Learning about Language with Normalizing Flows Graham Neubig Language Technologies Institute, Carnegie Mellon University Chunting Zhou Junxian He Di Wang, Xuezhe Ma, Daniel Spokoyny, Taylor Berg-Kirkpatrick
Learning about Language? 2
Learning about Language? Syntactic structure 2
Learning about Language? Syntactic structure The cat sat on a green wall 2
Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN 2
Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN Dependency: 2
Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN Dependency: Cross-lingual correspondences 2
Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN Dependency: Cross-lingual correspondences a cat green on sat the wall のは上壁猫緑座った 2
Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN Dependency: Cross-lingual correspondences a cat green on sat the wall のは上壁猫緑座った 2
Supervised Approaches X Y 3
Supervised Approaches X Supervised Learning Y X Y 4
Supervised Approaches X Supervised Learning Y X Y 5
Unsupervised Approaches X 6
Unsupervised Approaches X Learning language models P(X) 6
Unsupervised Approaches X Learning language models P(X) Learning continuous features from language models (e.g. word2vec, skipthought, BERT) 6
Unsupervised Approaches X Learning language models P(X) Learning continuous features from language models (e.g. word2vec, skipthought, BERT) But how do we turn this into interpretable structure? 6
Unsupervised Approaches X Learning language models P(X) Learning continuous features from language models (e.g. word2vec, skipthought, BERT) But how do we turn this into interpretable structure? How do we do it while taking advantage of continuous features? 6
Latent Variable Approaches X Unsupervised Y Y???? X 7
Latent Variable Approaches X Unsupervised Y Y???? X 8
Density Matching for Bilingual Word Embedding Chunting Zhou, Xuezhe Ma, Di Wang, Graham Neubig (NAACL 2019) 9
Bilingual Word Embedding 10
Bilingual Word Embedding apple pear professor school canine piazza dog cat basketball planet earth 10
Bilingual Word Embedding apple pear professor school 梨梨りんご 学校 教授 canine piazza ピアッツ 犬 dog cat basketball planet earth 猫バスケット 星 地球 10
Bilingual Word Embedding apple pear professor school 梨梨 教授 りんご 学校 canine piazza ピアッツ 犬 dog cat basketball planet earth 猫 バスケット 星 地球 Map word embeddings from different languages into a single vector space 10
Bilingual Word Embedding apple pear professor school 梨梨 教授 りんご 学校 canine piazza ピアッツ 犬 dog cat basketball planet earth 猫 バスケット 星 地球 Map word embeddings from different languages into a single vector space -Cross-lingual transfer 10
Bilingual Word Embedding apple pear professor school 梨梨 教授 りんご 学校 canine piazza ピアッツ 犬 dog cat basketball planet earth 猫 バスケット 星 地球 Map word embeddings from different languages into a single vector space -Cross-lingual transfer -Cross-lingual NLP tasks 10
Previous Work on Unsupervised BWE
Previous Work on Unsupervised BWE Unsupervised methods of minimization some form of distance between distributions of discrete vector sets:
Previous Work on Unsupervised BWE Unsupervised methods of minimization some form of distance between distributions of discrete vector sets:
Previous Work on Unsupervised BWE Unsupervised methods of minimization some form of distance between distributions of discrete vector sets: No direct probabilistic interpretation, not a "typical" unsupervised generative model
Density Mapping for Bilingual Word Embedding (DeMa-BWE) 12
Density Mapping for Bilingual Word Embedding (DeMa-BWE) Japanese Space English Space dog canine mapping function cat bird 12
Density Mapping for Bilingual Word Embedding (DeMa-BWE) Japanese Space English Space dog canine mapping function cat bird Mapping function is learned with normalizing flow 12
Normalizing Flows 13
<latexit sha1_base64="x6e2kl/41zlzrutzrp09f040wuy=">aaab8nicbvbns8naej3ur1q/qh69lbahxkpsbt0wvxisygsgdwwz3brld7nhdyou0j/hxymixv013vw3btsctpxbwoo9gwbmrsln2rjut1naw9/y3cpvv3z29/ypqodhxs0zrwihsc6vh2fnoutoxzddqz8qikxe6wm0vp35j09uasatbznjasjwmgexi9hykfbrtzob2nx/vf+tuq13drrkviluoec7x/3qdstjbe0m4vjrwhnte+zyguy4nvz6maypjmm8piglcrzuh/n85ck6s8oaxvlzsgyaq78nciy0nojidgpsrnrzm4n/eufm4uswz0magzqqxai448hinpsfdziixpcjjzgozm9fziqvjsamvlehemsvr5jus+fdnjr3l7xwtrfhgu7gforgwrw04a7a0aecep7hfd4c47w4787horxkfdph8afo5w+sx5ai</latexit> Normalizing Flows X P (X) 13
Normalizing Flows X P (X) <latexit sha1_base64="x6e2kl/41zlzrutzrp09f040wuy=">aaab8nicbvbns8naej3ur1q/qh69lbahxkpsbt0wvxisygsgdwwz3brld7nhdyou0j/hxymixv013vw3btsctpxbwoo9gwbmrsln2rjut1naw9/y3cpvv3z29/ypqodhxs0zrwihsc6vh2fnoutoxzddqz8qikxe6wm0vp35j09uasatbznjasjwmgexi9hykfbrtzob2nx/vf+tuq13drrkviluoec7x/3qdstjbe0m4vjrwhnte+zyguy4nvz6maypjmm8piglcrzuh/n85ck6s8oaxvlzsgyaq78nciy0nojidgpsrnrzm4n/eufm4uswz0magzqqxai448hinpsfdziixpcjjzgozm9fziqvjsamvlehemsvr5jus+fdnjr3l7xwtrfhgu7gforgwrw04a7a0aecep7hfd4c47w4787horxkfdph8afo5w+sx5ai</latexit> Z N (0, I) <latexit sha1_base64="z/gr+uic+bj4zzxnm516ybitp9q=">aaab9xicbvbnswmxej31s9avqkcvwsjukljbbt0wvehfktgpbnestdm2nmkusvyps/+hfw+kepw/eppfmlz70nyha4/3zpizf0scaeo6387c4tlyympmlbu+sbm1ndvzrekwvorwschd1qiwppxjwjxmcnqifmui4lqedc7hfv2rks1cewegefuf7knwzqqbkz3co5zmat0u3gn0fdto5d2iowgaj15k8pci0s59ttohiqwvhncsddnzi+mnwblgob1lw7gmesyd3knnsyuwvpvj5oororrkb3vdzusanff/tyryad0uge0u2pt1rdcw//oaseme+wmtuwyojnnf3zgje6jxbkjdfcwgdy3brdf7kyj9rdaxnqisdcgbfxme1epf76ryuj3nly/sodkwdwdqaa/ooaxxuieqefdwdk/w5jw5l8678zftxxdsmt34a+fzb7aykk0=</latexit> 13
Normalizing Flows X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> X P (X) <latexit sha1_base64="x6e2kl/41zlzrutzrp09f040wuy=">aaab8nicbvbns8naej3ur1q/qh69lbahxkpsbt0wvxisygsgdwwz3brld7nhdyou0j/hxymixv013vw3btsctpxbwoo9gwbmrsln2rjut1naw9/y3cpvv3z29/ypqodhxs0zrwihsc6vh2fnoutoxzddqz8qikxe6wm0vp35j09uasatbznjasjwmgexi9hykfbrtzob2nx/vf+tuq13drrkviluoec7x/3qdstjbe0m4vjrwhnte+zyguy4nvz6maypjmm8piglcrzuh/n85ck6s8oaxvlzsgyaq78nciy0nojidgpsrnrzm4n/eufm4uswz0magzqqxai448hinpsfdziixpcjjzgozm9fziqvjsamvlehemsvr5jus+fdnjr3l7xwtrfhgu7gforgwrw04a7a0aecep7hfd4c47w4787horxkfdph8afo5w+sx5ai</latexit> Z N (0, I) <latexit sha1_base64="z/gr+uic+bj4zzxnm516ybitp9q=">aaab9xicbvbnswmxej31s9avqkcvwsjukljbbt0wvehfktgpbnestdm2nmkusvyps/+hfw+kepw/eppfmlz70nyha4/3zpizf0scaeo6387c4tlyympmlbu+sbm1ndvzrekwvorwschd1qiwppxjwjxmcnqifmui4lqedc7hfv2rks1cewegefuf7knwzqqbkz3co5zmat0u3gn0fdto5d2iowgaj15k8pci0s59ttohiqwvhncsddnzi+mnwblgob1lw7gmesyd3knnsyuwvpvj5oororrkb3vdzusanff/tyryad0uge0u2pt1rdcw//oaseme+wmtuwyojnnf3zgje6jxbkjdfcwgdy3brdf7kyj9rdaxnqisdcgbfxme1epf76ryuj3nly/sodkwdwdqaa/ooaxxuieqefdwdk/w5jw5l8678zftxxdsmt34a+fzb7aykk0=</latexit> 13
Normalizing Flows X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> X P (X) <latexit sha1_base64="x6e2kl/41zlzrutzrp09f040wuy=">aaab8nicbvbns8naej3ur1q/qh69lbahxkpsbt0wvxisygsgdwwz3brld7nhdyou0j/hxymixv013vw3btsctpxbwoo9gwbmrsln2rjut1naw9/y3cpvv3z29/ypqodhxs0zrwihsc6vh2fnoutoxzddqz8qikxe6wm0vp35j09uasatbznjasjwmgexi9hykfbrtzob2nx/vf+tuq13drrkviluoec7x/3qdstjbe0m4vjrwhnte+zyguy4nvz6maypjmm8piglcrzuh/n85ck6s8oaxvlzsgyaq78nciy0nojidgpsrnrzm4n/eufm4uswz0magzqqxai448hinpsfdziixpcjjzgozm9fziqvjsamvlehemsvr5jus+fdnjr3l7xwtrfhgu7gforgwrw04a7a0aecep7hfd4c47w4787horxkfdph8afo5w+sx5ai</latexit> Z N (0, I) <latexit sha1_base64="z/gr+uic+bj4zzxnm516ybitp9q=">aaab9xicbvbnswmxej31s9avqkcvwsjukljbbt0wvehfktgpbnestdm2nmkusvyps/+hfw+kepw/eppfmlz70nyha4/3zpizf0scaeo6387c4tlyympmlbu+sbm1ndvzrekwvorwschd1qiwppxjwjxmcnqifmui4lqedc7hfv2rks1cewegefuf7knwzqqbkz3co5zmat0u3gn0fdto5d2iowgaj15k8pci0s59ttohiqwvhncsddnzi+mnwblgob1lw7gmesyd3knnsyuwvpvj5oororrkb3vdzusanff/tyryad0uge0u2pt1rdcw//oaseme+wmtuwyojnnf3zgje6jxbkjdfcwgdy3brdf7kyj9rdaxnqisdcgbfxme1epf76ryuj3nly/sodkwdwdqaa/ooaxxuieqefdwdk/w5jw5l8678zftxxdsmt34a+fzb7aykk0=</latexit> Change of variable formula: @f (x) p (x) = pz (f (x)) det( ) @x <latexit sha1_base64="st0jwp7pyeca65l8lakjgjqile4=">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</latexit> 13
