Lecture 3 Flow Models

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University of Cincinnati, Main Campus *

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OPTIMIZATI

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Computer Science

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Oct 30, 2023

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pdf

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58

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±S294-158 ²eep Unsupervised !%Learning 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, ==:=VTVX (00-KNLNltKNLNir) ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] 6636² ´KNLNir]_KNLN^`KNLNy !%LeJcture 3 !%Likelihood &Models: °low &Models
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))muir ·bd_AA>A?^` 552bd_ILJAA>A?y ȅ ³bd_w ltbd_ PSQSVTVXlt AA>A? ILJKNLN`b]jsVTVXlty _abd_ILJKNLN^` wVTVXltRUSU JJGJHbd_`b]ltVTVX`b]mubd_mujs ȅ <<9<RUSUAA>A?lt ILJbd_ wKNLN wAA>A?`b]lt PSQSirbd__a ltRUSUVTVXjs _abd_ILJKNLN^`? ȅ ·bd_bd_ILJ PSQSVTVXlt ltbd_ ltRUSUKNLN ltirAA>A?VTVX`b]VTVX`b]QTRT ILJAA>A?ltAA>A? (irKNLNAA>A?^`^`y, ltRUSUKNLN mu`b]ILJKNLNir^`yVTVX`b]QTRT ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b]!) ȅ ¶bd_ir `b]KNLNw [ , AA>A?IIFIGVTVX^`VTVXlty ltbd_ KNLNvAA>A?^`muAA>A?ltKNLN ȅ ±IIFIGVTVX^`VTVXlty ltbd_ jsAA>A?_aikf^`KNLN PSQSirbd__a ȅ ±`b]ILJ, VTVXILJKNLNAA>A?^`^`y, AA>A? ^`AA>A?ltKNLN`b]lt irKNLNikfirKNLNjsKNLN`b]ltAA>A?ltVTVXbd_`b] ltRUSUAA>A?lt’js _aKNLNAA>A?`b]VTVX`b]QTRTPSQSmu^` 2
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))muir ·bd_AA>A?^` 552bd_ILJAA>A?y ȅ ³bd_w ltbd_ PSQSVTVXlt AA>A? ILJKNLN`b]jsVTVXlty _abd_ILJKNLN^` wVTVXltRUSU Ee_c]ltYc]mue_mujs ȅ <<9<RUSUAA>A?lt ILJbd_ wKNLN wAA>A?`b]lt PSQSirbd__a ltRUSUVTVXjs _abd_ILJKNLN^`? ȅ ·bd_bd_ILJ PSQSVTVXlt ltbd_ ltRUSUKNLN ltirAA>A?VTVX`b]VTVX`b]QTRT ILJAA>A?ltAA>A? (irKNLNAA>A?^`^`y, ltRUSUKNLN mu`b]ILJKNLNir^`yVTVX`b]QTRT ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b]!) ȅ ¶bd_ir `b]KNLNw [ , AA>A?IIFIGVTVX^`VTVXlty ltbd_ KNLNvAA>A?^`muAA>A?ltKNLN ȅ ±IIFIGVTVX^`VTVXlty ltbd_ jsAA>A?_aikf^`KNLN PSQSirbd__a ȅ ±`b]ILJ, VTVXILJKNLNAA>A?^`^`y, < a<ltQc]lt irQlfirQjsQc]lt<ltYe_c] ltRUSUAA>A?lt’js _aKNLNAA>A?`b]VTVX`b]QTRTPSQSmu^` ȅ µVTVXPSQSPSQSKNLNirKNLN`b]JJGJHKNLNjs PSQSirbd__a ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN &&bd_ILJKNLN^`js PSQSirbd__a ^`AA>A?jslt ^`KNLNJJGJHltmuirKNLN 3
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN Ŷ °oundations of °lows (1-±) ȅ 2-µ ¶^`bd_wjs ȅ ''-µ ¶^`bd_wjs ȅ µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 4
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 22/muVTVXJJGJH]_ 330KNLNPSQSirKNLNjsRUSUKNLNir: 00-irbd_IIFIGAA>A?IIFIGVTVX^`VTVXlty µKNLN`b]jsVTVXlty &&bd_ILJKNLN^`js 5
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ³bd_w ltbd_ PSQSVTVXlt AA>A? ILJKNLN`b]jsVTVXlty _abd_ILJKNLN^`? 6 ±ontinuous data 0 22159854 0 84525919 0 09121633 0 364252 0 30738086 0 32240615 0 24371194 0 22400792 0 39181847 0 16407012 0 84685229 0 15944969 0 79142357 0 6505366 0 33123603 0 81409325 0 74042126 0 67950372 0 74073271 0 37091554 0 83476616 0 38346571 0 33561352 0 74100048 0 32061713 0 09172335 0 39037131 0 80496586 0 80301971 0 32048452 0 79428266 0 6961708 0 20183965 0 82621227 0 367292 0 76095756 0 10125199 0 41495427 0 85999877 0 23004346 0 28881973 0 41211802 0 24764836 0 72743029 0 20749136 0 29877091 0 75781455 0 29219608 0 79681589 0 86823823 0 29936483 0 02948181 0 78528968 0 84015573 0 40391632 0 77816356 0 75039186 0 84709016 0 76950307 0 29772759 0 41163966 0 24862007 0 34249207 0 74363912 0 38303383 ĺ &Maximum !%Likelihood: ³quivalently:
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¸xAA>A?_aikf^`KNLN µKNLN`b]jsVTVXlty &&bd_ILJKNLN^`: &&VTVXxltmuirKNLNjs bd_PSQS ·AA>A?mujsjsVTVXAA>A?`b]js 00-AA>A?irAA>A?_aKNLNltKNLNirjs: _aKNLNAA>A?`b]js AA>A?`b]ILJ vAA>A?irVTVXAA>A?`b]JJGJHKNLNjs bd_PSQS JJGJHbd__aikfbd_`b]KNLN`b]ltjs, _aVTVXxltmuirKNLN wKNLNVTVXQTRTRUSUltjs 7
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ±jsVTVXILJKNLN bd_`b] &&VTVXxltmuirKNLNjs bd_PSQS ·AA>A?mujsjsVTVXAA>A?`b]js µbd_ _aVTVXxltmuirKNLNjs bd_PSQS ·AA>A?mujsjsVTVXAA>A?`b]js wbd_ir]_ PSQSbd_ir RUSUVTVXQTRTRUSU-ILJVTVX_aKNLN`b]jsVTVXbd_`b]AA>A?^` ILJAA>A?ltAA>A?? ''bd_lt irKNLNAA>A?^`^`y. 552RUSUKNLN jsAA>A?_aikf^`VTVX`b]QTRT ikfirbd_JJGJHKNLNjsjs VTVXjs: 1. 00-VTVXJJGJH]_ AA>A? JJGJH^`mujsltKNLNir JJGJHKNLN`b]ltKNLNir 2. ±ILJILJ ·AA>A?mujsjsVTVXAA>A?`b] `b]bd_VTVXjsKNLN ¹_aAA>A?QTRTVTVX`b]KNLN ltRUSUVTVXjs PSQSbd_ir _abd_ILJKNLN^`VTVX`b]QTRT `b]AA>A?ltmuirAA>A?^` VTVX_aAA>A?QTRTKNLNjs! 552RUSUKNLN bd_`b]^`y wAA>A?y AA>A? irKNLNAA>A?^`VTVXjsltVTVXJJGJH VTVX_aAA>A?QTRTKNLN JJGJHAA>A?`b] IIFIGKNLN QTRTKNLN`b]KNLNirAA>A?ltKNLNILJ VTVXjs VTVXPSQS VTVXlt VTVXjs AA>A? JJGJH^`mujsltKNLNir JJGJHKNLN`b]ltKNLNir, VTVX.KNLN. VTVXPSQS VTVXlt VTVXjs AA>A?^`irKNLNAA>A?ILJy jsltbd_irKNLNILJ ILJVTVXirKNLNJJGJHlt^`y VTVX`b] ltRUSUKNLN ikfAA>A?irAA>A?_aKNLNltKNLNirjs. 8
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ³bd_w ltbd_ PSQSVTVXlt AA>A? QTRTKNLN`b]KNLNirAA>A?^` ILJKNLN`b]jsVTVXlty _abd_ILJKNLN^`? 9 ... ȅ ³bd_w ltbd_ KNLN`b]jsmuirKNLN ikfirbd_ikfKNLNir ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b]? ȅ ³bd_w ltbd_ jsAA>A?_aikf^`KNLN? ȅ !%%AA>A?ltKNLN`b]lt irKNLNikfirKNLNjsKNLN`b]ltAA>A?ltVTVXbd_`b]?
