏱔􏱎􏱑􏱒􏱍􏱔􏱇+􏱕.7,􏱉􏱓F􏱓􏱋􏱇E7􏱘􏱕.7,􏱉􏱓􏱎􏱋&􏱝􏱏􏱐􏱕􏱖􏱓􏱋&􏱝
Ó*(ÙÓ?=!􏱡U=􏱸Ó􏱓􏱠􏱍5􏱬( 0,P6􏱠]$􏱸Ó?=75RKJ=!ÒE5a5􏱬(!P!-􏱴􏱝’G*􏱡 􏱠*Î􏱬ær$3⁄4􏱡U= 􏱠1Ï6􏱠 K*(􏱴5r$?I􏱞- Ð?IUI 􏱠􏱏Ì9âU=rÌ􏱨$9Î7K3$EC 0􏱸Ó􏱸KJ=!Ò753⁄4$􏱏$7K1⁄40/Þ􏱘Ar􏱕􏱈􏱇 Ö 0ÙK􏱨$!(?I6􏱠.<>􏱡􏱏$ Ï?=􏱏$3$ ÏEC6􏱠JIÝ􏱋’K*􏱡10U=E5nÌ􏱨$7C?=􏱬GìÎ􏱏$􏱏$E5eK $*Ò753⁄4Ï!􏱡 0?I6􏱠􏱓$ ÏEC􏱏$ 075DKJ= 0 Ö¦=,-;C;$ <;$)Ñ.0!Òc-r􏱕􏱈􏱇 Ö 0DK􏱨$!(?I6􏱠.<>􏱡􏱏$ Ï?=􏱏$ IUI 􏱠􏱸Ó7K»􏱋 􏱝0Ö60􏱘K9􏱡 0?=75nÌ􏱨$7C?=􏱬GxÑ.0?IUI 􏱠*ÒMK@$!ÒE5@Ï!􏱡 0?I6􏱠R5eK1􏱠»Ö􏱘KJ=a$rÌ*􏱡;$ ÏEC􏱏$1qhg􏱘s 􏱤lqljki􏰌j¦i􏱞ges 􏱤lq1j¦ia_1oes􏲀°;d􏰌􏱩􏱨\n_eb qhg _1oesÙ_eb􏱤hg;\n_􏰋Z􏱍􏱎°;d􏰋gE_l􏱤k􏱥;\;􏱧6jW_􏱨^kiW_#􏱩>j􏱏\]geb;desEfE_’􏱧6jW_􏱨^kiW_􏰌_loesc}􏱸Zloes􏱵ieb􏱏d􏱨^k􏱥;\Á􏱣􏱽y􏱓_loes@b􏱏dl􏱾1*􏱋􏱌1N7M􏱡;$ Ï?=􏱏$􏱼ÒWÌ 􏱠􏱨$!-􏱏K9(lÒRÞ
􏱕-E􏱋 5+􏱏􏰍􏱏􏰍J􏱇I􏰌􏱇
-;KE5􏱬(!Î*􏱡U=􏰋$!ÒE5vÖ 0oæU=7CE5.􏱠􏱴Ó􏱑$􏱏Ì*􏱡6􏱠J=7C 􏱠.<.0rÌ]$!ÒE54K 􏱠􏰋$􏱸Ó6􏱠EK]$!Ò75@$,P􏰋$!Ò75 5!( 0@K*􏱡EC!(75􏰌$7K 􏱠􏱏Ì􏰋KJ=!ÒE5 􏱡?= 􏱡 􏱠?=7KEK;$􏱍Ø􏱴$!ÒRÞ􏲀A;KEC1075nÌ􏱨$ <􏱼Ï!􏱡?=!􏱡?=6􏱠WC75􏰍Ö 0@C􏱏$,PÙÓ!(>􏱡􏱴$!ÒE5:K?=íI!$EC􏱏$!ÒaÑ
􏱋􏱙{ 􏱉
􏱋 \ 􏱉
©􏱘􏱉􏱊s.j;\MsOg1j􏱨dWfc°􏱏d􏱘gW_ljki9y1!􏱇􏱈1N7M􏱡10rÌ􏱑5MK;$75MK 􏱠»Ö C10!î􏰍Az􏱇 Ö 03⁄4$!(?I6􏱠.<>􏱡􏱏$1ÏU=;$􏱴5eK􏱨$ ÏEC6􏱠JI#$!Ò75􏱓KJ= 􏱏 + 􏱉 􏱖 $EC􏱏$*ÒaÑ􏰀-1$rÌ*􏱡;$ ÏEC􏱏$1AW􏱒 I?I6􏱠aC 0ÝÖ F 􏱗 0 A $ AÀ=>- 􏱇 􏱗 􏱏 , 􏱖􏰊0 F 􏱇 􏱏 ÖU=􏱍$ ÏEC􏱏$10rÌ􏱓Ö60Á$1Ï!􏱡6􏱠*ÒWÌã􏱠ãÏ!􏱡?=>Í6􏱠ÁQ
􏱕 ” 􏱗#􏱋
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Learning rates:
η0 = 0.12 η1 = 0.03 η2 = 0.02
Hessian largest eigenvalue:
λ = 0.84
max
Weight space
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Maximum admissible Learning
rate (batch): −5
η
= 2.38
Log MSE (dB) 0
−10
−15
−20
0 1 2 3 4 5 6 7 8 9 10
epochs
max
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5MK 􏱠»Ö􏱸ÐEC􏱏$1<3⁄4􏱡 􏱠?=7KEK;$􏱍Ø:$ Ï 􏱠EC;$ <'􏱠]$!ÒE5/Ö60@$!(?I6􏱠.<>􏱡􏱏$1ÏU=;$45MK;$1Ï7C 􏱠?I􏰋$*Ò753⁄4􏱡U= 􏱠E5􏱬P10􏱴075D$ Ï6􏱠WC􏱏$ Ll<'􏱠3Ï!􏱡U=*􏱡!􏱡!(EC􏰌$!ÒE5􏱞Ö60􏰌$MK*(DK;$1Í6􏱠􏱴ÓÙ$?I!(EC􏱚Ï!􏱡?=aÑ.0UI?I 0ÝÖ􏰌$!ÒRÞ 􏱉􏱊>&􏱞 3&􏱞􏱚”􏱟􏱜2􏰍􏱠􏱜􏱘* 􏱓􏱔􏱑 􏱞 􏱖 %􏱡30″0.􏱡3 􏱈