Normalizing Flows X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> X P (X) <latexit sha1_base64="x6e2kl/41zlzrutzrp09f040wuy=">aaab8nicbvbns8naej3ur1q/qh69lbahxkpsbt0wvxisygsgdwwz3brld7nhdyou0j/hxymixv013vw3btsctpxbwoo9gwbmrsln2rjut1naw9/y3cpvv3z29/ypqodhxs0zrwihsc6vh2fnoutoxzddqz8qikxe6wm0vp35j09uasatbznjasjwmgexi9hykfbrtzob2nx/vf+tuq13drrkviluoec7x/3qdstjbe0m4vjrwhnte+zyguy4nvz6maypjmm8piglcrzuh/n85ck6s8oaxvlzsgyaq78nciy0nojidgpsrnrzm4n/eufm4uswz0magzqqxai448hinpsfdziixpcjjzgozm9fziqvjsamvlehemsvr5jus+fdnjr3l7xwtrfhgu7gforgwrw04a7a0aecep7hfd4c47w4787horxkfdph8afo5w+sx5ai</latexit> Z N (0, I) <latexit sha1_base64="z/gr+uic+bj4zzxnm516ybitp9q=">aaab9xicbvbnswmxej31s9avqkcvwsjukljbbt0wvehfktgpbnestdm2nmkusvyps/+hfw+kepw/eppfmlz70nyha4/3zpizf0scaeo6387c4tlyympmlbu+sbm1ndvzrekwvorwschd1qiwppxjwjxmcnqifmui4lqedc7hfv2rks1cewegefuf7knwzqqbkz3co5zmat0u3gn0fdto5d2iowgaj15k8pci0s59ttohiqwvhncsddnzi+mnwblgob1lw7gmesyd3knnsyuwvpvj5oororrkb3vdzusanff/tyryad0uge0u2pt1rdcw//oaseme+wmtuwyojnnf3zgje6jxbkjdfcwgdy3brdf7kyj9rdaxnqisdcgbfxme1epf76ryuj3nly/sodkwdwdqaa/ooaxxuieqefdwdk/w5jw5l8678zftxxdsmt34a+fzb7aykk0=</latexit> Change of variable formula: @f (x) p (x) = pz (f (x)) det( ) @x <latexit sha1_base64="st0jwp7pyeca65l8lakjgjqile4=">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</latexit> Intuitively, prevents degenerative mapping of everything to zero vector 13
Normalizing Flows X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> X P (X) <latexit sha1_base64="x6e2kl/41zlzrutzrp09f040wuy=">aaab8nicbvbns8naej3ur1q/qh69lbahxkpsbt0wvxisygsgdwwz3brld7nhdyou0j/hxymixv013vw3btsctpxbwoo9gwbmrsln2rjut1naw9/y3cpvv3z29/ypqodhxs0zrwihsc6vh2fnoutoxzddqz8qikxe6wm0vp35j09uasatbznjasjwmgexi9hykfbrtzob2nx/vf+tuq13drrkviluoec7x/3qdstjbe0m4vjrwhnte+zyguy4nvz6maypjmm8piglcrzuh/n85ck6s8oaxvlzsgyaq78nciy0nojidgpsrnrzm4n/eufm4uswz0magzqqxai448hinpsfdziixpcjjzgozm9fziqvjsamvlehemsvr5jus+fdnjr3l7xwtrfhgu7gforgwrw04a7a0aecep7hfd4c47w4787horxkfdph8afo5w+sx5ai</latexit> Z N (0, I) <latexit sha1_base64="z/gr+uic+bj4zzxnm516ybitp9q=">aaab9xicbvbnswmxej31s9avqkcvwsjukljbbt0wvehfktgpbnestdm2nmkusvyps/+hfw+kepw/eppfmlz70nyha4/3zpizf0scaeo6387c4tlyympmlbu+sbm1ndvzrekwvorwschd1qiwppxjwjxmcnqifmui4lqedc7hfv2rks1cewegefuf7knwzqqbkz3co5zmat0u3gn0fdto5d2iowgaj15k8pci0s59ttohiqwvhncsddnzi+mnwblgob1lw7gmesyd3knnsyuwvpvj5oororrkb3vdzusanff/tyryad0uge0u2pt1rdcw//oaseme+wmtuwyojnnf3zgje6jxbkjdfcwgdy3brdf7kyj9rdaxnqisdcgbfxme1epf76ryuj3nly/sodkwdwdqaa/ooaxxuieqefdwdk/w5jw5l8678zftxxdsmt34a+fzb7aykk0=</latexit> Change of variable formula: @f (x) p (x) = pz (f (x)) det( ) @x <latexit sha1_base64="st0jwp7pyeca65l8lakjgjqile4=">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</latexit> Intuitively, prevents degenerative mapping of everything to zero vector Normalizing Flow: A series of such invertible transformations f <latexit sha1_base64="aj1vrwgqsrqkgqj/blgtstmrk/w=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0mqomeif48t2fpoq9lsj+3azsbsboqs+gu8efdeqz/jm//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xt0s7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoepgtj25n/8irk81jem0mcfkshkoecuwolztgvv9yqowdzjv5okpcj0s9/9qyxsyouhgmqdddze+nnvbnobe5lvvrjqtmydrfrqaqraj+bhzolz1yzkdbwtqqhc/x3reyjrsdrydsjakz62zuj/3nd1itxfszlkhqubleotauxmzl9tqzcitniygllittbcrtrrzmx2zrscn7yy6ukxat6f9va87jsv8njkmijnmi5ehafdbidbrsaacizvmkb8+i8oo/ox6k14oqzx/ahzucpzd+m7g==</latexit> 13
DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird 14
DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird Notations: 14
DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird Notations: x 2 R d, y 2 R d : denote vectors in the src and tgt embedding space 14
DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird Notations: x 2 R d, y 2 R d : denote vectors in the src and tgt embedding space x i, y j : denote an actual word in src and tgt vocabularies 14
DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird Notations: x 2 R d, y 2 R d : denote vectors in the src and tgt embedding space x i, y j : denote an actual word in src and tgt vocabularies f xy, f yx : denote src->tgt, and tgt-src mapping functions 14
Prior Distribution Assumption on the monolingual word embedding space: Gaussian mixture model 鳥 猫 犬 15
Prior Distribution Assumption on the monolingual word embedding space: Gaussian mixture model 鳥 猫 犬 15
Prior Distribution Assumption on the monolingual word embedding space: Gaussian mixture model 鳥 猫 犬 15
DeMa-BWE: Density Matching 16
DeMa-BWE: Density Matching Sampling a continuous vector from the GMM 16
DeMa-BWE: Density Matching Sampling a continuous vector from the GMM 鳥 猫 x i (x i ) x p(x x i ) 犬 16
DeMa-BWE: Density Matching Sampling a continuous vector from the GMM 鳥 猫 x i (x i ) x p(x x i ) 犬 Apply the mapping function vector in the target space. f xy to obtain the transformed 16
DeMa-BWE: Density Matching 鳥 猫 Sampling a continuous vector from the GMM xi (xi ) x p (x xi ) Apply the mapping function fxy to obtain the transformed vector in the target space. fxy ( ) = Wxy 犬 dog canine cat bird 16
DeMa-BWE: Density Matching 鳥 猫 Sampling a continuous vector from the GMM xi (xi ) x p (x xi ) Apply the mapping function fxy to obtain the transformed vector in the target space. fxy ( ) = Wxy 犬 dog canine cat bird Computing the density of x in the mapped target space 16
DeMa-BWE: Density Matching 鳥 猫 Sampling a continuous vector from the GMM xi (xi ) x p (x xi ) Apply the mapping function fxy to obtain the transformed vector in the target space. fxy ( ) = Wxy 犬 dog canine cat bird Computing the density of x in the mapped target space 16
DeMa-BWE: Density Matching 鳥 猫 Sampling a continuous vector from the GMM xi (xi ) x p (x xi ) Apply the mapping function fxy to obtain the transformed vector in the target space. fxy ( ) = Wxy 犬 dog canine cat bird Computing the density of x in the mapped target space 16
DeMa-BWE: Density Matching 鳥 猫 Sampling a continuous vector from the GMM xi (xi ) x p (x xi ) Apply the mapping function fxy to obtain the transformed vector in the target space. fxy ( ) = Wxy 犬 dog canine cat bird Computing the density of x in the mapped target space Objective: 16
DeMa-BWE: Density Matching 鳥 猫 Sampling a continuous vector from the GMM xi (xi ) x p (x xi ) Apply the mapping function fxy to obtain the transformed vector in the target space. fxy ( ) = Wxy 犬 dog canine cat bird Computing the density of x in the mapped target space Objective: 16
Method Details 17
Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal 17
Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal 17
Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal Weak Supervision w/ Identical Strings: Take advantage of the fact that identical strings are usually the same word in both languages 17
Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal Weak Supervision w/ Identical Strings: Take advantage of the fact that identical strings are usually the same word in both languages 17
Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal Weak Supervision w/ Identical Strings: Take advantage of the fact that identical strings are usually the same word in both languages Alignment Selection Methods: Use cross-domain similarity local scaling (CSLS) 17
Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal Weak Supervision w/ Identical Strings: Take advantage of the fact that identical strings are usually the same word in both languages Alignment Selection Methods: Use cross-domain similarity local scaling (CSLS) 17
Dataset and Tasks Experiments Bilingual Lexicon Induction Task: MUSE dataset (Conneau el al., 2017) Cross-lingual Word Similarity Task: SemEval 2017 Languages Baseline languages: en - es, de, fr, ru, zh, ja Morphologically rich languages: en - et, fi, el, hu, pl, tr 18
Main Results on BLI (close languages) 85 Procrustes(R) MUSE (U+R) SL-unsup-ID DeMa-BWE 81.25 77.5 73.75 70 en-de de-en en-es es-en 19
Main Results on BLI (distant languages) 65 Procrustes(R) MUSE (U+R) DeMa-BWE 48.75 32.5 16.25 0 en-et et-en en-el el-en en-ja ja-en 20
Unsupervised Learning of Syntactic Structure w/ Invertible Neural Projections Junxian He, Graham Neubig, Taylor Berg-Kirkpatrick (EMNLP 2018) 21