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_wjs: &&AA>A?VTVX`b] ¹ILJKNLNAA>A? 10 ... ''bd_ir_aAA>A?^`VTVXzVTVX`b]QTRT ¶^`bd_w: ³bd_w ltbd_ ltirAA>A?VTVX`b]? ³bd_w ltbd_ KNLNvAA>A?^`muAA>A?ltKNLN ? ³bd_w ltbd_ jsAA>A?_aikf^`KNLN? ·KNLN`b]KNLNirAA>A?^`^`y:
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_wjs: 552irAA>A?VTVX`b]VTVX`b]QTRT 11 ...
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²RUSUAA>A?`b]QTRTKNLN bd_PSQS ;;8;AA>A?irVTVXAA>A?IIFIG^`KNLNjs 12 ''bd_ltKNLN: irKNLNmhqmuVTVXirKNLNjs VTVX`b]vKNLNirltVTVXIIFIG^`KNLN & ILJVTVXPSQSPSQSKNLNirKNLN`b]ltVTVXAA>A?IIFIG^`KNLN
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_wjs: 552irAA>A?VTVX`b]VTVX`b]QTRT 13 lj AA>A?jsjsmu_aVTVX`b]QTRT wKNLN RUSUAA>A?vKNLN AA>A?`b] KNLNxikfirKNLNjsjsVTVXbd_`b] PSQSbd_ir , ltRUSUVTVXjs JJGJHAA>A?`b] IIFIGKNLN bd_ikfltVTVX_aVTVXzKNLNILJ wVTVXltRUSU 441ltbd_JJGJHRUSUAA>A?jsltVTVXJJGJH ·irAA>A?ILJVTVXKNLN`b]lt µKNLNjsJJGJHKNLN`b]lt
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_wjs: 441AA>A?_aikf^`VTVX`b]QTRT 14 ... 441ltKNLNikf 1: jsAA>A?_aikf^`KNLN 441ltKNLNikf 2:
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs <<9<RUSUAA>A?lt ILJbd_ wKNLN `b]KNLNKNLNILJ ltbd_ ]_KNLNKNLNikf VTVX`b] _aVTVX`b]ILJ PSQSbd_ir PSQS? 15 ... 330KNLNJJGJHAA>A?^`^`, JJGJHRUSUAA>A?`b]QTRTKNLN bd_PSQS vAA>A?irVTVXAA>A?IIFIG^`KNLN PSQSbd_ir_amu^`AA>A? irKNLNmhqmuVTVXirKNLNjs - ¹`b]vKNLNirltVTVXIIFIG^`KNLN & ILJVTVXPSQSPSQSKNLNirKNLN`b]ltVTVXAA>A?IIFIG^`KNLN
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¸xAA>A?_aikf^`KNLN: ¶^`bd_w ltbd_ 6636`b]VTVXPSQSbd_ir_a z ´efore training µfter training 552irmuKNLN ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] bd_PSQS x ¶^`bd_w x lj z ¸_aikfVTVXirVTVXJJGJHAA>A?^` ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] bd_PSQS z 16
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¸xAA>A?_aikf^`KNLN: ¶^`bd_w ltbd_ ´KNLNltAA>A?(5,5) z ´efore training µfter training 552irmuKNLN ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] bd_PSQS x ¶^`bd_w x lj z ¸_aikfVTVXirVTVXJJGJHAA>A?^` ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] bd_PSQS z 17
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¸xAA>A?_aikf^`KNLN: ¶^`bd_w ltbd_ ·AA>A?mujsjsVTVXAA>A?`b] z ´efore training µfter training 552irmuKNLN ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] bd_PSQS x ¶^`bd_w x lj z ¸_aikfVTVXirVTVXJJGJHAA>A?^` ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] bd_PSQS z 18
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 00-irAA>A?JJGJHltVTVXJJGJHAA>A?^` 00-AA>A?irAA>A?_aKNLNltKNLNirVTVXzAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs 19 330KNLNmhqmuVTVXirKNLN_aKNLN`b]lt: ¹`b]vKNLNirltVTVXIIFIG^`KNLN AA>A?`b]ILJ µVTVXPSQSPSQSKNLNirKNLN`b]ltVTVXAA>A?IIFIG^`KNLN ȅ ²mu_amu^`AA>A?ltVTVXvKNLN µKNLN`b]jsVTVXlty ¶mu`b]JJGJHltVTVXbd_`b]js ȅ ¸.QTRT. ·AA>A?mujsjsVTVXAA>A?`b] _aVTVXxltmuirKNLN ILJKNLN`b]jsVTVXlty, _aVTVXxltmuirKNLN bd_PSQS ^`bd_QTRTVTVXjsltVTVXJJGJHjs ȅ ''KNLNmuirAA>A?^` ''KNLNlt ȅ ¹PSQS KNLNAA>A?JJGJHRUSU ^`AA>A?yKNLNir PSQS^`bd_w, ltRUSUKNLN`b] jsKNLNmhqmuKNLN`b]JJGJHVTVX`b]QTRT bd_PSQS ^`AA>A?yKNLNirjs = PSQS^`bd_w ȅ ¸AA>A?JJGJHRUSU ^`AA>A?yKNLNir: ȅ 330KNLN!%%6636? ȅ 441VTVXQTRT_abd_VTVXILJ? ȅ 552AA>A?`b]RUSU?