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􏱇 Ö 0RC 075rÌ;$1􏱡􏱏$1ÏU=;$􏰌Ï!􏱡?=􏱬G*􏱡109Î7K;$ECEC 0rÌ]$*Ò75􏱓075D$ ÏEC􏱏$1􏱡;$ Ï?=􏱏$*I 􏱠*Î?=rÌ*􏱡?=7C*Î􏱍$!Ò75]0754$ ÏEC􏱏$1j􏱏\>o􏰋􏱢􏱨􏱢;􏱢n~®°􏱏d)gesOgkihglj;dWf􏰍sE_hg@^1j¦i1jki;\MbhsR_lo®ul©WgebE_esE_􏰋Z􏱏\Mb􏱏\lq _W_ebU°o¤;􏱯;¢r~1⁄4􏱩>j􏱏\􏰋glj;d¦ies7fW_lj1j􏱨dWf3¤;􏱰􏱨􏱇£;􏱇}WgebE_.􏱶W\¦􏱥o¡åoes􏱵i􏱘􏲁3􏱫eb;d􏰀􏲁[sE_1j]s.o􏱨^kiW_®􏲁#􏱩n_ebr\>ohg􏱚\]b;d?°@glj􏱨dkiesr\Ms |1jW_hgE_eb1q􏱼jebE_eshsr\>q􏰍°;d􏱍b7_􏱬{:Zl􏱤6j]_loes􏱘°􏱏d#j􏱨dkiesEflj1􏱤J°􏱓\:g;\Ù_l􏱤k􏱥;\;􏱧6jW_􏱨^kiW_]_1oes􏰍°􏱏d#j􏱨dkies1􏱤k􏱥;d’􏱧/􏱉1,􏱇􏱍􏱎1N7Mj;\lo:􏱢􏱨􏱢;􏱰/°;d􏰍gesOgkihglj􏱨dEf)s7_hg@^ljkiljki􏱏\Mbhs _1o®u􏰀©7􏱋 􏱢n~ 􏱉 􏱢;􏱰#􏱉 £;¤􏱏􏱯 􏱉 􏱫eb;d®􏲁ÀsE_ljÁ􏱩n_esEfE_ljlj;dWfÁ􏱶􏱵􏱥k􏱥l􏱤J°􏱴\Ùb;d?°􏱓_es􏱏\Mbo^ljkiljebr\n_k􏱥􏱞􏱥;\]Zkies1q;d 􏱩W_esEfki1􏱩n_eb q 􏱩lj;\3⁄4_es􏱏\MbÁ^ljkiljebr\n_k􏱥zjW_W_®􏲁ÀsE_􏱬{d¦iesr\Mbo_1oes􏱍°;dçj􏱨dkiesEflj1􏱤J°􏱼\􏱴g􏱏\􏱴b;debhb7_z􏱩W_eb􏱏\>􏱤.􏱇gzj;\W_􏱽1*􏱇􏱍􏱎1N7MP􏱓G;$1<􏱏$?=!ÒWÌ 􏱠􏰍C107CEC􏱏$ $*Ò75􏱚0E5@$MKl0UI􏱏Ì3⁄4ÐEC􏱏$1<􏰋$7C 􏱠 ÒWÌ?=*ÒaÑRKEC107CEC􏱏$%I 􏱠*(􏱬GJ=MK􏱨$7C􏱍K;$ Ð’􏱠RÑ?I6􏱠a􏱕􏱚$75.􏱠WC/I 􏱠􏱴ÓU=E5􏱬Î 0􏱼G􏱨$75nÌJ=􏱬G􏱨$7C!Î $*Ò75:Ï!􏱡?=MK*( A $ AÀ=»􏱋􏱟􏱇􏰊5l􏱡6􏱠W5eK*􏱡10rÌ@Ï!􏱡?=MKl0l0*Ò)&6A􏱨5l􏱡6􏱠E5MK*􏱡10rÌ􏱓G;$!􏱡*â􏱏$􏱬G;$EC!Î:􏱠ÙÐ>P4$75.􏱠WC:Ï!􏱡U=*􏱡7C 􏱠;$?I􏰌G􏱨$75rÌ?=􏱬G L􏱏$7C*Î]$!Ò75􏱸Ï*􏱡U=>Ð?I!ÎU=E5¦I*(ÙÓoÐ>P3G;$E5􏱬(!Î􏱸Ó 0􏱏Ì􏱍K;$E5.􏱠WC􏱸Ï!􏱡U=*􏱡7C 􏱠;$?I3ÒE5¦=aÑ􏱍K*ÒWÌ􏱬09Î;$􏰀Ö 0􏱍C􏱏$,PÙÓ!(>􏱡3G;$􏱬æ!â􏱓􏱠
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2.5
2
1.5
1
0.5
0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
LEARNING RATE
PREDICTED OPTIMAL LEARNING RATE
1 epoch
2 epochs 3 epochs 4 epochs 5 epochs
MEAN SQUARED ERROR
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