<latexit sha1_base64="wplw8y/0iqfhjrazcsqahuwq07k=">aaacdxiczvdlsgmxfm34rpvvdekmwaqxpcyioo6kblxwcgyhlswtztrqzdikd8qy9b/erx6hk3hrn/gz/ogz6sxseyhck3ppdscniau34lo/zsrq2vrgzmmrvl2zu7dfoth8ncrrlplucaxbatfm8ij5wegwdqwzkyfgrwb8m81bt0wbrqihmmssj8kw4ignbczv7gyyfz72vx6l6tbdvpay8apqruu1+5xf7kdrrliiqcdgddw3hl5knhaq2ltctqylcr2tietygbhjtc/n/u7xqwugoftanghwzv7fsik0ksdikrnm5myza0oju7mqgmmszc/kdzorqs2qtuyktwgwjeb41ut5fcfaijrzesycg8jznnjanamgjhyqqrn9cqyjogkfm2dzzuqtjrim/pp6dd29v6g2boqwsugynaaz5kfl1eb3qil8rjfar+gnvtsvzofz6xznpctosxoe5sr5/gmr950d</latexit> <latexit 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sha1_base64="jbtr+w9vprfg2sai7yimoblknik=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucqmcoqu6mzlrccw2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dry0trahhjjeqe2bnoyuobww47cskyhfw2g7gn9m8/usvzjj6gelmfygheqszwwco++e+26/wniatl70m3aluufgtfvw3n5aketqcwrhwxdejwu+xaky4nvz6iayxjmm8pf0diyyo9tpc6tq+mczadquyjwi7z/9vpfhogwfklfntc7omasm5rhsvjetwsmfyu56iob6ieizsotqlric89fmwxqnqimx8ham3qdpzlpaakuqatwzardhzfzumsmiethgvk5g7mmgy8m4avw3n7rzwvc7ckqmjdixokysuubpdohbyeefd9ire0lv1yn1yn9bxtfqyip1dnffw9x/utzqw</latexit> 1 z<latexit sha1_base64="znkure23lzcjmnesh6crgwekxji=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lio4w2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dby0trahhjjeqe2bnoyuobww47cskyhfw+himr7p54xnvmsnohiyx9querixkbioh7p77zx615jscvoxl4baghopq96u/vyekiaarei617rpodh6kftdc6btsszsnmrnjie0aggfbtz/mvqf2iwegdiivorhyoft/i8vccwwjo8yanptldejjdd2oycsylj2t3/vebpva1dor0qfemalhhz+yke6armtmi0y4ddloyrehtfecfgiajoqzr9hkhbumymkrmizcxuswgddsxdac27na66oiq4yo0de6rs46ry10g9riqwqn0st6q+/wi/vhfvpfm2njknyo0vxz33/v75qx</latexit> 2 z 3 x 1 x<latexit sha1_base64="oue9nq3ny9iilmhpu3onq4ttfro=">aaacdxiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lco4ttepjpjk2njkmyr2xdp0hcavf4urc+g1+hn9gzjol214i9+tcc8pjcwlondjoj1vaw9/y3cpvv3z29/ypqodhj1omilcpsc5vn8cachzrdxhw2o0vxslgtbnmbrn554kqzwt0anoy+gkpihyygsfq3x4g0ufzodmo1pygk5e9ctwc1fbr7uh1tz+ujbe0askx1j3xicfpsqjgoj1v+ommmsytpki9aymsqpbt3o/mpjpm0a6lmicco2f/b6ryaifhbjrz0wuzjaepua4bfyxf1rjn8rueiqaeihomujrus0ygvpjtfsuj0ijmfyqjt0hawtb2kclkge8nweqx8xwbjlhcbeycfzoru5zikvcajeugc39ra90uyzxrctpf58hfl6if7labeyggjl7rg3q3xqwp69p6mktlvrfzjbbk+v4dlzmdba==</latexit> 2 x 3 The cat sat 22
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sha1_base64="jbtr+w9vprfg2sai7yimoblknik=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucqmcoqu6mzlrccw2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dry0trahhjjeqe2bnoyuobww47cskyhfw2g7gn9m8/usvzjj6gelmfygheqszwwco++e+26/wniatl70m3aluufgtfvw3n5aketqcwrhwxdejwu+xaky4nvz6iayxjmm8pf0diyyo9tpc6tq+mczadquyjwi7z/9vpfhogwfklfntc7omasm5rhsvjetwsmfyu56iob6ieizsotqlric89fmwxqnqimx8ham3qdpzlpaakuqatwzardhzfzumsmiethgvk5g7mmgy8m4avw3n7rzwvc7ckqmjdixokysuubpdohbyeefd9ire0lv1yn1yn9bxtfqyip1dnffw9x/utzqw</latexit> 1 z<latexit sha1_base64="znkure23lzcjmnesh6crgwekxji=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lio4w2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dby0trahhjjeqe2bnoyuobww47cskyhfw+himr7p54xnvmsnohiyx9querixkbioh7p77zx615jscvoxl4baghopq96u/vyekiaarei617rpodh6kftdc6btsszsnmrnjie0aggfbtz/mvqf2iwegdiivorhyoft/i8vccwwjo8yanptldejjdd2oycsylj2t3/vebpva1dor0qfemalhhz+yke6armtmi0y4ddloyrehtfecfgiajoqzr9hkhbumymkrmizcxuswgddsxdac27na66oiq4yo0de6rs46ry10g9riqwqn0st6q+/wi/vhfvpfm2njknyo0vxz33/v75qx</latexit> 2 z 3 x 1 x<latexit sha1_base64="oue9nq3ny9iilmhpu3onq4ttfro=">aaacdxiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lco4ttepjpjk2njkmyr2xdp0hcavf4urc+g1+hn9gzjol214i9+tcc8pjcwlondjoj1vaw9/y3cpvv3z29/ypqodhj1omilcpsc5vn8cachzrdxhw2o0vxslgtbnmbrn554kqzwt0anoy+gkpihyygsfq3x4g0ufzodmo1pygk5e9ctwc1fbr7uh1tz+ujbe0askx1j3xicfpsqjgoj1v+ommmsytpki9aymsqpbt3o/mpjpm0a6lmicco2f/b6ryaifhbjrz0wuzjaepua4bfyxf1rjn8rueiqaeihomujrus0ygvpjtfsuj0ijmfyqjt0hawtb2kclkge8nweqx8xwbjlhcbeycfzoru5zikvcajeugc39ra90uyzxrctpf58hfl6if7labeyggjl7rg3q3xqwp69p6mktlvrfzjbbk+v4dlzmdba==</latexit> 2 x 3 The cat sat x i N (µ zi, zi ) [Lin et al. 2015] 23
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sha1_base64="jbtr+w9vprfg2sai7yimoblknik=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucqmcoqu6mzlrccw2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dry0trahhjjeqe2bnoyuobww47cskyhfw2g7gn9m8/usvzjj6gelmfygheqszwwco++e+26/wniatl70m3aluufgtfvw3n5aketqcwrhwxdejwu+xaky4nvz6iayxjmm8pf0diyyo9tpc6tq+mczadquyjwi7z/9vpfhogwfklfntc7omasm5rhsvjetwsmfyu56iob6ieizsotqlric89fmwxqnqimx8ham3qdpzlpaakuqatwzardhzfzumsmiethgvk5g7mmgy8m4avw3n7rzwvc7ckqmjdixokysuubpdohbyeefd9ire0lv1yn1yn9bxtfqyip1dnffw9x/utzqw</latexit> 1 z<latexit sha1_base64="znkure23lzcjmnesh6crgwekxji=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lio4w2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dby0trahhjjeqe2bnoyuobww47cskyhfw+himr7p54xnvmsnohiyx9querixkbioh7p77zx615jscvoxl4baghopq96u/vyekiaarei617rpodh6kftdc6btsszsnmrnjie0aggfbtz/mvqf2iwegdiivorhyoft/i8vccwwjo8yanptldejjdd2oycsylj2t3/vebpva1dor0qfemalhhz+yke6armtmi0y4ddloyrehtfecfgiajoqzr9hkhbumymkrmizcxuswgddsxdac27na66oiq4yo0de6rs46ry10g9riqwqn0st6q+/wi/vhfvpfm2njknyo0vxz33/v75qx</latexit> 2 z 3 x 1 x<latexit sha1_base64="oue9nq3ny9iilmhpu3onq4ttfro=">aaacdxiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lco4ttepjpjk2njkmyr2xdp0hcavf4urc+g1+hn9gzjol214i9+tcc8pjcwlondjoj1vaw9/y3cpvv3z29/ypqodhj1omilcpsc5vn8cachzrdxhw2o0vxslgtbnmbrn554kqzwt0anoy+gkpihyygsfq3x4g0ufzodmo1pygk5e9ctwc1fbr7uh1tz+ujbe0askx1j3xicfpsqjgoj1v+ommmsytpki9aymsqpbt3o/mpjpm0a6lmicco2f/b6ryaifhbjrz0wuzjaepua4bfyxf1rjn8rueiqaeihomujrus0ygvpjtfsuj0ijmfyqjt0hawtb2kclkge8nweqx8xwbjlhcbeycfzoru5zikvcajeugc39ra90uyzxrctpf58hfl6if7labeyggjl7rg3q3xqwp69p6mktlvrfzjbbk+v4dlzmdba==</latexit> 2 x 3 The cat sat x i N (µ zi, zi ) [Lin et al. 2015] 23
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Latent Embeddings w/ Neural Projection Markov Structure zi <latexit sha1_base64="a0van9bpw9kdxhs9rkub77rptae=">aaacdniczvdltgixfo3gc/gfunttsexcedjjtnqd0y1lteqwmcgd0oggdjpp75jghi8wbvu7xbm3/okf4r/ygvki3ks5p+ee25yeibbcgov+oiwv1bx1jejmawt7z3evvh/wyfsikwtsjzrub8qwwspwba6ctwpniaweawwjm2zeemlacbxdwzhmvisdiiecerdu43op467hevfkfbfmtgsvay8hfzrxo1f+7fyvtsslgapitmdzy/btooftwsalbmjytoiidfjhwohizvx0anictyztx6hs9ksap+z/jzriiwkmrtjrzm6wmacumfwrgqhmwvbm9g7gmqggspqjtanngheil/2ur3eclkizh2eimcichyp7xdmkymwbozrbr2a6jjpqsbgwbebeyillohlwu6q5d+ev+nuevhedown0ijx0gerofjvqe1ek0st6q+/oi/phfdpfm2nbyxco0vw5339maj0j</latexit> ei N (µzi, zi ) z1 z2 z3 e1 e2 e3 <latexit 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Dependency Model with Valence DT NN VBD DT NN IN NN The cat stopped a dog in Paris [Klein and Manning 2004] 25
Dependency Model with Valence DT NN VBD DT NN IN NN [Klein and Manning 2004] 26
Dependency Parse Induction from POS? DT NN VBD DT NN IN NN 27
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Learning and Inference dim(x) = dim(e) and f is invertible <latexit sha1_base64="yxqgwog/cf5s2bxoodztuzlxmic=">aaacshiczvdlsgmxfm3uv61vxbozlikluqyiqdvrjcskvgudujlpnty0jyg5o5zhvsctfpvf4ge4endm2i5svrduybnnhpmtjyjbdijpr7s0vlk6vl6vbgxube/s7u0/wj0abi2mhtbtifoqxeeloqpojwaojaq8rqobyv74bmzyre5xnebx0ohimwcuhxux93arqt2ylp8fngagsmbv7o15p2ffs1scqiaotz1gkga3owy5e5bxwtrcqtmidqdjokisbdebom39y8f0/vgbdxt6e/bvrkallrshtlk0ozcrgnra2jpt4vawrxhmcrdjgduiwstexsz2wqjgf92mqyrfugzqi06fj9ovuvh73abdmxaamspdv3w2piyydnnnmyhkxqmecp6zlpkqfha+5fkyyewlz+d5mplgejdyyzht/6b1wr+sb3dn1avrwdxlckioyalpkhnyrw5jk7qii0beyrt59z68l+/b+5lks95s54dmvan0c2jls4w=</latexit> 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Learning and Inference dim(x) = dim(e) and f is invertible <latexit sha1_base64="yxqgwog/cf5s2bxoodztuzlxmic=">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</latexit> 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sha1_base64="utagcrqx87kpw4tgkv92duqd8gc=">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</latexit> <latexit sha1_base64="4v3leopic1acf3czlm/0mgv/hee=">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</latexit> p(xi zi ;, ) 1 @f = p(f 1 (xi ) zi ; ) det @xi <latexit 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Learning and Inference dim(x) = dim(e) and f is invertible <latexit sha1_base64="yxqgwog/cf5s2bxoodztuzlxmic=">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</latexit> 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sha1_base64="lmxtcczivcl5koy83kcxgg87iye=">aaacpxiczvhlsgnbeoz1gepbq5dfetyespgi3gqvhhwmedwqzie9znaey0yvjiz7bv71q/wcp8otehm2ycgyhqfrqruhmuokk8jrfh0gc4tlyyurtbx6+sbm1vbo7sadm7nl2ozggnufmidsagytiin3muwmeomd5omyqnee0tph9c2nmuwq9qhfkjgjt92kvz3dqbmni5whrsk4hglc93adk7hvek5qe5fmuydwlfg3yjyel1jw49xhxvgte8qhdzvt6lrfwgkzhnmmh6bg+qmphln/jwqmngi08j1vcjo1iifjpgv4lhqoklxpjo9upjjgohpvk3wp+yee0rnuixswe2o+0zhmmiqtvq6efwgrkxx5wlgv/ishhzdlohnvzkqkqqzxy40v3cjfdl+ih2urj6oyluusjxmepe9dbv33db60t5rnzegmghrswwecqwto4qku4brawahhfd7gpfgivolvytowgule9mamgp9fnzky2a==</latexit> sha1_base64="unn/qvlr88pxzv/bcsyxqupjxxo=">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</latexit> zi <latexit