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ³bd_w QTRTKNLN`b]KNLNirAA>A?^` AA>A?irKNLN PSQS^`bd_wjs? - ²AA>A?`b] KNLNvKNLNiry (js_abd_bd_ltRUSU) ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b] IIFIGKNLN irKNLNikfirKNLNjsKNLN`b]ltKNLNILJ IIFIGy AA>A? (`b]bd_ir_aAA>A?^`VTVXzVTVX`b]QTRT) PSQS^`bd_w? [JJGJHbd_`b]jsVTVXILJKNLNirVTVX`b]QTRT 1-µ PSQSbd_ir `b]bd_w] 20
6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 330KNLNPSQSirKNLNjsRUSUKNLNir: ²mu_amu^`AA>A?ltVTVXvKNLN µKNLN`b]jsVTVXlty ¶mu`b]JJGJHltVTVXbd_`b] (²µ¶) 21
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 441AA>A?_aikf^`VTVX`b]QTRT vVTVXAA>A? VTVX`b]vKNLNirjsKNLN ²µ¶ 441AA>A?_aikf^`VTVX`b]QTRT PSQSirbd__a ltRUSUKNLN _abd_ILJKNLN^`: 552RUSUKNLN ²µ¶ VTVXjs AA>A?`b] VTVX`b]vKNLNirltVTVXIIFIG^`KNLN, ILJVTVXPSQSPSQSKNLNirKNLN`b]ltVTVXAA>A?IIFIG^`KNLN _aAA>A?ikf PSQSirbd__a ILJAA>A?ltAA>A? ltbd_ [0, 1] 22
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ³bd_w QTRTKNLN`b]KNLNirAA>A?^` AA>A?irKNLN PSQS^`bd_wjs? - ²µ¶ ltmuir`b]js AA>A?`b]y ILJKNLN`b]jsVTVXlty VTVX`b]ltbd_ mu`b]VTVXPSQSbd_ir_a - ¹`b]vKNLNirjsKNLN PSQS^`bd_w VTVXjs PSQS^`bd_w 23 ĺ Jcan turn any (smooth) p(x) into any (smooth) p(z)
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN ȅ ¶bd_mu`b]ILJAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs (1-µ) Ŷ 2-± °lows ȅ ''-µ ¶^`bd_wjs ȅ µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 24
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 2-µ ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_w 25
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 2-µ ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_w: 552wbd_ &&bd_bd_`b]js ±irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLN: ȅ ´AA>A?jsKNLN ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b]: 6636`b]VTVXPSQSbd_ir_a[0,1]^2 ȅ x1: _aVTVXxltmuirKNLN bd_PSQS 5 ·AA>A?mujsjsVTVXAA>A?`b]js ȅ x2: _aVTVXxltmuirKNLN bd_PSQS 5 ·AA>A?mujsjsVTVXAA>A?`b]js, JJGJHbd_`b]ILJVTVXltVTVXbd_`b]KNLNILJ bd_`b] x1 26
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 2-µ ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_w: ¶AA>A?JJGJHKNLN ±irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLN: ȅ ´AA>A?jsKNLN ILJVTVXjsltirVTVXIIFIGmultVTVXbd_`b]: 6636`b]VTVXPSQSbd_ir_a[0,1]^2 ȅ x1: _aVTVXxltmuirKNLN bd_PSQS 5 ·AA>A?mujsjsVTVXAA>A?`b]js ȅ x2: _aVTVXxltmuirKNLN bd_PSQS 5 ·AA>A?mujsjsVTVXAA>A?`b]js, JJGJHbd_`b]ILJVTVXltVTVXbd_`b]KNLNILJ bd_`b] x1 27
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN ȅ ¶bd_mu`b]ILJAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs (1-µ) ȅ 2-µ ¶^`bd_wjs Ŷ N-± °lows ȅ µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 28
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ³VTVXQTRTRUSU-ILJVTVX_aKNLN`b]jsVTVXbd_`b]AA>A?^` ILJAA>A?ltAA>A? PSQS (VTVX`b]PSQSKNLNirKNLN`b]JJGJHKNLN) PSQS -1 (jsAA>A?_aikf^`VTVX`b]QTRT) x and z must have the same dimension 29
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN ȅ ¶bd_mu`b]ILJAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs (1-µ) ȅ 2-µ ¶^`bd_wjs Ŷ N-± °lows Ŷ ²utoregressive °lows and Inverse ²utoregressive °lows ȅ 330KNLNAA>A?^`'';;8;00- (^`VTVX]_KNLN) AA>A?irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLNjs ȅ ·^`bd_w, ¶^`bd_w++, ¶¶°))330µ ȅ µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 30
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_wjs ȅ 552RUSUKNLN jsAA>A?_aikf^`VTVX`b]QTRT ikfirbd_JJGJHKNLNjsjs bd_PSQS AA>A? ´AA>A?yKNLNjs `b]KNLNlt VTVXjs AA>A? PSQS^`bd_w ȅ ¹PSQS AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN, ltRUSUVTVXjs PSQS^`bd_w VTVXjs JJGJHAA>A?^`^`KNLNILJ AA>A?`b] autoregressive flow ȅ 441AA>A?_aikf^`VTVX`b]QTRT VTVXjs AA>A?`b] invertiIble _aAA>A?ikfikfVTVX`b]QTRT PSQSirbd__a z ltbd_ x 31
