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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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-cG􏱟$756􏱖!G,Î*(RK?=RC 0E5nÌ􏱟$ <>􏱘􏱁$1ÏU=;$􏱊$!Ò75 Ö60Ù$ Ï6􏱖WC􏱁$1<'􏱖#Ï!􏱘U=*􏱘!􏱘!(EC􏱅$*Ò75 G*􏱘6􏱖åG;$E5􏱣(!Î􏱯Ó 0􏱁Ì􏱊K?=@􏱇;􏱇 􏱘 􏰷 􏰷 F 􏱇 􏱕 􏱘 􏰷 􏰷􏱉􏰷1􏱍 􏰶 􏰳 $􏱁Ì*􏱘􏱁$EC􏱁$􏱷VW=!G􏱎$!Ò754A􏱣" -%G;$EC 075MK􏱊$7C 􏱖R􏱇 􏱕 􏱘 􏰷 􏰷􏱈KE5l􏱘;$U=􏱣G1􏱖EC1Ï􏱊$!Ò753⁄4G*􏱘 􏱖aC 075rÌ;$1ÍWÌ 􏱖9P#G*􏱘6􏱖􏱳GWC6􏱖 􏱅RÑ􏱆􏱃􏱄􏱁7C 􏰹10ÝÖ􏱫􏱖aA 􏰵 – 􏱘 C1075rÌ;$ ÍWÌ1􏱖*P3G*􏱘6􏱖3GEC 􏱖4ÑEC 0ÝÖ􏱔􏱖,-;5􏱣(!ÎE5􏱣( 03G􏱟$7C?=MK􏱟$􏱣GÁÒ75U=aÑ3⁄4G;$75l􏱘;$MK􏱟$7C!Î􏱃K?=􏱳􏱘EC􏱁$E5M5.􏱖9Î􏱃5􏱣(!Î!􏱘?=􏱳􏱘6􏱖aA E -􏱊􏰴􏰸Ö603􏱘 0?=756􏱖;ë?=UI 􏱖?=E5¦=*􏱘U=􏰹C10ÝÖ􏱎􏱘;$MKl0!ÒWÌ􏰹KJ=4C1075rÌ;$ 􏱘􏱁$1ÏU=;$􏱊$!Ò75>-1$􏱁ë?=7C 􏱖􏱯Ó􏱯Ó!(7K􏱫0/Þ
􏱇􏰵&#􏰷 5􏱇􏱇􏱘􏰷􏱘􏱐􏰴 F􏱇􏱁􏱄􏱄􏰺􏰳􏰴􏱁􏱍􏱎7􏱘􏰷 􏱘􏱐􏰴􏱃 􏱍 7􏰴􏱇􏰸􏰹F􏰳􏰷 􏱔􏱊􏰶􏰷.􏰴 􏱑􏰴 􏰺􏱁􏰳􏰴 􏱁􏱓􏱑􏰴􏰳
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 􏰴
A􏱟5r$7K􏱫Ï!􏱘U=*􏱘U= 􏱖ECE5@$!Ò753􏱘?=3⁄4􏱘EC􏱁$E5M5.􏱖9Î3⁄4ÒWÌ1􏱖;$aC 0ÝÖaK*Î􏱁$E5eKo􏱘10U=E5.􏱖lÏ6􏱖*Î107C!ÎoGEC 􏱖4Ñ>ÍWÌ1􏱖*P􏱯0)Ñe5oG*􏱘6􏱖 GEC 􏱖4ÑEC 0ÝÖ:04Ñe5vK;$ECU=!(lÜ􏱟$7C4$7C*(􏱣G;$􏱁Ì>0EC!Î􏱒KJ=!ÒE5*Ö60􏱅􏱘 0?=756􏱖EC;$75U=R$!􏱘DB1A;5l􏱘6􏱖W5eK9􏱘 0rÌ!I?I6􏱖􏱫Ó7K@􏱖􏱒K?=#􏱍 $EC􏱁$!ÒaÑ
􏱇􏱋&#􏰷 􏱃 􏱇􏱇􏱘􏰷 􏱘􏱐􏰴 F􏱇􏱁􏱄􏱄􏰺􏰳􏰴􏰹􏱁􏱍 7 􏱘􏰷 􏱘%􏰴􏱃 􏱍 􏱔 􏱊􏰶􏰷.􏰴 􏱑􏰴􏰺􏰳􏰴􏱑􏰴􏰳
􏰳 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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}nb;des􏱬i1􏱠n_W}*􏱠􏱟\;\1􏱚a􏱠¦i6􏱞n\apaj[Y;©Wglj;dkies􏱁\]Zki 1⁄2rdeb qlq􏱁\􏰠fkiesOg;\lo7f.desOgR􏱠lj;\,g􏱊Zloes􏱬ieb;d;^¦􏱜􏱁\/_ljkik􏱜lj4􏱰6©*􏱛􏱟deshs􏱁d􏰠􏱡n©%y􏰹© 􏰳 ©6¡􏱟􏱮􏱟􏱮W~
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