sha1_base64="4gmlgsvwrggpxtovfy6uyjcrnuq=">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</latexit> sha1_base64="jbazhnaj26j1i9dtzz4so1plzqy=">aaacoxiczvhltsmwehtkq4ty6jllbeliufvpl8aniqthkahuihvyna216kdkb6bvlb/gylfxcxwfj8qvp+2blpwshc/uwupzjbfcyhh+eo219y3nrea2v9pyd/f2d1r3vhegqcs00gaquauck4iqo4bbbodkrmbdmr6q6w8vyczx6g6noqwlfvy844yio26edo7dbjil4d/olcaxwcrt2+vhqwafbivmugsfe2gow5ia5exa5cefhzyymx2grwcvlwch5uxnfzw4jg0ybdxrgmzyvxmllvzshlnootmlws2g1sj2xbeozj3qz2z3o5vjj5gdusnyzk4iwex8whkvfwikzxvkhqhqb7unqconmbrtbygz3h0lycnqkepn3jkirfa+hyt4zvpkqtiynlrlnmhyulxlpmx5clwzubdq6x8q9bsx3fa2je1ysi7ikemrm3jjrskniqgjkxkj796h9+v9z3fr8bzlazol8h5+ay4ysxk=</latexit> sha1_base64="yd99wiwenul3nlx8oscdffa1yqk=">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</latexit> sha1_base64="utagcrqx87kpw4tgkv92duqd8gc=">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</latexit> <latexit sha1_base64="4v3leopic1acf3czlm/0mgv/hee=">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</latexit> p(xi zi ;, ) 1 @f = p(f 1 (xi ) zi ; ) det @xi <latexit sha1_base64="vugvam4zx8treom4e38jgvl+oae=">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</latexit> <latexit sha1_base64="hdczgh5isca9u7++4o6kholgwk0=">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</latexit> ei <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> sha1_base64="jbazhnaj26j1i9dtzz4so1plzqy=">aaacoxiczvhltsmwehtkq4ty6jllbeliufvpl8aniqthkahuihvyna216kdkb6bvlb/gylfxcxwfj8qvp+2blpwshc/uwupzjbfcyhh+eo219y3nrea2v9pyd/f2d1r3vhegqcs00gaquauck4iqo4bbbodkrmbdmr6q6w8vyczx6g6noqwlfvy844yio26edo7dbjil4d/olcaxwcrt2+vhqwafbivmugsfe2gow5ia5exa5cefhzyymx2grwcvlwch5uxnfzw4jg0ybdxrgmzyvxmllvzshlnootmlws2g1sj2xbeozj3qz2z3o5vjj5gdusnyzk4iwex8whkvfwikzxvkhqhqb7unqconmbrtbygz3h0lycnqkepn3jkirfa+hyt4zvpkqtiynlrlnmhyulxlpmx5clwzubdq6x8q9bsx3fa2je1ysi7ikemrm3jjrskniqgjkxkj796h9+v9z3fr8bzlazol8h5+ay4ysxk=</latexit> sha1_base64="znhkautewogmapwollzuede8uty=">aaacqhiczve9swnbej3z2/gzw5vdifieclfro8hgmojrwqthbznnfvfj2j1tw3f/wvz/lb3/w0os3etsgdowzns3m8vbn0kmhamo+gwwfpewv1bx1msbm1vbo7v1zrtncsuxy4009i5hdqxq2cvbeu8yi0wlem+tx4uqfvue1gmjr2muyu+xby1swrlvvix90d9trk1ohoe8ae9ba6br6ded43hgek5qe5fmuft2lfgvyjyel1jw4txhxvgje8b7dzvt6hrfwgwzhnpmekbg+qmphln/jwqmngi09j1vcjo1iifjpgv6lhqqklxpjo9upjjmoppvk3wp+yee0tneixswe2o+0zhmmiqtvsaea2grkxx5wlgv/ishhzllohn7zkqkqqzvyo3p3cjf9kcix8oitltxupazpe549lx3uf3f03nqpw6dtakrcnzghw7gcnpwaudwcr3oaochvmibvacfwvfwpvnhqjddyx7mrpdzcypztby=</latexit> sha1_base64="virgb/9jv45ewvl+qymhz6smazo=">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</latexit> x f 1 (e; ) <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">aaacwxiczvbnsynbeo2m7hrhdy169niybbdimjgfvbyixjwqgbwcbojp1jjg/hi6a9qwzd/xqv9j8mfyk+rgtkcp169ena9fmkvhmiregshc4o+fs83lcoxx6u+11vrgltof5ddjrhp7kzihumjoouajn7kfplij1+n9at2/fgdrhngxom6hr9idfpngdd01alwy/+vexo0mcestfct+dfrtqbtnin4h8qy0yazob+un/wroekfai5fmuds4yrffmoucs6jcphcqm37p7udwq80uuh45sv7rhc8mawaspxrphp28utllfmorv9bnzc1qbo2rrunvofj1q5+z3n1ypz1udwqrdzn7ygszg34pdf4gad71krwsoqf1thqolhcuyw8yt8j/hfirs4yjd3porkqqmew0phkjfnpdmnmqyirv5vnvzfmw5chzpux4a6bfqw+/e9inlv62j09mctfjftkmuyqm/8gxospnpec4esdp5iw8nt6diggg4vqangy7m2sugs0pzfk2fa==</latexit> Determinant of Jacobian matrix <latexit sha1_base64="yfyvkxezekmeahc8b85drlg5ulo=">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</latexit> xi <latexit 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Learning and Inference dim(x) = dim(e) and f is invertible <latexit sha1_base64="yxqgwog/cf5s2bxoodztuzlxmic=">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</latexit> 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sha1_base64="utagcrqx87kpw4tgkv92duqd8gc=">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</latexit> <latexit sha1_base64="4v3leopic1acf3czlm/0mgv/hee=">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</latexit> p(xi zi ;, ) 1 @f = p(f 1 (xi ) zi ; ) det @xi <latexit 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sha1_base64="hdczgh5isca9u7++4o6kholgwk0=">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</latexit> ei <latexit 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sha1_base64="virgb/9jv45ewvl+qymhz6smazo=">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</latexit> x f <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">aaacwxiczvbnsynbeo2m7hrhdy169niybbdimjgfvbyixjwqgbwcbojp1jjg/hi6a9qwzd/xqv9j8mfyk+rgtkcp169ena9fmkvhmiregshc4o+fs83lcoxx6u+11vrgltof5ddjrhp7kzihumjoouajn7kfplij1+n9at2/fgdrhngxom6hr9idfpngdd01alwy/+vexo0mcestfct+dfrtqbtnin4h8qy0yazob+un/wroekfai5fmuds4yrffmoucs6jcphcqm37p7udwq80uuh45sv7rhc8mawaspxrphp28utllfmorv9bnzc1qbo2rrunvofj1q5+z3n1ypz1udwqrdzn7ygszg34pdf4gad71krwsoqf1thqolhcuyw8yt8j/hfirs4yjd3porkqqmew0phkjfnpdmnmqyirv5vnvzfmw5chzpux4a6bfqw+/e9inlv62j09mctfjftkmuyqm/8gxospnpec4esdp5iw8nt6diggg4vqangy7m2sugs0pzfk2fa==</latexit> 1 (e; ) Gaussian distribution Determinant of Jacobian matrix <latexit 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sha1_base64="skzyiwujm36ifl+bt7gkjngfpcg=">aaaclhicdzfbixmxfmftws9rvxx1wqdfgkuqrgvmlnvdb2hzrraewcg6c5s6zdkznmwuq3jmbrnzofw04pt+ezntrst4iotkdy6c809eseegjr93op1r12/c3l3vvx3n7r37vb0hh52plemtzqsx5zl1xarnjyba8vpkcqpyyc/yy5m2fnbfrrngf4blxaekzrqobamqunz7i59j4mpfpjmryzh7qoavock4efjayhpsuqucslx+al4kpgtglrz/w8kie35vmvx68fdwyd8d7en4gmfjje1ajx2pxo5wekhrfbsx02yvk5lcsfpxduxs5y6suijp03zmkvsuqr2vklukm34rxe0vd9nmtbxhtwmpcglsobrwiv5d0vdlfiv5ygwvtxvrcrgj3sbkwvy1v9tm9xzllq9ynwitrcvdp4naetbthk5q4jqt5yhrichgvmfccmszygvwklmirilznaytifzd1hc58t0u0fwzm0prxtrkeytnvbpwfpvzneebb5v/s4n/70zs4eewfj/qhx1v5n5fj9et9awlaiyo0bt0iiaioa/og/qbfkapolfrsfr6nrp1njup0zzf734bef7pyq==</latexit> Learning and Inference z<latexit 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sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> 1 when is not invertible f<latexit 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sha1_base64="n3cjhtjw9tlldu5m9ej8nnibiqk=">aaacdniczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiibiakuzrq0e4n7r0tmuejjfv9bneujbsx/oib/8e/sdowelhjc0/ppbc5px7emqbh+bzyk6tr6xv5zclw9s7uxnh/4e7lwbhqesmlavtyu85c6gidtturolj4nlb80xu6bz1qpzkmb2ecuu/gqcgcrjay6r7ri4roeqzqk5acipovvqyqm1cqhtv/2gv9s9er/nb7ksschka41rptdslweqyaeu4nhw6saytjca9ox8aqc6q9jdm8su8n07cdqcwjwc7y/xsjflpggbpl2vtclgvasq7lrgvdkbb0meyux8iv+6kcipqo9iircc69hivrddqkux9bzg2qdhqo3wekeubjazbrzhzfjkosmaetyzprdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioieeklp6mv6tn6sd+tjks1zs51dnffw1x/syqdz</latexit> <latexit sha1_base64="n3cjhtjw9tlldu5m9ej8nnibiqk=">aaacdniczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiibiakuzrq0e4n7r0tmuejjfv9bneujbsx/oib/8e/sdowelhjc0/ppbc5px7emqbh+bzyk6tr6xv5zclw9s7uxnh/4e7lwbhqesmlavtyu85c6gidtturolj4nlb80xu6bz1qpzkmb2ecuu/gqcgcrjay6r7ri4roeqzqk5acipovvqyqm1cqhtv/2gv9s9er/nb7ksschka41rptdslweqyaeu4nhw6saytjca9ox8aqc6q9jdm8su8n07cdqcwjwc7y/xsjflpggbpl2vtclgvasq7lrgvdkbb0meyux8iv+6kcipqo9iircc69hivrddqkux9bzg2qdhqo3wekeubjazbrzhzfjkosmaetyzprdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioieeklp6mv6tn6sd+tjks1zs51dnffw1x/syqdz</latexit> Learning with Inverse Projection e i e i Inverse f<latexit sha1_base64="wkf9utthcqfbfvs3vq9hzy9dmzu=">aaacbniczvdltgixfl3je/gfuttfrgligpdbjbojunejismkqeindkchnu7aoyzkwgcyt/odrowu/q13/oj/yadycnykuafnntucniaw3kdnftsrq2vrg5u5rfz2zu7efuhg8mgorfpmuywubgbemmej5inhwzqxzkqggjwc4w02bzwybbik7neus44k/yihnbk0vd3sfope2zuuuwwqm1csnnzwfwcg1i38tnukjpjfsauxplxxyuykrcongo3z7cswmnah6bowhrgrzhtsidgxe2aznhsqbu+e7ot9v5esastbgvvmzcznmgaveqzkvtiqwcuemdznsaalqjyyktahwtcc4vun5vgciivo1eeycbevm4xi9rhmfmxiaki1t19x6yboqtfgl7czvrytwqb+rfm67nvtvjcwrrwcwymcqwuuoqp3uamfkdb4hhd4dz6cn+fd+zhkv5zzzhhmlfp1b+ntndk=</latexit> sha1_base64="nqdpikguhm9/+kirn5wz6hvhruk=">aaacbniczvdltgixfo3gc/gfutsyictebsezbtqd0y1lsbwhaui6pqmn7xts3jehe5yujfv9dldglv6g3+dkp7azsbc4sxnpzz23ot1+xjkgx/m2ciura+sb+c3c1vbo7l5x/+bby1gr6hhjpwr6wfpoquoba06bkajy+jw2/oftom88uqwzdo9hfngowp2qbyxgmfq96bzltsxjyl4g7gyuqset+s/tyatwlf62e5legozaona65tordbksgbfox4v2rgmeyrd3acvaeauqo0lmdgyfgaznb1kze4kdsf83eiy0wdawyrtpuvnkgjrcl40kbijt6tpzxy+ex/zforuphegfixbcdriwrjhqkex9bdg3qdppkhapkuqajwzardhzfzsmsmiethqfk5g7mmgy8c4q1xwnbrk6qdpkoyn0is6riy5rfd2hgviqqrs9off0zj1b79ah9tmv5qzzzigak+vrd/imnz8=</latexit> f 1 (x i ) x i x i 38