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_wjs ȅ ³bd_w ltbd_ PSQSVTVXlt AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_wjs? ȅ &&AA>A?ikf x ltbd_ z ȅ ¶mu^`^`y ikfAA>A?irAA>A?^`^`KNLN^`VTVXzAA>A?IIFIG^`KNLN ȅ ''bd_ltVTVXJJGJHKNLN ȅ x ĺ z RUSUAA>A?js ltRUSUKNLN jsAA>A?_aKNLN jsltirmuJJGJHltmuirKNLN AA>A?js ltRUSUKNLN log likelihood JJGJHbd__aikfmultAA>A?ltVTVXbd_`b] bd_PSQS AA>A?`b] AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN _abd_ILJKNLN^` ȅ z ĺ x RUSUAA>A?js ltRUSUKNLN jsAA>A?_aKNLN jsltirmuJJGJHltmuirKNLN AA>A?js ltRUSUKNLN sampling ikfirbd_JJGJHKNLNILJmuirKNLN bd_PSQS AA>A?`b] AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN _abd_ILJKNLN^` 32
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¹`b]vKNLNirjsKNLN AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_wjs ȅ 552RUSUKNLN VTVX`b]vKNLNirjsKNLN bd_PSQS AA>A?`b] AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_w VTVXjs AA>A?^`jsbd_ AA>A? PSQS^`bd_w, JJGJHAA>A?^`^`KNLNILJ ltRUSUKNLN inverse autoregressive flow (¶µ°) ȅ x ĺ z RUSUAA>A?js ltRUSUKNLN jsAA>A?_aKNLN jsltirmuJJGJHltmuirKNLN AA>A?js ltRUSUKNLN sampling VTVX`b] AA>A?`b] AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN _abd_ILJKNLN^` ȅ z ĺ x RUSUAA>A?js ltRUSUKNLN jsAA>A?_aKNLN jsltirmuJJGJHltmuirKNLN AA>A?js log likelihood JJGJHbd__aikfmultAA>A?ltVTVXbd_`b] bd_PSQS AA>A?`b] AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN _abd_ILJKNLN^`. 441bd_, ¶µ° sampling is fast 33
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ±¶ vjs ¹±¶ ȅ ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_w ȅ °ast KNLNvAA>A?^`muAA>A?ltVTVXbd_`b] bd_PSQS ikf(x) PSQSbd_ir AA>A?irIIFIGVTVXltirAA>A?iry x ȅ Slow jsAA>A?_aikf^`VTVX`b]QTRT ȅ ¹`b]vKNLNirjsKNLN AA>A?multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN PSQS^`bd_w ȅ Slow KNLNvAA>A?^`muAA>A?ltVTVXbd_`b] bd_PSQS ikf(x) PSQSbd_ir AA>A?irIIFIGVTVXltirAA>A?iry x, jsbd_ ltirAA>A?VTVX`b]VTVX`b]QTRT ILJVTVXirKNLNJJGJHlt^`y IIFIGy _aAA>A?xVTVX_amu_a ^`VTVX]_KNLN^`VTVXRUSUbd_bd_ILJ VTVXjs js^`bd_w. ȅ °ast jsAA>A?_aikf^`VTVX`b]QTRT ȅ °ast KNLNvAA>A?^`muAA>A?ltVTVXbd_`b] bd_PSQS ikf(x) VTVXPSQS x VTVXjs AA>A? jsAA>A?_aikf^`KNLN ȅ 552RUSUKNLNirKNLN AA>A?irKNLN _abd_ILJKNLN^`js (00-AA>A?irAA>A?^`^`KNLN^` <<9<AA>A?vKNLN''KNLNlt, ¹±¶-;;8;±¸) ltRUSUAA>A?lt KNLNxikf^`bd_VTVXlt ¹±¶’js PSQSAA>A?jslt jsAA>A?_aikf^`VTVX`b]QTRT 34
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ±¶ AA>A?`b]ILJ ¹±¶ ''AA>A?VTVXvKNLN^`y, IIFIGbd_ltRUSU KNLN`b]ILJ muikf IIFIGKNLNVTVX`b]QTRT AA>A?js ILJKNLNKNLNikf AA>A?js ltRUSUKNLN `b]mu_aIIFIGKNLNir bd_PSQS vAA>A?irVTVXAA>A?IIFIG^`KNLNjs! - ¸.QTRT. 1&&00- VTVX_aAA>A?QTRTKNLN lj 1&& ^`AA>A?yKNLNirjs ĺ ²AA>A?`b] ILJbd_ ikfAA>A?irAA>A?_aKNLNltKNLNir jsRUSUAA>A?irVTVX`b]QTRT AA>A?js VTVX`b] ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN &&bd_ILJKNLN^`js PSQSirbd__a ^`KNLNJJGJHltmuirKNLN 2 [KNLN.QTRT. 330'''', _aAA>A?js]_VTVX`b]QTRT] 35
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN ȅ ¶bd_mu`b]ILJAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs (1-µ) ȅ 2-µ ¶^`bd_wjs Ŷ N-± °lows ȅ ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_wjs AA>A?`b]ILJ ¹`b]vKNLNirjsKNLN ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_wjs Ŷ RealNVP (like) architectures ȅ ·^`bd_w, ¶^`bd_w++, ¶¶°))330µ ȅ µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 36