Learning with Inverse Projection ei <latexit sha1_base64="n3cjhtjw9tlldu5m9ej8nnibiqk=">aaacdniczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiibiakuzrq0e4n7r0tmuejjfv9bneujbsx/oib/8e/sdowelhjc0/ppbc5px7emqbh+bzyk6tr6xv5zclw9s7uxnh/4e7lwbhqesmlavtyu85c6gidtturolj4nlb80xu6bz1qpzkmb2ecuu/gqcgcrjay6r7ri4roeqzqk5acipovvqyqm1cqhtv/2gv9s9er/nb7ksschka41rptdslweqyaeu4nhw6saytjca9ox8aqc6q9jdm8su8n07cdqcwjwc7y/xsjflpggbpl2vtclgvasq7lrgvdkbb0meyux8iv+6kcipqo9iircc69hivrddqkux9bzg2qdhqo3wekeubjazbrzhzfjkosmaetyzprdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioieeklp6mv6tn6sd+tjks1zs51dnffw1x/syqdz</latexit> sha1_base64="mrqipjqbeebmku3laa4lyqylndy=">aaacdniczvdltgixfl2dl8qxaulgzuri4okqwy26i7pxcykibgh2sgca2umkvwncjnyecavf4cq49rfc+a/+gr1gixct5p6ee25zevxicioe9+1kvlbx1jeym7mt7z3dvfz+wb1rsaastpvquuktwwqpwr05ctamncpsf6zhd2/seeojacnveiejihuk6yc84jsgpr7avkzyumtz3xzbk3mtcpdbeqyklapazymavlv533zp0viyekkgxrtkxosdhgjkvlbxrh0bfhe6jh3wsjakkplomje8dk8t03mdpe0j0z2w/zcsio0koldktjm5wcqgusiurqohmm3pm5o7gum/6mtiktimmatgmljsjdymymqhnfoiyugictnw3b7xjkiywuco5vyrlh0qtsjacnomyoujlip6eemq5nvsvtcwrswcwwmcqrkuoak3uiu6ujdwaq/w5jw7786h8zmvzpzzzihmlfp1b2cznx0=</latexit> (1) hi,l = xi,l <latexit sha1_base64="ecox4vbbnenfsay0sy5zwnwjfdg=">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</latexit> (1) ei hi,r = xi,r + g(xi,l ) <latexit sha1_base64="rekaecskl6vtpqb6kre5yd3huvc=">aaacdxiczvdlsgmxfm3uv62vqks3wsk4kguqgrorunfzwbgfdiiznnogjpmhusouof8gbvu7xilbv8hp8a/mtgdh2wvhnpx7bjg5qsy4adf9cupr6xubw+xtys7u3v5b9fdoyaheu+zrjztubsqwwspmaqfburfmraacdyljxtbvpdntuioeyrozx5jrxenocviq2w9kymydpqjw3iabf14fzqluufhtqfw3p1q0kswckogxvaybg58sdzwknqv0e8niqidkxhowrkqy46e53xk+s8wqh0rbewho2f8bkzfgehhbzdbmwixjqclh6lyfy5m17jn8bqyyqaeynom0cc2seqiv/zrhcqisonmfysiwkjxlg4dcmwpiagghmtuvydommlcwcvzsrs3lrfabd9g4abgpl7xwbrfwgz2gu3somugktda9aimpustqk3pd786l8+f8ol9zackpdo7rqjnff2gonsg=</latexit> <latexit sha1_base64="gqpvuajfnzzo3nggi6xcpttqyce=">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</latexit> Inverse f (1) <latexit sha1_base64="nqdpikguhm9/+kirn5wz6hvhruk=">aaacbniczvdltgixfo3gc/gfutsyictebsezbtqd0y1lsbwhaui6pqmn7xts3jehe5yujfv9dldglv6g3+dkp7azsbc4sxnpzz23ot1+xjkgx/m2ciura+sb+c3c1vbo7l5x/+bby1gr6hhjpwr6wfpoquoba06bkajy+jw2/oftom88uqwzdo9hfngowp2qbyxgmfq96bzltsxjyl4g7gyuqset+s/tyatwlf62e5legozaona65tordbksgbfox4v2rgmeyrd3acvaeauqo0lmdgyfgaznb1kze4kdsf83eiy0wdawyrtpuvnkgjrcl40kbijt6tpzxy+ex/zforuphegfixbcdriwrjhqkex9bdg3qdppkhapkuqajwzardhzfzsmsmiethqfk5g7mmgy8c4q1xwnbrk6qdpkoyn0is6riy5rfd2hgviqqrs9off0zj1b79ah9tmv5qzzzigak+vrd/imnz8=</latexit> sha1_base64="wkf9utthcqfbfvs3vq9hzy9dmzu=">aaacbniczvdltgixfl3je/gfuttfrgligpdbjbojunejismkqeindkchnu7aoyzkwgcyt/odrowu/q13/oj/yadycnykuafnntucniaw3kdnftsrq2vrg5u5rfz2zu7efuhg8mgorfpmuywubgbemmej5inhwzqxzkqggjwc4w02bzwybbik7neus44k/yihnbk0vd3sfope2zuuuwwqm1csnnzwfwcg1i38tnukjpjfsauxplxxyuykrcongo3z7cswmnah6bowhrgrzhtsidgxe2aznhsqbu+e7ot9v5esastbgvvmzcznmgaveqzkvtiqwcuemdznsaalqjyyktahwtcc4vun5vgciivo1eeycbevm4xi9rhmfmxiaki1t19x6yboqtfgl7czvrytwqb+rfm67nvtvjcwrrwcwymcqwuuoqp3uamfkdb4hhd4dz6cn+fd+zhkv5zzzhhmlfp1b+ntndk=</latexit> f <latexit sha1_base64="fjhixwws58gsuvpwxj5bws9qhg8=">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</latexit> sha1_base64="t5o2w5fdbe9yvbrkg7tbseu/cmg=">aaaci3iczvbnlwrbek3xbx0tjiq6reky1owlbslfkcqiswvs09tjo7qnj901yjozix8jrvwoj3fxcpat/am9uw4+kunuq1evoluvsqww6ptv3sdg0pdi6nh4zwjyanqmojt3anvmgg8wlbu5j6jluis8gqilp08npyqs/cy6pij7zzfcwkgte+ymvkxovsjiwsg6kqwuxmhejftetduikc7zjabyk+vbihtryxxfr/u9ip9b8a1w9jy/lggahixvz2zbs0zxbjmk1l4efoqtnboutpki0swstym7plf8wsgekm5bee+kgqw6pk1ibdxlkptynxm5vvzr7dhlmeyvxsmg1tlwnao7qkzln73adlvui1stfbkb2z+lylztykwszsgt1t8jzirbturhsfsyzlb2hadmchckyr1qkepna8v5fpx15d9obnv36/6x82of+jegc7amaxdanuzbirxbaxjcwqm8wpn37z17l95rxzrgfc/mw6/w3r8aawenqq==</latexit> 1 (xi ) (1) hi,l hi,r <latexit sha1_base64="bsenpziye1ec45vcyn+ot1hadn0=">aaacgniczvdlsgmxfm3uv62vapdugkwoumqmcoqu6mzlbwslbs2znnogjpmhusouyt5f3op3ubk3bvwm/8bm24vtl4r7cu654et4keagxpfhya2tb2xu5bclo7t7+wffw6nho2jnwzmqoxtbj4yjhrimcbcshwlgpc9yyx/fzvpwm9ogq/abjhhrstimecapauv1i6wul5nr2k94fyv0kal4z2m/whzr7rtwkvdmoizm1egxf7sdrwpjqqccgnpx3ah6cdhaqwbporsbfhe6jkpwstakkplemjwf4lpldhcgtd0h4cn7fymh0kgci6vmmlmyzqwojuzvqmaks5y9m72bifsrvqxmim0cs2qegqtewsmobhbsmy8gfhguzolca64zbtgxgfdn7vcwhrfnkng4czyjbzmrvda8r13x3pulcv1mhlyehamtveeeukr1dicaqikomqbx9ibenrfnw/l0vmbsndpfkagfcr7/ahw0ovi=</latexit> sha1_base64="d20bzpfnqohstvpbd1jyhfyujwm=">aaab93iczvdlsgmxfm3uvx2r1rwbybfcldj1o+4eny4rolbqdiwtybsheqzjhaemxqtu/q5x4sf4gf6bmwkxtr0q7sm554ate2ecwwich6+2s7u3f1a/9i8a/vhjabpxynvukaupftomymkz4iqfwegwqwyykbfg/xj2um77r8xyrtuzzdmwstjrpowugkn642yr6arv4w3qxyewwtw4+ttknm0lu0afsxbydtkicmkau8ew/ii3lcn0rizs6kaiktmoqgwu8kvjepxq444cxlh/nwoirsqwdcqy2bvzyydwwradcqaybouz1d3ozdyozbsugzvadsoq3kyfv1kotngljzqxgdqui8ejn4ycmdtaqohuk5hoiseuxhc+i6i7gcg2ck87d53gkub1di4u0bxqoht0jx5rd4wiogs9ow/vzfv0vpzj1rxvpgdorbzvp0dvlyg=</latexit> sha1_base64="uqjbtkil1ol91wcjvitdnrmxj54=">aaacd3iczvdlsgmxfl3js9aqvzdugkvqkgxgjbot3lisyg2hrswtztpgmhmso0iz5lperx6hk/el/az/wezbhbyxwj0599xwcsjecou+/+2trk6tb2ywtsrblz3dvep+5chq1ddeylpq0wmp5vleviucje8khlmvst4on26kefuzgyt0fi+thpcvhcuieoyiowbvw16osne+yesdypwxow3o8kg15jf8azflemxbdebvhfr/ekpnusvjzjja2w38bpsznsiy5hm5l1qeupzer7zryewvt/1saj4nj44zkkgbd2iku/bvrkavvrthtlk0+29wmki1thwnwreqwvhm9g4nkqyhql6iji3sghgmlvuzijmuecxmpqjuetskciomheem5cqbyoxwxyfsta1l6oisu4ycxuswqeu8cdxw73wowrecwykecahxcatnaagdcbzcg7x7l96h9zklc8wbp3oa/8r7+gwmo5/n</latexit> sha1_base64="3qhylx9qlp+05qt4pk5toemfbus=">aaacgniczvdltgixfo34rhyhln00ehnmcjlxo+6iblxi4ggjiomudjs000l7x4rm5lomw/0ov8atgz/dp7adlaru0tztc89ttk8qc27adx+ctfwnza3twk5xd2//4lb0dpxovkip86kssrcdypjgefobg2dtwdmia8fawfg2n7eemtzcrq8wivlpkmheq04jwkpfkncdmy6yfsprwgrpadu7z/qlilt3p4vxgtchftsvzr/02x0omkgwarxemi7nxtbliqzobcuk3cswmnaxgbkohrgrzptsqfkmn1lmgeol7ykat9n/gymrrhiywwxezmisz0apywpwbsozt/yz6d1mzfalzc0xaroajsmqxvvshsujsijofisjwkbwhhqecm0oiikfhgpuv4lpighcwczztbl5y4msav+ifl13791k42yevggdofnurr66ra10h5rirxrn0ct6q+/oi/phfdpfm+mam98po4vyvv8aevshtg==</latexit> <latexit sha1_base64="m6axfmcurf5oagaaj1n7fjbneki=">aaacgniczvdlsgmxfm3uv62vapdugkwoumqmcoqu6mzlbwslbs2znnogjpmhusouyt5f3op3ubk3bvwm/8bm24vtl4r7cu654et4keagxpfhya2tb2xu5bclo7t7+wffw6nho2jnwzmqoxtbj4yjhrimcbcshwlgpc9yyx/fzvpwm9ogq/abjhhrstimecapauv1i6wul5nr2k94fev0kal4z2m/whzr7rtwkvdmoizm1egxf7sdrwpjqqccgnpx3ah6cdhaqwbporsbfhe6jkpwstakkplemjwf4lpldhcgtd0h4cn7fymh0kgci6vmmlmyzqwojuzvqmaks5y9m72bifsrvqxmim0cs2qegqtewsmobhbsmy8gfhguzolca64zbtgxgfdn7vcwhrfnkng4czyjbzmrvda8r13x3pulcv1mhlyehamtveeeukr1dicaqikomqbx9ibenrfnw/l0vmbsndpfkagfcr7/aiyqovg=</latexit> = + <latexit sha1_base64="j710iawjldt3kcch2yc+lhtgmcu=">aaace3iczvbnswmxem3wr7p+vt16wsych1k2xtrdsejfywvrc+1astlsg5pslmrwkev/hnhv/4ynetx5fn+g/8bs24ntb8k8vhkzzdw/5kyd6/5yuaxlldw1/lq9sbm1vvpy3bvtmlgenojkurv8rclnew0aa05bsajy+jw2/cfvvm8+ukwzjg5hgfnp4f7eqkywgoq+40se6kewka2ouowiw3bh4sycyhqulz7tavz6bde7hd9oiekiaasey63bftcgl8ukgof0zhcstwnmbrhh2wzgwfdtpeotr86ryqinlmq8cjwx+78jxuildh2jzjkeqwumsml1yaigl7kujrn/s4nkvihliqvdpbcihgdeyqi4arqryr5hwh2qtuaqezbfcfchazgozk5xsb8rtmd4abupkvoolilgsfm87n64xdolmkqehabddiwq6btv0dwqowyisken9ixerefrzxq3pibsndxt2uczyx39asohoxk=</latexit> <latexit sha1_base64="tt7shfqoco2k4yfwnueei7nc9mq=">aaace3iczvdnsgmxej6tf7x+vt16wsych1k2xtrdsejfywvrc20t2ttbhiabjzkvyllwkcsrpocnetz5ad6gb2c29wdbgtbfvvlmmpn8shcdnvftzjawv1bxsuu5jc2t7z387t6dubgmre6vulrpe8med1kdoqrwjdqj0hes4q+v0nrjgwndvxilo4h1jomhpocuokxu274sptosniwvctdf8erejnxfup4dhyvpxouragrd/e+7p2gswyhuegnazs/ctki0ciryoneodysihzi+a1kyeslmj5lspxaplnnza6xtc9gdsp87eiknjdiwyjszmvrkoflcfk0kbzjn6zjjpz2o6mtiktimmholyhdwsxgyxchcot0jiiwlyk0dcntcm4pizaghmtttxdogmlc0puasr+v5rxzb/ar0xvjuvel1eqarhqm4hgmowylu4rpquackgp7hbv6dj+fnexc+ptkm89ezdzphfp0cdochpq==</latexit> <latexit sha1_base64="9nq30mfpfbgzwoip8ja7qqu/idq=">aaace3iczvdnssnaen7uv1r/qh69bisgweoqgnorevfywdhcg8tms2mx7mbd7kqooa8hxvu5pilxh8dh8a3cpdnydmczb7/5zpj5/jgzdy7zy5vwvtfwn8qbla3tnd296v7bo5ajitqlkkvv9bgmnexubqacdmnfsfa57fjj26zeeazkmxk9wcsmnsddiiwmydduu9+xpnatyvj6nh1ua07dycnebs0c1far7uh1tx9ikggaaefy617ticflsqjgoj1w+ommmszjpkq9aymsqpbsfoupfwkywa6lmi8co2f/d6ryaifhzjrz0no1jaepua4bfyxelrix+t87qo6leizsotqli0b45ausihogezntesbcbmlndtkbu5qanxiaiwlmfjumsmieji8v41fz0zfl4j43rhvo/uwtdvoyvuzh6bidoia6rc10h9rirqqp9ire0lv1yn1yn9bxtfqyip5dnbfw9x9lxz/k</latexit> <latexit xi <latexit sha1_base64="jhj0nqdq+87fbqs5prps5rhlnou=">aaacbniczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny5bcgyhhuomzbshywri7ghl6aeiw/0ov+lw3/az/amz01ny9kk4j+eeg05oeatuwhv/nlx1jc2t7djoexdv/+cwcnt8zfsikfooekp3a2ky4bhzging3vgzigpboshkppt3npk2xewpmi2zl8ko4ignbczvhg0qvbfu5ovxqamavvrua1d57q8vtsslgapitk/hxucnraongs3k/cswmnajgbgehrgrzphpbnsgzy0zxkhs9ksac/b/rkqkkqtgvpk1szdlgfbkmjpvwvhmlxsmv5updgqbrguibukzzatcgz/luzwai+jcr5gidapnoeah14ycmfpaqob2k5ioisyubhrlm1fjozfv4f3wb+tu+6ravcvckqftdiyuuandoyz6qc3kiyoyekvv6n15ct6ct+drll1zip0ttfdo9x+yv5n5</latexit> g <latexit