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²RUSUAA>A?`b]QTRTKNLN bd_PSQS &&±''>>;>< vAA>A?irVTVXAA>A?IIFIG^`KNLNjs ¶bd_ir z ~ ikf(z), jsAA>A?_aikf^`VTVX`b]QTRT ikfirbd_JJGJHKNLNjsjs PSQS -1 ^`VTVX`b]KNLNAA>A?ir^`y ltirAA>A?`b]jsPSQSbd_ir_ajs AA>A? js_aAA>A?^`^` JJGJHmuIIFIGKNLN ILJz ltbd_ AA>A? js_aAA>A?^`^` ikfAA>A?irAA>A?^`^`KNLN^`KNLNikfVTVXikfKNLNILJ ILJx. 00-irbd_IIFIGAA>A?IIFIGVTVX^`VTVXlty VTVXjs JJGJHbd_`b]jsKNLNirvKNLNILJ: ¶ntuition : x VTVXjs ^`VTVX]_KNLN^`y VTVXPSQS VTVXlt _aAA>A?ikfjs ltbd_ AA>A? “^`AA>A?irQTRTKNLN” irKNLNQTRTVTVXbd_`b] VTVX`b] z jsikfAA>A?JJGJHKNLN 37
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_w _abd_ILJKNLN^`js: ltirAA>A?VTVX`b]VTVX`b]QTRT ±hange-of-variaIbles formula ^`KNLNltjs mujs JJGJHbd__aikfmultKNLN ltRUSUKNLN ILJKNLN`b]jsVTVXlty bd_vKNLNir x: 552irAA>A?VTVX`b] wVTVXltRUSU _aAA>A?xVTVX_amu_a ^`VTVX]_KNLN^`VTVXRUSUbd_bd_ILJ: 'New key requirement : ltRUSUKNLN °AA>A?JJGJHbd_IIFIGVTVXAA>A?`b] ILJKNLNltKNLNir_aVTVX`b]AA>A?`b]lt _amujslt IIFIGKNLN KNLNAA>A?jsy ltbd_ JJGJHAA>A?^`JJGJHmu^`AA>A?ltKNLN AA>A?`b]ILJ ILJVTVXPSQSPSQSKNLNirKNLN`b]ltVTVXAA>A?ltKNLN! 38
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²bd_`b]jsltirmuJJGJHltVTVX`b]QTRT PSQS^`bd_wjs: JJGJHbd__aikfbd_jsVTVXltVTVXbd_`b] ȅ °lows Jcan Ibe Jcomposed x lj PSQS 1 lj PSQS 2 lj ĺ PSQS ]_ lj z ȅ ¸AA>A?jsy wAA>A?y ltbd_ VTVX`b]JJGJHirKNLNAA>A?jsKNLN KNLNxikfirKNLNjsjsVTVXvKNLN`b]KNLNjsjs 39
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ±PSQSPSQSVTVX`b]KNLN PSQS^`bd_wjs ȅ ±`b]bd_ltRUSUKNLNir `b]AA>A?_aKNLN PSQSbd_ir AA>A?PSQSPSQSVTVX`b]KNLN PSQS^`bd_w: _amu^`ltVTVXvAA>A?irVTVXAA>A?ltKNLN ·AA>A?mujsjsVTVXAA>A?`b]. ȅ 00-AA>A?irAA>A?_aKNLNltKNLNirjs: AA>A?`b] VTVX`b]vKNLNirltVTVXIIFIG^`KNLN _aAA>A?ltirVTVXx ± AA>A?`b]ILJ AA>A? vKNLNJJGJHltbd_ir IIFIG ȅ PSQS(x) = ± -1 (x - IIFIG) ȅ 441AA>A?_aikf^`VTVX`b]QTRT: x = ±z + IIFIG, wRUSUKNLNirKNLN z ~ ''(0, ¹) ȅ !%%bd_QTRT ^`VTVX]_KNLN^`VTVXRUSUbd_bd_ILJ VTVXjs KNLNxikfKNLN`b]jsVTVXvKNLN wRUSUKNLN`b] ILJVTVX_aKNLN`b]jsVTVXbd_`b] VTVXjs ^`AA>A?irQTRTKNLN. ȅ 552RUSUKNLN °AA>A?JJGJHbd_IIFIGVTVXAA>A?`b] bd_PSQS PSQS VTVXjs ± -1 ȅ !%%bd_QTRT ^`VTVX]_KNLN^`VTVXRUSUbd_bd_ILJ VTVX`b]vbd_^`vKNLNjs JJGJHAA>A?^`JJGJHmu^`AA>A?ltVTVX`b]QTRT ILJKNLNlt(±) 40
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¸^`KNLN_aKNLN`b]ltwVTVXjsKNLN PSQS^`bd_wjs ȅ !%%bd_ltjs bd_PSQS PSQSirKNLNKNLNILJbd__a VTVX`b] KNLN^`KNLN_aKNLN`b]ltwVTVXjsKNLN PSQS^`bd_w ȅ ²AA>A?`b] mujsKNLN KNLN^`KNLN_aKNLN`b]ltwVTVXjsKNLN AA>A?PSQSPSQSVTVX`b]KNLN PSQSmu`b]JJGJHltVTVXbd_`b]js bd_ir ²µ¶ PSQS^`bd_wjs. ȅ 552RUSUKNLN °AA>A?JJGJHbd_IIFIGVTVXAA>A?`b] VTVXjs ILJVTVXAA>A?QTRTbd_`b]AA>A?^`, jsbd_ ltRUSUKNLN ILJKNLNltKNLNir_aVTVX`b]AA>A?`b]lt VTVXjs KNLNAA>A?jsy ltbd_ KNLNvAA>A?^`muAA>A?ltKNLN. 41
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ''¹²¸/330KNLNAA>A?^`'';;8;00- ±PSQSPSQSVTVX`b]KNLN JJGJHbd_muikf^`VTVX`b]QTRT ^`AA>A?yKNLNir ȅ 441ikf^`VTVXlt vAA>A?irVTVXAA>A?IIFIG^`KNLNjs VTVX`b] RUSUAA>A?^`PSQS: x 1:ILJ/2 , x ILJ/2+1:ILJ ȅ ¹`b]vKNLNirltVTVXIIFIG^`KNLN! ''bd_ltKNLN ltRUSUAA>A?lt js ʨ AA>A?`b]ILJ lt ʨ JJGJHAA>A?`b] IIFIGKNLN AA>A?irIIFIGVTVXltirAA>A?iry `b]KNLNmuirAA>A?^` `b]KNLNltjs wVTVXltRUSU no restriJctions . ȅ 552RUSUVTVX`b]]_ bd_PSQS ltRUSUKNLN_a AA>A?js data-parameterized elementwise flows . 42
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ''¹²¸/330KNLNAA>A?^`'';;8;00- ȅ ¹lt AA>A?^`jsbd_ RUSUAA>A?js AA>A? ltirAA>A?JJGJHltAA>A?IIFIG^`KNLN °AA>A?JJGJHbd_IIFIGVTVXAA>A?`b] ILJKNLNltKNLNir_aVTVX`b]AA>A?`b]lt ȅ 552RUSUKNLN °AA>A?JJGJHbd_IIFIGVTVXAA>A?`b] VTVXjs ltirVTVXAA>A?`b]QTRTmu^`AA>A?ir, jsbd_ VTVXltjs ILJKNLNltKNLNir_aVTVX`b]AA>A?`b]lt VTVXjs ltRUSUKNLN ikfirbd_ILJmuJJGJHlt bd_PSQS ILJVTVXAA>A?QTRTbd_`b]AA>A?^` KNLN`b]ltirVTVXKNLNjs. 43