sha1_base64="ytthjcjf7thl3c1aric7iyn4ixu=">aaacdxiczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny4rolbqdiwtztrqzdikd8qy9b/erx6hk3hrn/gz/ogz6sxseyhck3ppdscniau34lo/zsrq2vrgzmmrvl2zu7dfoth8ncrrlhlucau7atfm8ih5wegwtqwzkyfg7wb8m83bt0wbrqihmmtml2qy8zbtapbq9akzpk/7vf+punu3l7wmggwooqja/cpvb6boilkevbbjug03bj8lgjgvbfrujybfhi7jkhutjihkxk9zv1n8apkbdpw2jwkcs/83uiknjdcyyqyzuvnggflc1kwkrjjr2tp53uxkuatklrnpe5ofixbe+smp4gryrgc+wkrgudjlbg+4zhtexajcnbdfwxrenkfgeyzbjbqliswd77x+xxfvl6rnmyksejpgj+gmndalaqi71eieokigv/sg3p0x58p5dl5m0hwn2dlcc+v8/wghz507</latexit> xi,l <latexit sha1_base64="vwq6y+pohkbnqh+fpepzilaedao=">aaaceniczvdltgixfo3gc/gfunttsexctmhgtnqd0y1lteriyei6pqmn7xrs7xjjzd7dunxvcgxc+gn+hn9gb1gi3ks5p+ee25yeibbcgof9oiwv1bx1jejmawt7z3evvh/wyfsikwtsjzrub8qwwspwba6ctwpniaweawwjm3zeemlacbxdwzhmvisdiiecercu3w1k+pz1uu5ikfxkfa/qtqovg9omvncsgr3yb7evacjzbfqqyzo1lwy/jro4fswrdrpdykjhzma6fkzemuone9mzprfmh4dk2xmbnrd/n1iijsqwtmq8mblzzobswrhwbuozt/yzyd2mzeag0s1f2ormwqiel37kozgbftgpjzargbtoa8j9rhkfmbaaum3tvzadek0o2bhlnqpayillohlwvap6d+ev+vusrci6qsfofnxqbaqjw9ratutri3pfb+jdexe+ne/nayotolodqzrxzvcffosfhq==</latexit> xi,r <latexit sha1_base64="nykeauiuyivai+ip12gc3twldk8=">aaaceniczvdltgixfo3gc/gfunttsexctmhgtnqd0y1lteriyei6pqmn7xrs7xjjzd7dunxvcgxc+gn+hn9gb1gi3ks5p+ee25yeibbcgof9oiwv1bx1jejmawt7z3evvh/wyfsikwtsjzrub8qwwspwba6ctwpniaweawwjm3zeemlacbxdwzhmvisdiiecercu3w1k+pz1uu5infxkfa/qtqovg9omvncsgr3yb7evacjzbfqqyzo1lwy/jro4fswrdrpdykjhzma6fkzemuone9mzprfmh4dk2xmbnrd/n1iijsqwtmq8mblzzobswrhwbuozt/yzyd2mzeag0s1f2ormwqiel37kozgbftgpjzargbtoa8j9rhkfmbaaum3tvzadek0o2bhlnqpayillohlwvap6d+ev+vusrci6qsfofnxqbaqjw9ratutri3pfb+jdexe+ne/nayotolodqzrxzvcfifafiw==</latexit> xi <latexit sha1_base64="qlouskqawl0c7ub9u7+xlyfg6ya=">aaacdxiczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiicuxip3sgoz1o2jtgmuefjfv9bbeujcstv8gn/+af2blycnykuafnntuchj/itipjffu5ldw19y38zmfre2d3r7h/ckdlrah1ierstx2skwchdyebp+1iusx8tlv+6dqdt+6p0kygtzcoqcfwigqbixgm1e76inmy9fivwhiqtlb2mqjoqkl21pxhr/xprq/42+1legsaaufy607vicblsajgoj0uurgmesyjpkada0msqpaszo/epjvm3w6kmiceo2p/byryaifhajrp03ozlaepus4bfqxf2tjnsrsec7/si3iqujrqc0ygupqsfkyx0jbmfqqxt0haatz2nylkgi8nweqx8xwbdlhcbeycbznrdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioi4eklp6mv6tn6sd+tjks1zs51dnffw1x/s7ady</latexit> sha1_base64="cabdscof/szjsejxwej+zulgkro=">aaacdxiczvdltgixfl2dl8qxaulgzuri4okqwy26i7pxcykicrdsla40tnnje8dijvydcavf4cq49rvc+a/+gr1gixct5p6ee25zevxicioe9+1kvlbx1jeym7mt7z3dvfz+wb1rsaastpvquuktwwqpwr05ctamncpsf6zhd2/seeoracnveiejihuk6yc84jsgpzptxyzp4y7v5gteyzuuuwzkm1cohnv+hgcg2s3/tnukxpkfsauxplx2iuwkrcongo1z7diwinah6bowhsgrzhssid+xe2qznhsobu+i7ot9v5eqastbgvwmzcznugaveqzovtiqauufmdznsppfxxztktabwtccwwun4weuiwvp1ecqcxevm2bj9rhmfmxiaki1t19x6yboqtemmlmzlrctwqb189jvyavzrk5hwlk4hhm4gzjcqavuoqp1ocdgbv7hzxl23p0p53mqztiznuoyk+frd02knxw=</latexit> [Dinh et al. 2014] 39
Learning with Inverse Projection ei <latexit sha1_base64="n3cjhtjw9tlldu5m9ej8nnibiqk=">aaacdniczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiibiakuzrq0e4n7r0tmuejjfv9bneujbsx/oib/8e/sdowelhjc0/ppbc5px7emqbh+bzyk6tr6xv5zclw9s7uxnh/4e7lwbhqesmlavtyu85c6gidtturolj4nlb80xu6bz1qpzkmb2ecuu/gqcgcrjay6r7ri4roeqzqk5acipovvqyqm1cqhtv/2gv9s9er/nb7ksschka41rptdslweqyaeu4nhw6saytjca9ox8aqc6q9jdm8su8n07cdqcwjwc7y/xsjflpggbpl2vtclgvasq7lrgvdkbb0meyux8iv+6kcipqo9iircc69hivrddqkux9bzg2qdhqo3wekeubjazbrzhzfjkosmaetyzprdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioieeklp6mv6tn6sd+tjks1zs51dnffw1x/syqdz</latexit> sha1_base64="mrqipjqbeebmku3laa4lyqylndy=">aaacdniczvdltgixfl2dl8qxaulgzuri4okqwy26i7pxcykibgh2sgca2umkvwncjnyecavf4cq49rfc+a/+gr1gixct5p6ee25zevxicioe9+1kvlbx1jeym7mt7z3dvfz+wb1rsaastpvquuktwwqpwr05ctamncpsf6zhd2/seeojacnveiejihuk6yc84jsgpr7avkzyumtz3xzbk3mtcpdbeqyklapazymavlv533zp0viyekkgxrtkxosdhgjkvlbxrh0bfhe6jh3wsjakkplomje8dk8t03mdpe0j0z2w/zcsio0koldktjm5wcqgusiurqohmm3pm5o7gum/6mtiktimmatgmljsjdymymqhnfoiyugictnw3b7xjkiywuco5vyrlh0qtsjacnomyoujlip6eemq5nvsvtcwrswcwwmcqrkuoak3uiu6ujdwaq/w5jw7786h8zmvzpzzzihmlfp1b2cznx0=</latexit> (1) hi,l = xi,l <latexit sha1_base64="ecox4vbbnenfsay0sy5zwnwjfdg=">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</latexit> (1) ei hi,r = xi,r + g(xi,l ) <latexit sha1_base64="rekaecskl6vtpqb6kre5yd3huvc=">aaacdxiczvdlsgmxfm3uv62vqks3wsk4kguqgrorunfzwbgfdiiznnogjpmhusouof8gbvu7xilbv8hp8a/mtgdh2wvhnpx7bjg5qsy4adf9cupr6xubw+xtys7u3v5b9fdoyaheu+zrjztubsqwwspmaqfburfmraacdyljxtbvpdntuioeyrozx5jrxenocviq2w9kymydpqjw3iabf14fzqluufhtqfw3p1q0kswckogxvaybg58sdzwknqv0e8niqidkxhowrkqy46e53xk+s8wqh0rbewho2f8bkzfgehhbzdbmwixjqclh6lyfy5m17jn8bqyyqaeynom0cc2seqiv/zrhcqisonmfysiwkjxlg4dcmwpiagghmtuvydommlcwcvzsrs3lrfabd9g4abgpl7xwbrfwgz2gu3somugktda9aimpustqk3pd786l8+f8ol9zackpdo7rqjnff2gonsg=</latexit> <latexit sha1_base64="gqpvuajfnzzo3nggi6xcpttqyce=">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</latexit> Inverse f <latexit sha1_base64="nqdpikguhm9/+kirn5wz6hvhruk=">aaacbniczvdltgixfo3gc/gfutsyictebsezbtqd0y1lsbwhaui6pqmn7xts3jehe5yujfv9dldglv6g3+dkp7azsbc4sxnpzz23ot1+xjkgx/m2ciura+sb+c3c1vbo7l5x/+bby1gr6hhjpwr6wfpoquoba06bkajy+jw2/oftom88uqwzdo9hfngowp2qbyxgmfq96bzltsxjyl4g7gyuqset+s/tyatwlf62e5legozaona65tordbksgbfox4v2rgmeyrd3acvaeauqo0lmdgyfgaznb1kze4kdsf83eiy0wdawyrtpuvnkgjrcl40kbijt6tpzxy+ex/zforuphegfixbcdriwrjhqkex9bdg3qdppkhapkuqajwzardhzfzsmsmiethqfk5g7mmgy8c4q1xwnbrk6qdpkoyn0is6riy5rfd2hgviqqrs9off0zj1b79ah9tmv5qzzzigak+vrd/imnz8=</latexit> sha1_base64="wkf9utthcqfbfvs3vq9hzy9dmzu=">aaacbniczvdltgixfl3je/gfuttfrgligpdbjbojunejismkqeindkchnu7aoyzkwgcyt/odrowu/q13/oj/yadycnykuafnntucniaw3kdnftsrq2vrg5u5rfz2zu7efuhg8mgorfpmuywubgbemmej5inhwzqxzkqggjwc4w02bzwybbik7neus44k/yihnbk0vd3sfope2zuuuwwqm1csnnzwfwcg1i38tnukjpjfsauxplxxyuykrcongo3z7cswmnah6bowhrgrzhtsidgxe2aznhsqbu+e7ot9v5esastbgvvmzcznmgaveqzkvtiqwcuemdznsaalqjyyktahwtcc4vun5vgciivo1eeycbevm4xi9rhmfmxiaki1t19x6yboqtfgl7czvrytwqb+rfm67nvtvjcwrrwcwymcqwuuoqp3uamfkdb4hhd4dz6cn+fd+zhkv5zzzhhmlfp1b+ntndk=</latexit> f <latexit sha1_base64="fjhixwws58gsuvpwxj5bws9qhg8=">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</latexit> sha1_base64="t5o2w5fdbe9yvbrkg7tbseu/cmg=">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</latexit> 1 (xi ) (1) hi,l (1) hi,r <latexit sha1_base64="bsenpziye1ec45vcyn+ot1hadn0=">aaacgniczvdlsgmxfm3uv62vapdugkwoumqmcoqu6mzlbwslbs2znnogjpmhusouyt5f3op3ubk3bvwm/8bm24vtl4r7cu654et4keagxpfhya2tb2xu5bclo7t7+wffw6nho2jnwzmqoxtbj4yjhrimcbcshwlgpc9yyx/fzvpwm9ogq/abjhhrstimecapauv1i6wul5nr2k94fyv0kal4z2m/whzr7rtwkvdmoizm1egxf7sdrwpjqqccgnpx3ah6cdhaqwbporsbfhe6jkpwstakkplemjwf4lpldhcgtd0h4cn7fymh0kgci6vmmlmyzqwojuzvqmaks5y9m72bifsrvqxmim0cs2qegqtewsmobhbsmy8gfhguzolca64zbtgxgfdn7vcwhrfnkng4czyjbzmrvda8r13x3pulcv1mhlyehamtveeeukr1dicaqikomqbx9ibenrfnw/l0vmbsndpfkagfcr7/ahw0ovi=</latexit> sha1_base64="d20bzpfnqohstvpbd1jyhfyujwm=">aaab93iczvdlsgmxfm3uvx2r1rwbybfcldj1o+4eny4rolbqdiwtybsheqzjhaemxqtu/q5x4sf4gf6bmwkxtr0q7sm554ate2ecwwich6+2s7u3f1a/9i8a/vhjabpxynvukaupftomymkz4iqfwegwqwyykbfg/xj2um77r8xyrtuzzdmwstjrpowugkn642yr6arv4w3qxyewwtw4+ttknm0lu0afsxbydtkicmkau8ew/ii3lcn0rizs6kaiktmoqgwu8kvjepxq444cxlh/nwoirsqwdcqy2bvzyydwwradcqaybouz1d3ozdyozbsugzvadsoq3kyfv1kotngljzqxgdqui8ejn4ycmdtaqohuk5hoiseuxhc+i6i7gcg2ck87d53gkub1di4u0bxqoht0jx5rd4wiogs9ow/vzfv0vpzj1rxvpgdorbzvp0dvlyg=</latexit> sha1_base64="uqjbtkil1ol91wcjvitdnrmxj54=">aaacd3iczvdlsgmxfl3js9aqvzdugkvqkgxgjbot3lisyg2hrswtztpgmhmso0iz5lperx6hk/el/az/wezbhbyxwj0599xwcsjecou+/+2trk6tb2ywtsrblz3dvep+5chq1ddeylpq0wmp5vleviucje8khlmvst4on26kefuzgyt0fi+thpcvhcuieoyiowbvw16osne+yesdypwxow3o8kg15jf8azflemxbdebvhfr/ekpnusvjzjja2w38bpsznsiy5hm5l1qeupzer7zryewvt/1saj4nj44zkkgbd2iku/bvrkavvrthtlk0+29wmki1thwnwreqwvhm9g4nkqyhql6iji3sghgmlvuzijmuecxmpqjuetskciomheem5cqbyoxwxyfsta1l6oisu4ycxuswqeu8cdxw73wowrecwykecahxcatnaagdcbzcg7x7l96h9zklc8wbp3oa/8r7+gwmo5/n</latexit> sha1_base64="3qhylx9qlp+05qt4pk5toemfbus=">aaacgniczvdltgixfo34rhyhln00ehnmcjlxo+6iblxi4ggjiomudjs000l7x4rm5lomw/0ov8atgz/dp7adlaru0tztc89ttk8qc27adx+ctfwnza3twk5xd2//4lb0dpxovkip86kssrcdypjgefobg2dtwdmia8fawfg2n7eemtzcrq8wivlpkmheq04jwkpfkncdmy6yfsprwgrpadu7z/qlilt3p4vxgtchftsvzr/02x0omkgwarxemi7nxtbliqzobcuk3cswmnaxgbkohrgrzptsqfkmn1lmgeol7ykat9n/gymrrhiywwxezmisz0apywpwbsozt/yz6d1mzfalzc0xaroajsmqxvvshsujsijofisjwkbwhhqecm0oiikfhgpuv4lpighcwczztbl5y4msav+ifl13791k42yevggdofnurr66ra10h5rirxrn0ct6q+/oi/phfdpfm+mam98po4vyvv8aevshtg==</latexit> <latexit sha1_base64="m6axfmcurf5oagaaj1n7fjbneki=">aaacgniczvdlsgmxfm3uv62vapdugkwoumqmcoqu6mzlbwslbs2znnogjpmhusouyt5f3op3ubk3bvwm/8bm24vtl4r7cu654et4keagxpfhya2tb2xu5bclo7t7+wffw6nho2jnwzmqoxtbj4yjhrimcbcshwlgpc9yyx/fzvpwm9ogq/abjhhrstimecapauv1i6wul5nr2k94fev0kal4z2m/whzr7rtwkvdmoizm1egxf7sdrwpjqqccgnpx3ah6cdhaqwbporsbfhe6jkpwstakkplemjwf4lpldhcgtd0h4cn7fymh0kgci6vmmlmyzqwojuzvqmaks5y9m72bifsrvqxmim0cs2qegqtewsmobhbsmy8gfhguzolca64zbtgxgfdn7vcwhrfnkng4czyjbzmrvda8r13x3pulcv1mhlyehamtveeeukr1dicaqikomqbx9ibenrfnw/l0vmbsndpfkagfcr7/aiyqovg=</latexit> = + <latexit sha1_base64="j710iawjldt3kcch2yc+lhtgmcu=">aaace3iczvbnswmxem3wr7p+vt16wsych1k2xtrdsejfywvrc+1astlsg5pslmrwkev/hnhv/4ynetx5fn+g/8bs24ntb8k8vhkzzdw/5kyd6/5yuaxlldw1/lq9sbm1vvpy3bvtmlgenojkurv8rclnew0aa05bsajy+jw2/cfvvm8+ukwzjg5hgfnp4f7eqkywgoq+40se6kewka2ouowiw3bh4sycyhqulz7tavz6bde7hd9oiekiaasey63bftcgl8ukgof0zhcstwnmbrhh2wzgwfdtpeotr86ryqinlmq8cjwx+78jxuildh2jzjkeqwumsml1yaigl7kujrn/s4nkvihliqvdpbcihgdeyqi4arqryr5hwh2qtuaqezbfcfchazgozk5xsb8rtmd4abupkvoolilgsfm87n64xdolmkqehabddiwq6btv0dwqowyisken9ixerefrzxq3pibsndxt2uczyx39asohoxk=</latexit> <latexit sha1_base64="tt7shfqoco2k4yfwnueei7nc9mq=">aaace3iczvdnsgmxej6tf7x+vt16wsych1k2xtrdsejfywvrc20t2ttbhiabjzkvyllwkcsrpocnetz5ad6gb2c29wdbgtbfvvlmmpn8shcdnvftzjawv1bxsuu5jc2t7z387t6dubgmre6vulrpe8med1kdoqrwjdqj0hes4q+v0nrjgwndvxilo4h1jomhpocuokxu274sptosniwvctdf8erejnxfup4dhyvpxouragrd/e+7p2gswyhuegnazs/ctki0ciryoneodysihzi+a1kyeslmj5lspxaplnnza6xtc9gdsp87eiknjdiwyjszmvrkoflcfk0kbzjn6zjjpz2o6mtiktimmholyhdwsxgyxchcot0jiiwlyk0dcntcm4pizaghmtttxdogmlc0puasr+v5rxzb/ar0xvjuvel1eqarhqm4hgmowylu4rpquackgp7hbv6dj+fnexc+ptkm89ezdzphfp0cdochpq==</latexit> xi g... <latexit sha1_base64="ytthjcjf7thl3c1aric7iyn4ixu=">aaacdxiczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny4rolbqdiwtztrqzdikd8qy9b/erx6hk3hrn/gz/ogz6sxseyhck3ppdscniau34lo/zsrq2vrgzmmrvl2zu7dfoth8ncrrlhlucau7atfm8ih5wegwtqwzkyfg7wb8m83bt0wbrqihmmtml2qy8zbtapbq9akzpk/7vf+punu3l7wmggwooqja/cpvb6boilkevbbjug03bj8lgjgvbfrujybfhi7jkhutjihkxk9zv1n8apkbdpw2jwkcs/83uiknjdcyyqyzuvnggflc1kwkrjjr2tp53uxkuatklrnpe5ofixbe+smp4gryrgc+wkrgudjlbg+4zhtexajcnbdfwxrenkfgeyzbjbqliswd77x+xxfvl6rnmyksejpgj+gmndalaqi71eieokigv/sg3p0x58p5dl5m0hwn2dlcc+v8/wghz507</latexit> xi,l <latexit sha1_base64="vwq6y+pohkbnqh+fpepzilaedao=">aaaceniczvdltgixfo3gc/gfunttsexctmhgtnqd0y1lteriyei6pqmn7xrs7xjjzd7dunxvcgxc+gn+hn9gb1gi3ks5p+ee25yeibbcgof9oiwv1bx1jejmawt7z3evvh/wyfsikwtsjzrub8qwwspwba6ctwpniaweawwjm3zeemlacbxdwzhmvisdiiecercu3w1k+pz1uu5ikfxkfa/qtqovg9omvncsgr3yb7evacjzbfqqyzo1lwy/jro4fswrdrpdykjhzma6fkzemuone9mzprfmh4dk2xmbnrd/n1iijsqwtmq8mblzzobswrhwbuozt/yzyd2mzeag0s1f2ormwqiel37kozgbftgpjzargbtoa8j9rhkfmbaaum3tvzadek0o2bhlnqpayillohlwvap6d+ev+vusrci6qsfofnxqbaqjw9ratutri3pfb+jdexe+ne/nayotolodqzrxzvcffosfhq==</latexit> 1 1 <latexit sha1_base64="9nq30mfpfbgzwoip8ja7qqu/idq=">aaace3iczvdnssnaen7uv1r/qh69bisgweoqgnorevfywdhcg8tms2mx7mbd7kqooa8hxvu5pilxh8dh8a3cpdnydmczb7/5zpj5/jgzdy7zy5vwvtfwn8qbla3tnd296v7bo5ajitqlkkvv9bgmnexubqacdmnfsfa57fjj26zeeazkmxk9wcsmnsddiiwmydduu9+xpnatyvj6nh1ua07dycnebs0c1far7uh1tx9ikggaaefy617ticflsqjgoj1w+ommmszjpkq9aymsqpbsfoupfwkywa6lmi8co2f/d6ryaifhzjrz0no1jaepua4bfyxelrix+t87qo6leizsotqli0b45ausihogezntesbcbmlndtkbu5qanxiaiwlmfjumsmieji8v41fz0zfl4j43rhvo/uwtdvoyvuzh6bidoia6rc10h9rirqqp9ire0lv1yn1yn9bxtfqyip5dnbfw9x9lxz/k</latexit> <latexit <latexit sha1_base64="jhj0nqdq+87fbqs5prps5rhlnou=">aaacbniczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny5bcgyhhuomzbshywri7ghl6aeiw/0ov+lw3/az/amz01ny9kk4j+eeg05oeatuwhv/nlx1jc2t7djoexdv/+cwcnt8zfsikfooekp3a2ky4bhzging3vgzigpboshkppt3npk2xewpmi2zl8ko4ignbczvhg0qvbfu5ovxqamavvrua1d57q8vtsslgapitk/hxucnraongs3k/cswmnajgbgehrgrzphpbnsgzy0zxkhs9ksac/b/rkqkkqtgvpk1szdlgfbkmjpvwvhmlxsmv5updgqbrguibukzzatcgz/luzwai+jcr5gidapnoeah14ycmfpaqob2k5ioisyubhrlm1fjozfv4f3wb+tu+6ravcvckqftdiyuuandoyz6qc3kiyoyekvv6n15ct6ct+drll1zip0ttfdo9x+yv5n5</latexit> 1 xi,r <latexit sha1_base64="nykeauiuyivai+ip12gc3twldk8=">aaaceniczvdltgixfo3gc/gfunttsexctmhgtnqd0y1lteriyei6pqmn7xrs7xjjzd7dunxvcgxc+gn+hn9gb1gi3ks5p+ee25yeibbcgof9oiwv1bx1jejmawt7z3evvh/wyfsikwtsjzrub8qwwspwba6ctwpniaweawwjm3zeemlacbxdwzhmvisdiiecercu3w1k+pz1uu5infxkfa/qtqovg9omvncsgr3yb7evacjzbfqqyzo1lwy/jro4fswrdrpdykjhzma6fkzemuone9mzprfmh4dk2xmbnrd/n1iijsqwtmq8mblzzobswrhwbuozt/yzyd2mzeag0s1f2ormwqiel37kozgbftgpjzargbtoa8j9rhkfmbaaum3tvzadek0o2bhlnqpayillohlwvap6d+ev+vusrci6qsfofnxqbaqjw9ratutri3pfb+jdexe+ne/nayotolodqzrxzvcfifafiw==</latexit> xi 0 1 1 1 <latexit sha1_base64="qlouskqawl0c7ub9u7+xlyfg6ya=">aaacdxiczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiicuxip3sgoz1o2jtgmuefjfv9bbeujcstv8gn/+af2blycnykuafnntuchj/itipjffu5ldw19y38zmfre2d3r7h/ckdlrah1ierstx2skwchdyebp+1iusx8tlv+6dqdt+6p0kygtzcoqcfwigqbixgm1e76inmy9fivwhiqtlb2mqjoqkl21pxhr/xprq/42+1legsaaufy607vicblsajgoj0uurgmesyjpkada0msqpaszo/epjvm3w6kmiceo2p/byryaifhajrp03ozlaepus4bfqxf2tjnsrsec7/si3iqujrqc0ygupqsfkyx0jbmfqqxt0haatz2nylkgi8nweqx8xwbdlhcbeycbznrdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioi4eklp6mv6tn6sd+tjks1zs51dnffw1x/s7ady</latexit> sha1_base64="cabdscof/szjsejxwej+zulgkro=">aaacdxiczvdltgixfl2dl8qxaulgzuri4okqwy26i7pxcykicrdsla40tnnje8dijvydcavf4cq49rvc+a/+gr1gixct5p6ee25zevxicioe9+1kvlbx1jeym7mt7z3dvfz+wb1rsaastpvquuktwwqpwr05ctamncpsf6zhd2/seeoracnveiejihuk6yc84jsgpzptxyzp4y7v5gteyzuuuwzkm1cohnv+hgcg2s3/tnukxpkfsauxplx2iuwkrcongo1z7diwinah6bowhsgrzhssid+xe2qznhsobu+i7ot9v5eqastbgvwmzcznugaveqzovtiqauufmdznsppfxxztktabwtccwwun4weuiwvp1ecqcxevm2bj9rhmfmxiaki1t19x6yboqtemmlmzlrctwqb189jvyavzrk5hwlk4hhm4gzjcqavuoqp1ocdgbv7hzxl23p0p53mqztiznuoyk+frd02knxw=</latexit> [Dinh et al. 2014] 39
Learning with Inverse Projection ei <latexit sha1_base64="n3cjhtjw9tlldu5m9ej8nnibiqk=">aaacdniczvdltgixfo3gc/gfmrhxm5gyucbkckpucg5cquiibiakuzrq0e4n7r0tmuejjfv9bneujbsx/oib/8e/sdowelhjc0/ppbc5px7emqbh+bzyk6tr6xv5zclw9s7uxnh/4e7lwbhqesmlavtyu85c6gidtturolj4nlb80xu6bz1qpzkmb2ecuu/gqcgcrjay6r7ri4roeqzqk5acipovvqyqm1cqhtv/2gv9s9er/nb7ksschka41rptdslweqyaeu4nhw6saytjca9ox8aqc6q9jdm8su8n07cdqcwjwc7y/xsjflpggbpl2vtclgvasq7lrgvdkbb0meyux8iv+6kcipqo9iircc69hivrddqkux9bzg2qdhqo3wekeubjazbrzhzfjkosmaetyzprdtgrzecev64qttnkvuftyqnjdiloubvdobq6qq3kioieeklp6mv6tn6sd+tjks1zs51dnffw1x/syqdz</latexit> sha1_base64="mrqipjqbeebmku3laa4lyqylndy=">aaacdniczvdltgixfl2dl8qxaulgzuri4okqwy26i7pxcykibgh2sgca2umkvwncjnyecavf4cq49rfc+a/+gr1gixct5p6ee25zevxicioe9+1kvlbx1jeym7mt7z3dvfz+wb1rsaastpvquuktwwqpwr05ctamncpsf6zhd2/seeojacnveiejihuk6yc84jsgpr7avkzyumtz3xzbk3mtcpdbeqyklapazymavlv533zp0viyekkgxrtkxosdhgjkvlbxrh0bfhe6jh3wsjakkplomje8dk8t03mdpe0j0z2w/zcsio0koldktjm5wcqgusiurqohmm3pm5o7gum/6mtiktimmatgmljsjdymymqhnfoiyugictnw3b7xjkiywuco5vyrlh0qtsjacnomyoujlip6eemq5nvsvtcwrswcwwmcqrkuoak3uiu6ujdwaq/w5jw7786h8zmvzpzzzihmlfp1b2cznx0=</latexit> ei,r ei,l <latexit 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sha1_base64="ilr7g1a/jbsohtyqp1jeuivghq4=">aaaceniczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny4rolbqdiwtztrqzdimd4qyzgeiw/0ov+lwh/az/amz0y5seyhck3ppdscniau34lo/zsrq2vrgzmmrvl2zu7dfoth8ncrrlhlucau7atfm8ih5wegwtqwzkyfg7wb8m8/bz0wbrqihmmtml2qy8zbtapbye4fmwdzpeq3rrf+punw3klwmgjnqrbnq9su/vygiiwqruegm6tbcgpyuaobuskzcswylcr2tietagbhjjj8wpjn8apkbdpw2jwjcsp83uiknjdcyyryzuvnogflc1kwkrjjv+tpf3uxkuatklrdpe5ofixbe+smp4gryrkc+wkrgudgpca+4zhtexajcnbdfwxrenkfgyyzbjbqliswd77x+xxfvl6rnm1lyjxsmttazaqbl1er3qiu8rnetekvv6n15ct6ct+drkl1xzjthak6c7z8auj8q</latexit> <latexit sha1_base64="ga/1hto7lilujmrbyffquanefxa=">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</latexit> sha1_base64="zndhdgiburmtjyqp3+wy268rqpk=">aaacgniczvbntwixej31e/al5ehlizhxqmjirb0rvxjexbusingtxwhot5t21ors+ceejff9hz6mvy/+dp+bxeagmekzr2/etdmviau36hk/ztr6xubwdi5f2nnd2z8ohh49gjvoynyqhnktgbgmemr85chyk9amyecwzjc6yernj6ynv9e9jmpwlwqq8zbtgpbqfuudqim+guub0g5yycykvyx7vw8a7iqozug5no+fhwgg0sv+dvqkjpjfsauxpl3zyuymrcongk0kncswmnargbc2hrgxv3tt6fat99qyftdu2p4i3sn7vyml0kicq6vmklmozqwqjuzfqnaos5q9m71nw1ucwcle2ormaraml7spj+iewurnc4sjcfg5mvfun2tguywtifrzu4plh0qtitbogvwotuzikvdpq1dv7856dq2zymexnmaz1oac6naldfcbwhhe4q3enrfnw/l0vmbsnwfeu4kfcl7/aonopc4=</latexit> (1) hi,l = xi,l <latexit sha1_base64="ecox4vbbnenfsay0sy5zwnwjfdg=">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</latexit> + = <latexit sha1_base64="j710iawjldt3kcch2yc+lhtgmcu=">aaace3iczvbnswmxem3wr7p+vt16wsych1k2xtrdsejfywvrc+1astlsg5pslmrwkev/hnhv/4ynetx5fn+g/8bs24ntb8k8vhkzzdw/5kyd6/5yuaxlldw1/lq9sbm1vvpy3bvtmlgenojkurv8rclnew0aa05bsajy+jw2/cfvvm8+ukwzjg5hgfnp4f7eqkywgoq+40se6kewka2ouowiw3bh4sycyhqulz7tavz6bde7hd9oiekiaasey63bftcgl8ukgof0zhcstwnmbrhh2wzgwfdtpeotr86ryqinlmq8cjwx+78jxuildh2jzjkeqwumsml1yaigl7kujrn/s4nkvihliqvdpbcihgdeyqi4arqryr5hwh2qtuaqezbfcfchazgozk5xsb8rtmd4abupkvoolilgsfm87n64xdolmkqehabddiwq6btv0dwqowyisken9ixerefrzxq3pibsndxt2uczyx39asohoxk=</latexit> <latexit