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 330KNLNAA>A?^`'';;8;00- ȅ 552AA>A?]_KNLNAA>A?wAA>A?y: JJGJHbd_muikf^`VTVX`b]QTRT ^`AA>A?yKNLNirjs AA>A?^`^`bd_w mu`b]irKNLNjsltirVTVXJJGJHltKNLNILJ `b]KNLNmuirAA>A?^` `b]KNLNltjs ltbd_ IIFIGKNLN mujsKNLNILJ VTVX`b] PSQS^`bd_wjs, wRUSUVTVX^`KNLN ikfirKNLNjsKNLNirvVTVX`b]QTRT VTVX`b]vKNLNirltVTVXIIFIGVTVX^`VTVXlty AA>A?`b]ILJ ltirAA>A?JJGJHltAA>A?IIFIGVTVX^`VTVXlty [µVTVX`b]RUSU KNLNlt AA>A?^`. µKNLN`b]jsVTVXlty KNLNjsltVTVX_aAA>A?ltVTVXbd_`b] mujsVTVX`b]QTRT 330KNLNAA>A?^` '';;8;00-. ¹²!%%330 2017] 44
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 330KNLNAA>A?^`'';;8;00- ±irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLN ¹`b]ikfmult x: 32x32xJJGJH VTVX_aAA>A?QTRTKNLN ȅ !%%AA>A?yKNLNir 1: (²RUSUKNLNJJGJH]_KNLNirIIFIGbd_AA>A?irILJ x3, JJGJHRUSUAA>A?`b]`b]KNLN^` jsmhqmuKNLNKNLNzKNLN, JJGJHRUSUAA>A?`b]`b]KNLN^` x3) ȅ 441ikf^`VTVXlt irKNLNjsmu^`lt ltbd_ QTRTKNLNlt x1: 16x16x2JJGJH AA>A?`b]ILJ z1: 16x16x2JJGJH (PSQSVTVX`b]KNLN-QTRTirAA>A?VTVX`b]KNLNILJ ^`AA>A?ltKNLN`b]ltjs) ȅ !%%AA>A?yKNLNir 2: (²RUSUKNLNJJGJH]_KNLNirIIFIGbd_AA>A?irILJ x3, JJGJHRUSUAA>A?`b]`b]KNLN^` jsmhqmuKNLNKNLNzKNLN, JJGJHRUSUAA>A?`b]`b]KNLN^` x3) ȅ 441ikf^`VTVXlt irKNLNjsmu^`lt ltbd_ QTRTKNLNlt x2: 8x8x4JJGJH AA>A?`b]ILJ z2: 8x8x4JJGJH (JJGJHbd_AA>A?irjsKNLNir ^`AA>A?ltKNLN`b]ltjs) ȅ !%%AA>A?yKNLNir 3: (²RUSUKNLNJJGJH]_KNLNirIIFIGbd_AA>A?irILJ x3, JJGJHRUSUAA>A?`b]`b]KNLN^` jsmhqmuKNLNKNLNzKNLN, JJGJHRUSUAA>A?`b]`b]KNLN^` x3) ȅ ·KNLNlt z3: 4x4x16JJGJH (^`AA>A?ltKNLN`b]ltjs PSQSbd_ir RUSUVTVXQTRTRUSUKNLNjslt-^`KNLNvKNLN^` ILJKNLNltAA>A?VTVX^`js) 45
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs 330KNLNAA>A?^`'';;8;00-: ³bd_w ltbd_ ikfAA>A?irltVTVXltVTVXbd_`b] vAA>A?irVTVXAA>A?IIFIG^`KNLNjs? 46
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²RUSUKNLNJJGJH]_KNLNirIIFIGbd_AA>A?irILJ x4; JJGJHRUSUAA>A?`b]`b]KNLN^` jsmhqmuKNLNKNLNzKNLN; JJGJHRUSUAA>A?`b]`b]KNLN^` x3; JJGJHRUSUAA>A?`b]`b]KNLN^` mu`b]jsmhqmuKNLNKNLNzKNLN; JJGJHRUSUKNLNJJGJH]_KNLNirIIFIGbd_AA>A?irILJ x3 ·bd_bd_ILJ vjs ´AA>A?ILJ 00-AA>A?irltVTVXltVTVXbd_`b]VTVX`b]QTRT 47 (&&AA>A?js]_ ltbd_ikf RUSUAA>A?^`PSQS; _aAA>A?js]_ IIFIGbd_ltltbd__a RUSUAA>A?^`PSQS; _aAA>A?js]_ ^`KNLNPSQSlt RUSUAA>A?^`PSQS; _aAA>A?js]_ irVTVXQTRTRUSUlt RUSUAA>A?^`PSQS) x2
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN ȅ ¶bd_mu`b]ILJAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs (1-µ) ȅ 2-µ ¶^`bd_wjs Ŷ N-± °lows ȅ ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_wjs AA>A?`b]ILJ ¹`b]vKNLNirjsKNLN ±multbd_irKNLNQTRTirKNLNjsjsVTVXvKNLN ¶^`bd_wjs ȅ 330KNLNAA>A?^`'';;8;00- (^`VTVX]_KNLN) AA>A?irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLNjs Ŷ ³low, °low++, °°JOR± ȅ µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 48
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²RUSUbd_VTVXJJGJHKNLN bd_PSQS JJGJHbd_muikf^`VTVX`b]QTRT ltirAA>A?`b]jsPSQSbd_ir_aAA>A?ltVTVXbd_`b] ȅ ± ´AA>A?yKNLNjs `b]KNLNlt ILJKNLNPSQSVTVX`b]KNLNjs JJGJHbd_muikf^`VTVX`b]QTRT ILJKNLNikfKNLN`b]ILJKNLN`b]JJGJHy, IIFIGmult wRUSUAA>A?lt VTVX`b]vKNLNirltVTVXIIFIG^`KNLN ltirAA>A?`b]jsPSQSbd_ir_aAA>A?ltVTVXbd_`b] PSQS ltbd_ mujsKNLN VTVXjs AA>A? ILJKNLNjsVTVXQTRT`b] mhqmuKNLNjsltVTVXbd_`b] ȅ ±PSQSPSQSVTVX`b]KNLN ltirAA>A?`b]jsPSQSbd_ir_aAA>A?ltVTVXbd_`b] VTVXjs ltRUSUKNLN _abd_jslt JJGJHbd__a_abd_`b]^`y mujsKNLNILJ bd_`b]KNLN (''¹²¸, 330KNLNAA>A?^`'';;8;00-, ¹±¶-;;8;±¸, ĺ ) ȅ &&bd_irKNLN JJGJHbd__aikf^`KNLNx, `b]bd_`b]^`VTVX`b]KNLNAA>A?ir ltirAA>A?`b]jsPSQSbd_ir_aAA>A?ltVTVXbd_`b]js -> IIFIGKNLNltltKNLNir ikfKNLNirPSQSbd_ir_aAA>A?`b]JJGJHKNLN ȅ ²µ¶js AA>A?`b]ILJ VTVX`b]vKNLNirjsKNLN ²µ¶js PSQSbd_ir &&VTVXxltmuirKNLN bd_PSQS ·AA>A?mujsjsVTVXAA>A?`b]js bd_ir !%%bd_QTRTVTVXjsltVTVXJJGJHjs (¶^`bd_w++) ȅ 00-VTVXKNLNJJGJHKNLNwVTVXjsKNLN ^`VTVX`b]KNLNAA>A?ir/mhqmuAA>A?ILJirAA>A?ltVTVXJJGJH PSQSmu`b]JJGJHltVTVXbd_`b]js (''KNLNmuirAA>A?^` ¹_aikfbd_irltAA>A?`b]JJGJHKNLN 441AA>A?_aikf^`VTVX`b]QTRT) 49
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs '''' AA>A?irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLN AA>A?^`jsbd_ _aAA>A?ltltKNLNirjs 50 ȅ ¶^`bd_w++ = &&bd_!%% ltirAA>A?`b]jsPSQSbd_ir_aAA>A?ltVTVXbd_`b] + jsKNLN^`PSQS-AA>A?ltltKNLN`b]ltVTVXbd_`b] VTVX`b] '''' ȅ ´AA>A?yKNLNjs `b]KNLNlt (JJGJHbd_muikf^`VTVX`b]QTRT ILJKNLNikfKNLN`b]ILJKNLN`b]JJGJHy), ltirAA>A?`b]jsPSQSbd_ir_aAA>A?ltVTVXbd_`b] PSQSmu`b]JJGJHltVTVXbd_`b] JJGJH^`AA>A?jsjs, '''' AA>A?irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLN AA>A?^`^` ikf^`AA>A?y AA>A? irbd_^`KNLN VTVX`b] AA>A? PSQS^`bd_w’js ikfKNLNirPSQSbd_ir_aAA>A?`b]JJGJHKNLN. 441ltVTVX^`^` AA>A?`b]
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ȅ ·^`bd_w ( ^`VTVX`b]]_ ) ȅ ¹`b]vKNLNirltVTVXIIFIG^`KNLN 1x1 JJGJHbd_`b]vbd_^`multVTVXbd_`b]js ȅ !%%AA>A?irQTRTKNLN-jsJJGJHAA>A?^`KNLN ltirAA>A?VTVX`b]VTVX`b]QTRT ȅ ²bd_`b]ltVTVX`b]mubd_mujs ltVTVX_aKNLN PSQS^`bd_wjs (¶¶°))330µ) ȅ ±^`^`bd_wjs PSQSbd_ir mu`b]irKNLNjsltirVTVXJJGJHltKNLNILJ AA>A?irJJGJHRUSUVTVXltKNLNJJGJHltmuirKNLNjs. ¹`b]vKNLNirltVTVXIIFIGVTVX^`VTVXlty AA>A?`b]ILJ PSQSAA>A?jslt ^`bd_QTRT ikfirbd_IIFIGAA>A?IIFIGVTVX^`VTVXlty JJGJHbd__aikfmultAA>A?ltVTVXbd_`b] QTRTmuAA>A?irAA>A?`b]ltKNLNKNLNILJ. ))ltRUSUKNLNir JJGJH^`AA>A?jsjsKNLNjs bd_PSQS PSQS^`bd_wjs 51
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ))mult^`VTVX`b]KNLN ȅ ¶bd_mu`b]ILJAA>A?ltVTVXbd_`b]js bd_PSQS ¶^`bd_wjs (1-µ) ȅ 2-µ ¶^`bd_wjs ȅ ''-µ ¶^`bd_wjs Ŷ ±equantization 52
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_w bd_`b] µVTVXjsJJGJHirKNLNltKNLN µAA>A?ltAA>A? <<9<VTVXltRUSUbd_mult µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b]... 53
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²bd_`b]ltVTVX`b]mubd_mujs PSQS^`bd_wjs PSQSbd_ir ILJVTVXjsJJGJHirKNLNltKNLN ILJAA>A?ltAA>A? ȅ ± ikfirbd_IIFIG^`KNLN_a AA>A?irVTVXjsKNLNjs wRUSUKNLN`b] PSQSVTVXltltVTVX`b]QTRT JJGJHbd_`b]ltVTVX`b]mubd_mujs ILJKNLN`b]jsVTVXlty _abd_ILJKNLN^`js ltbd_ ILJVTVXjsJJGJHirKNLNltKNLN ILJAA>A?ltAA>A?: ILJKNLNQTRTKNLN`b]KNLNirAA>A?JJGJHy ȅ <<9<RUSUKNLN`b] ltRUSUKNLN ILJAA>A?ltAA>A? AA>A?irKNLN 3-IIFIGVTVXlt ikfVTVXxKNLN^` vAA>A?^`muKNLNjs, ȅ <<9<RUSUAA>A?lt ILJKNLN`b]jsVTVXlty ILJbd_KNLNjs AA>A? _abd_ILJKNLN^` AA>A?jsjsVTVXQTRT`b] ltbd_ vAA>A?^`muKNLNjs IIFIGKNLNltwKNLNKNLN`b] IIFIGVTVX`b]js ^`VTVX]_KNLN 0.4, 0.42 ĺ ? ȅ ²bd_irirKNLNJJGJHlt jsKNLN_aAA>A?`b]ltVTVXJJGJHjs: wKNLN wAA>A?`b]lt ltRUSUKNLN VTVX`b]ltKNLNQTRTirAA>A?^` bd_PSQS ikfirbd_IIFIGAA>A?IIFIGVTVX^`VTVXlty ILJKNLN`b]jsVTVXlty wVTVXltRUSUVTVX`b] AA>A? ILJVTVXjsJJGJHirKNLNltKNLN VTVX`b]ltKNLNirvAA>A?^` ltbd_ AA>A?ikfikfirbd_xVTVX_aAA>A?ltKNLN ILJVTVXjsJJGJHirKNLNltKNLN ikfirbd_IIFIGAA>A?IIFIGVTVX^`VTVXlty _aAA>A?jsjs 54
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ²bd_`b]ltVTVX`b]mubd_mujs PSQS^`bd_wjs PSQSbd_ir ILJVTVXjsJJGJHirKNLNltKNLN ILJAA>A?ltAA>A? ȅ 441bd_^`multVTVXbd_`b]: ²equantization . ±ILJILJ `b]bd_VTVXjsKNLN ltbd_ ILJAA>A?ltAA>A?. ȅ ȅ <<9<KNLN ILJirAA>A?w `b]bd_VTVXjsKNLN mu mu`b]VTVXPSQSbd_ir_a^`y PSQSirbd__a 55 >7OLPZ± 2VYK± %L[ONL± ²³´µD
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶^`bd_w bd_`b] µVTVXjsJJGJHirKNLNltKNLN µAA>A?ltAA>A? <<9<VTVXltRUSU µKNLNmhqmuAA>A?`b]ltVTVXzAA>A?ltVTVXbd_`b] 56
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ¶multmuirKNLN ILJVTVXirKNLNJJGJHltVTVXbd_`b]js ȅ 552RUSUKNLN mu^`ltVTVX_aAA>A?ltKNLN QTRTbd_AA>A?^`: AA>A? ^`VTVX]_KNLN^`VTVXRUSUbd_bd_ILJ-IIFIGAA>A?jsKNLNILJ _abd_ILJKNLN^` wVTVXltRUSU ȅ PSQSAA>A?jslt jsAA>A?_aikf^`VTVX`b]QTRT ȅ PSQSAA>A?jslt VTVX`b]PSQSKNLNirKNLN`b]JJGJHKNLN ȅ PSQSAA>A?jslt ltirAA>A?VTVX`b]VTVX`b]QTRT ȅ QTRTbd_bd_ILJ jsAA>A?_aikf^`KNLNjs ȅ QTRTbd_bd_ILJ JJGJHbd__aikfirKNLNjsjsVTVXbd_`b] ȅ ¶^`bd_wjs jsKNLNKNLN_a ltbd_ ^`KNLNlt mujs AA>A?JJGJHRUSUVTVXKNLNvKNLN jsbd__aKNLN bd_PSQS ltRUSUKNLNjsKNLN JJGJHirVTVXltKNLNirVTVXAA>A?. ȅ ´mult RUSUbd_w KNLNxAA>A?JJGJHlt^`y ILJbd_ wKNLN ILJKNLNjsVTVXQTRT`b] AA>A?`b]ILJ JJGJHbd__aikfbd_jsKNLN PSQS^`bd_wjs PSQSbd_ir QTRTirKNLNAA>A?lt ikfKNLNirPSQSbd_ir_aAA>A?`b]JJGJHKNLN? 552RUSUAA>A?lt’js AA>A?`b] bd_ikfKNLN`b] mhqmuKNLNjsltVTVXbd_`b]. 57
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6636² ´KNLNir]_KNLN^`KNLNy -- 441ikfirVTVX`b]QTRT 2020 -- µKNLNKNLNikf 6636`b]jsmuikfKNLNirvVTVXjsKNLNILJ !%%KNLNAA>A?ir`b]VTVX`b]QTRT -- 00-VTVXKNLNltKNLNir ±IIFIGIIFIGKNLNKNLN^`, 00-KNLNltKNLNir ²RUSUKNLN`b], °bd_`b]AA>A?ltRUSUAA>A?`b] ³bd_, ±irAA>A?vVTVX`b]ILJ 441irVTVX`b]VTVXvAA>A?js, ±^`KNLNx !%%VTVX, <<9<VTVX^`jsbd_`b] >>;><AA>A?`b] -- !%%3 ¶^`bd_wjs ´VTVXIIFIG^`VTVXbd_QTRTirAA>A?ikfRUSUy ''¹²¸: 'PUO± /H\YLU[± 'H]PK .Y\LNLY± HUK <VZO\H %LUNPV¶ ·1,&(¸ 1VU¹SPULHY PUKLWLUKLU[ JVTWVULU[Z LZ[PTH[PVU¶· IIr@@iv XprMMeXpriVnt IIr@@iv+"%"!±)&"' º²³´»¼¶ 330KNLNAA>A?^`'';;8;00-: 'PUO± /H\YLU[± -HZJOH 6VOS¹'PJRZ[LPU± HUK 6HT` %LUNPV¶ ·'LUZP[` LZ[PTH[PVU \ZPUN 5LHS 193¶· IIr@@iv XprMMeXpriVnt IIr@@iv+"'!&±!))!$ º²³´µ¼¶ ±¶: &OLU± ;P± L[ HS¶ ·9HYPH[PVUHS SVZZ` H\[VLUJVKLY¶· IIr@@iv XprMMeXpriVnt IIr@@iv+"'""±!#($" º²³´µ¼¶ 3HWHTHRHYPVZ± *LVYNL± 7OLV 3H]SHRV\± HUK ,HPU 0\YYH`¶ ·0HZRLK H\[VYLNYLZZP]L MSV^ MVY KLUZP[` LZ[PTH[PVU¶· $LLdvIIVnKKcMMes iVn 6MMeurIITl 1VnNNfWorUmIItiWoVn 8rWoKKcMMessiVnOg ;;ystMMeUms ¶ ²³´½¶ ¹±¶: .PUNTH± '\YR 3¶± L[ HS¶ ·,TWYV]LK ]HYPH[PVUHS PUMLYLUJL ^P[O PU]LYZL H\[VYLNYLZZP]L MSV^¶· $LLdvIIVnKKcMMes iVn VnMMeurIITl iVnNNfWorUmIItiWoVn XprWoKKcMMessiVnOg systMMeUms ²³´µ¶ ¶^`bd_w++: +V± -VUH[OHU± L[ HS¶ ·)SV^¾¾¸ ,TWYV]PUN )SV^¹%HZLK *LULYH[P]L 0VKLSZ ^P[O 9HYPH[PVUHS 'LX\HU[Pa|H[PVU HUK $YJOP[LJ[\YL 'LZPNU¶· IIr@@iv XprMMeXpriVnt IIr@@iv+"*!#±!!#(& º²³´¿¼¶ ''KNLNmuirAA>A?^` ¹_aikfbd_irltAA>A?`b]JJGJHKNLN 441AA>A?_aikf^`VTVX`b]QTRT: 0 SSLY± 7OVTHZ± L[ HS¶ ·1L\YHS PTWVY[HUJL ZHTWSPUN¶· IIr@@iv XprMMeXpriVnt IIr@@iv+")!)±!$ ·^`bd_w: .PUNTH± '\YR 3¶± HUK 3YHM\SSH 'OHYP^HS¶ ·*SV^¸ *LULYH[P]L MSV^ ^P[O PU]LY[PISL ´_´ JVU]VS\[PVUZ¶· $LLdvIIVnKKcMMes iVn 6MMeurIITl 1VnNNfWorUmIItiWoVn 8rWoKKcMMessiVnOg ;;ystMMeUms ¶ ²³´À¶ ¶¶°))330µ: *YH[O^VOS± :PSS± L[ HS¶ ·)MQVYK¸ )YLL¹MVYT JVU[PU\V\Z K`UHTPJZ MVY ZJHSHISL YL]LYZPISL NLULYH[P]L TVKLSZ¶· IIr@@iv XprMMeXpriVnt IIr@@iv+")"!±!"$'( º²³´À¼¶ 1L\YHS $\[VYLNYLZZP]L )SV^¸ +\HUN± &OPU¹:LP± L[ HS¶ ·1L\YHS H\[VYLNYLZZP]L MSV^Z¶· HY;P] WYLWYPU[ HY;P]¸´À³»¶³³½½¿ º²³´À¼ 5LZPK\HS )SV^Z MVY PU]LY[PISL NLULYH[P]L TVKLSPUN± 5PJR` 7¶ 4¶ &OLU± -LUZ %LOYTHUU± 'H]PK '\]LUH\K± -|YU¹+LUYPR -HJVIZLU± HY;P]¸ ´¿³µ¶³²½Á 1VYTHSPa|PUN )SV^Z MVY 3YVIHIPSPZ[PJ 0VKLSPUN HUK ,UMLYLUJL± *LVYNL 3HWHTHRHYPVZ± (YPJ 1HSPZUPJR± 'HUPSV -PTLULa| 5La|LUKL± 6OHRPY 0VOHTLK± %HSHQP /HRZOTPUHYH`HUHU O[[WZ¸ÃÃHY_P]¶VYNÃHIZô¿´²¶³²½µ² >[\[VYPHSD 58
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