sha1_base64="tt7shfqoco2k4yfwnueei7nc9mq=">aaace3iczvdnsgmxej6tf7x+vt16wsych1k2xtrdsejfywvrc20t2ttbhiabjzkvyllwkcsrpocnetz5ad6gb2c29wdbgtbfvvlmmpn8shcdnvftzjawv1bxsuu5jc2t7z387t6dubgmre6vulrpe8med1kdoqrwjdqj0hes4q+v0nrjgwndvxilo4h1jomhpocuokxu274sptosniwvctdf8erejnxfup4dhyvpxouragrd/e+7p2gswyhuegnazs/ctki0ciryoneodysihzi+a1kyeslmj5lspxaplnnza6xtc9gdsp87eiknjdiwyjszmvrkoflcfk0kbzjn6zjjpz2o6mtiktimmholyhdwsxgyxchcot0jiiwlyk0dcntcm4pizaghmtttxdogmlc0puasr+v5rxzb/ar0xvjuvel1eqarhqm4hgmowylu4rpquackgp7hbv6dj+fnexc+ptkm89ezdzphfp0cdochpq==</latexit> <latexit sha1_base64="3duua+im21knd6mo/uljro+f6gy=">aaacc3iczvdltgixfo3gc8cx6tlnrgligpdbjbowet24xmqbeickuzpqaaet9o4jmfalxq1u/qvxxq3xe4yf4r/ygvgi3ks5p+ee25wep+jmg+v+wlml5zxvtfy6vbg5tb1t2n2raxkrqj0iuvrnh2vkwug9ymbpm1iuc5/thj+8tuenb6o0k+edjclaebgfsoardiaqt0lpgu4wim7zzcpzbjupkf5+2rfr67dd6xz+2z1jykfdibxr3aq4exqsriartsd2o9y0wmsi+7rlyigf1z0kczt2jgztcwkpzanbydj/gwkwwmayggxa9mwszubkrktgbqortvsz7k5hwi/5opskla70nbeizjojc6myaegmpokyoycdnbmnxxqlwecgykky+ypdblhhaiy/22rumu9kexgn5foye+swq1dounl0ga7rmaqgu1rfn6igpetqpxpcz+jferterhfryylnwdodftrt1ucfeoefeg==</latexit> sha1_base64="n3mwcm7/bmwunjxbig9vkyiruu0=">aaacc3iczvdltgixfl2dl8qx6tlnrgligpdbjbowet24xmqbeickuzpqaaet9o4jisr+gxgr3+hkudv+g5/hh9gbfwi3ae7puec2pyeibtfoed9ozml5zxutu57b2nza3snv7twmsjrlplvc6uzadbm8yj5yfkwra0zkifg9gfyn8/od04ar6a6hmwtl0ot4yclbs9vatkvqdpifr+rnyl0e5t9qupzkxtwcqlwt/2l1fu0ki5akykyz7mxyhhgnnao2zrusw2jcb6thmhzgrdlthk3cjt0jy3tdugl7inqn7p+nezfgeuxbzdrmzcxluclhilaffzm29jnj3qxluaxkmrvpe5o5ixietuc8ihnkez36cbphonltznwu14yigfpaqob2ky7te00o2vxynqpyfcklwd8pnze8w69quyjpzeeadueyynakfbibkvha4r6e4qvensfnzxl3pqbsjpo3sw8z5xz+al1gngy=</latexit> (2) hi,l <latexit 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sha1_base64="rekaecskl6vtpqb6kre5yd3huvc=">aaacdxiczvdlsgmxfm3uv62vqks3wsk4kguqgrorunfzwbgfdiiznnogjpmhusouof8gbvu7xilbv8hp8a/mtgdh2wvhnpx7bjg5qsy4adf9cupr6xubw+xtys7u3v5b9fdoyaheu+zrjztubsqwwspmaqfburfmraacdyljxtbvpdntuioeyrozx5jrxenocviq2w9kymydpqjw3iabf14fzqluufhtqfw3p1q0kswckogxvaybg58sdzwknqv0e8niqidkxhowrkqy46e53xk+s8wqh0rbewho2f8bkzfgehhbzdbmwixjqclh6lyfy5m17jn8bqyyqaeynom0cc2seqiv/zrhcqisonmfysiwkjxlg4dcmwpiagghmtuvydommlcwcvzsrs3lrfabd9g4abgpl7xwbrfwgz2gu3somugktda9aimpustqk3pd786l8+f8ol9zackpdo7rqjnff2gonsg=</latexit> (1) ei <latexit sha1_base64="9nq30mfpfbgzwoip8ja7qqu/idq=">aaace3iczvdnssnaen7uv1r/qh69bisgweoqgnorevfywdhcg8tms2mx7mbd7kqooa8hxvu5pilxh8dh8a3cpdnydmczb7/5zpj5/jgzdy7zy5vwvtfwn8qbla3tnd296v7bo5ajitqlkkvv9bgmnexubqacdmnfsfa57fjj26zeeazkmxk9wcsmnsddiiwmydduu9+xpnatyvj6nh1ua07dycnebs0c1far7uh1tx9ikggaaefy617ticflsqjgoj1w+ommmszjpkq9aymsqpbsfoupfwkywa6lmi8co2f/d6ryaifhzjrz0no1jaepua4bfyxelrix+t87qo6leizsotqli0b45ausihogezntesbcbmlndtkbu5qanxiaiwlmfjumsmieji8v41fz0zfl4j43rhvo/uwtdvoyvuzh6bidoia6rc10h9rirqqp9ire0lv1yn1yn9bxtfqyip5dnbfw9x9lxz/k</latexit> <latexit <latexit sha1_base64="cjnetq2nhpm5e1hwblqizx9c5vc=">aaacgniczvdlsgmxfm34rpu12qwbybeqlditgrorunfzwdpcw0smzbshywri7ghlme8rt/odrsstgz/dpzdtdmhbc+genhtuodl+jlgbz/tx1ty3nre2czv53b39g0p36pjrqfht1qrkkn32iwgch6wjharrr5or6qvw8se32bz1zlthknyascr6kgxdhnbkwfj9t9d1ztjk+wkvy5e+jaxaedp3i17fmxzebdu5kkj5nfrub3egacxzcfqqyzpvl4jeqjrwklia78agrysoyzb1layjzkaxtm2n+mwyaxwobu8iemr+30iinjlaycqzzhzmgqnkcvo2khjjrgxpto9miv2yl8uzsjvalbmb4kqx8dckgyv05ioibqafs6dwggtgquwsifrz+xvmr0qtcjbovm2oupzikmjwktcv7/6iwl+zh5vdj+gulvavxai6ukmn1equtdarekpvzovz4xw6xzppmjpfkacfcr7/ah3yovm=</latexit> hi,r = xi,r + g(xi,l ) <latexit sha1_base64="gqpvuajfnzzo3nggi6xcpttqyce=">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</latexit> (2) hi,r <latexit sha1_base64="pld+o/nimrkoifavqzrglso/hiw=">aaacgniczvdlsgmxfm34rpu12qwbybeqlditgrorunfzwdpcw0smzbshywri7ghlme8rt/odrsstgz/dpzdtdmhbc+genhtuodl+jlgbz/tx1ty3nre2czv53b39g0p36pjrqfht1qrkkn32iwgch6wjharrr5or6qvw8se32bz1zlthknyascr6kgxdhnbkwfj9t9d1ztjk+wkvy50+jaxaedp3i17fmxzebdu5kkj5nfrub3egacxzcfqqyzpvl4jeqjrwklia78agrysoyzb1layjzkaxtm2n+mwyaxwobu8iemr+30iinjlaycqzzhzmgqnkcvo2khjjrgxpto9miv2yl8uzsjvalbmb4kqx8dckgyv05ioibqafs6dwggtgquwsifrz+xvmr0qtcjbovm2oupzikmjwktcv7/6iwl+zh5vdj+gulvavxai6ukmn1equtdarekpvzovz4xw6xzppmjpfkacfcr7/aifoovk=</latexit> + Inverse f <latexit 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sha1_base64="wkf9utthcqfbfvs3vq9hzy9dmzu=">aaacbniczvdltgixfl3je/gfuttfrgligpdbjbojunejismkqeindkchnu7aoyzkwgcyt/odrowu/q13/oj/yadycnykuafnntucniaw3kdnftsrq2vrg5u5rfz2zu7efuhg8mgorfpmuywubgbemmej5inhwzqxzkqggjwc4w02bzwybbik7neus44k/yihnbk0vd3sfope2zuuuwwqm1csnnzwfwcg1i38tnukjpjfsauxplxxyuykrcongo3z7cswmnah6bowhrgrzhtsidgxe2aznhsqbu+e7ot9v5esastbgvvmzcznmgaveqzkvtiqwcuemdznsaalqjyyktahwtcc4vun5vgciivo1eeycbevm4xi9rhmfmxiaki1t19x6yboqtfgl7czvrytwqb+rfm67nvtvjcwrrwcwymcqwuuoqp3uamfkdb4hhd4dz6cn+fd+zhkv5zzzhhmlfp1b+ntndk=</latexit> f <latexit sha1_base64="fjhixwws58gsuvpwxj5bws9qhg8=">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</latexit> sha1_base64="t5o2w5fdbe9yvbrkg7tbseu/cmg=">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</latexit> 1 (xi ) = <latexit sha1_base64="j710iawjldt3kcch2yc+lhtgmcu=">aaace3iczvbnswmxem3wr7p+vt16wsych1k2xtrdsejfywvrc+1astlsg5pslmrwkev/hnhv/4ynetx5fn+g/8bs24ntb8k8vhkzzdw/5kyd6/5yuaxlldw1/lq9sbm1vvpy3bvtmlgenojkurv8rclnew0aa05bsajy+jw2/cfvvm8+ukwzjg5hgfnp4f7eqkywgoq+40se6kewka2ouowiw3bh4sycyhqulz7tavz6bde7hd9oiekiaasey63bftcgl8ukgof0zhcstwnmbrhh2wzgwfdtpeotr86ryqinlmq8cjwx+78jxuildh2jzjkeqwumsml1yaigl7kujrn/s4nkvihliqvdpbcihgdeyqi4arqryr5hwh2qtuaqezbfcfchazgozk5xsb8rtmd4abupkvoolilgsfm87n64xdolmkqehabddiwq6btv0dwqowyisken9ixerefrzxq3pibsndxt2uczyx39asohoxk=</latexit> <latexit sha1_base64="tt7shfqoco2k4yfwnueei7nc9mq=">aaace3iczvdnsgmxej6tf7x+vt16wsych1k2xtrdsejfywvrc20t2ttbhiabjzkvyllwkcsrpocnetz5ad6gb2c29wdbgtbfvvlmmpn8shcdnvftzjawv1bxsuu5jc2t7z387t6dubgmre6vulrpe8med1kdoqrwjdqj0hes4q+v0nrjgwndvxilo4h1jomhpocuokxu274sptosniwvctdf8erejnxfup4dhyvpxouragrd/e+7p2gswyhuegnazs/ctki0ciryoneodysihzi+a1kyeslmj5lspxaplnnza6xtc9gdsp87eiknjdiwyjszmvrkoflcfk0kbzjn6zjjpz2o6mtiktimmholyhdwsxgyxchcot0jiiwlyk0dcntcm4pizaghmtttxdogmlc0puasr+v5rxzb/ar0xvjuvel1eqarhqm4hgmowylu4rpquackgp7hbv6dj+fnexc+ptkm89ezdzphfp0cdochpq==</latexit> <latexit sha1_base64="9nq30mfpfbgzwoip8ja7qqu/idq=">aaace3iczvdnssnaen7uv1r/qh69bisgweoqgnorevfywdhcg8tms2mx7mbd7kqooa8hxvu5pilxh8dh8a3cpdnydmczb7/5zpj5/jgzdy7zy5vwvtfwn8qbla3tnd296v7bo5ajitqlkkvv9bgmnexubqacdmnfsfa57fjj26zeeazkmxk9wcsmnsddiiwmydduu9+xpnatyvj6nh1ua07dycnebs0c1far7uh1tx9ikggaaefy617ticflsqjgoj1w+ommmszjpkq9aymsqpbsfoupfwkywa6lmi8co2f/d6ryaifhzjrz0no1jaepua4bfyxelrix+t87qo6leizsotqli0b45ausihogezntesbcbmlndtkbu5qanxiaiwlmfjumsmieji8v41fz0zfl4j43rhvo/uwtdvoyvuzh6bidoia6rc10h9rirqqp9ire0lv1yn1yn9bxtfqyip5dnbfw9x9lxz/k</latexit> <latexit (1) hi,l <latexit sha1_base64="jhj0nqdq+87fbqs5prps5rhlnou=">aaacbniczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny5bcgyhhuomzbshywri7ghl6aeiw/0ov+lw3/az/amz01ny9kk4j+eeg05oeatuwhv/nlx1jc2t7djoexdv/+cwcnt8zfsikfooekp3a2ky4bhzging3vgzigpboshkppt3npk2xewpmi2zl8ko4ignbczvhg0qvbfu5ovxqamavvrua1d57q8vtsslgapitk/hxucnraongs3k/cswmnajgbgehrgrzphpbnsgzy0zxkhs9ksac/b/rkqkkqtgvpk1szdlgfbkmjpvwvhmlxsmv5updgqbrguibukzzatcgz/luzwai+jcr5gidapnoeah14ycmfpaqob2k5ioisyubhrlm1fjozfv4f3wb+tu+6ravcvckqftdiyuuandoyz6qc3kiyoyekvv6n15ct6ct+drll1zip0ttfdo9x+yv5n5</latexit> g (1) hi,r <latexit sha1_base64="bsenpziye1ec45vcyn+ot1hadn0=">aaacgniczvdlsgmxfm3uv62vapdugkwoumqmcoqu6mzlbwslbs2znnogjpmhusouyt5f3op3ubk3bvwm/8bm24vtl4r7cu654et4keagxpfhya2tb2xu5bclo7t7+wffw6nho2jnwzmqoxtbj4yjhrimcbcshwlgpc9yyx/fzvpwm9ogq/abjhhrstimecapauv1i6wul5nr2k94fyv0kal4z2m/whzr7rtwkvdmoizm1egxf7sdrwpjqqccgnpx3ah6cdhaqwbporsbfhe6jkpwstakkplemjwf4lpldhcgtd0h4cn7fymh0kgci6vmmlmyzqwojuzvqmaks5y9m72bifsrvqxmim0cs2qegqtewsmobhbsmy8gfhguzolca64zbtgxgfdn7vcwhrfnkng4czyjbzmrvda8r13x3pulcv1mhlyehamtveeeukr1dicaqikomqbx9ibenrfnw/l0vmbsndpfkagfcr7/ahw0ovi=</latexit> sha1_base64="d20bzpfnqohstvpbd1jyhfyujwm=">aaab93iczvdlsgmxfm3uvx2r1rwbybfcldj1o+4eny4rolbqdiwtybsheqzjhaemxqtu/q5x4sf4gf6bmwkxtr0q7sm554ate2ecwwich6+2s7u3f1a/9i8a/vhjabpxynvukaupftomymkz4iqfwegwqwyykbfg/xj2um77r8xyrtuzzdmwstjrpowugkn642yr6arv4w3qxyewwtw4+ttknm0lu0afsxbydtkicmkau8ew/ii3lcn0rizs6kaiktmoqgwu8kvjepxq444cxlh/nwoirsqwdcqy2bvzyydwwradcqaybouz1d3ozdyozbsugzvadsoq3kyfv1kotngljzqxgdqui8ejn4ycmdtaqohuk5hoiseuxhc+i6i7gcg2ck87d53gkub1di4u0bxqoht0jx5rd4wiogs9ow/vzfv0vpzj1rxvpgdorbzvp0dvlyg=</latexit> sha1_base64="uqjbtkil1ol91wcjvitdnrmxj54=">aaacd3iczvdlsgmxfl3js9aqvzdugkvqkgxgjbot3lisyg2hrswtztpgmhmso0iz5lperx6hk/el/az/wezbhbyxwj0599xwcsjecou+/+2trk6tb2ywtsrblz3dvep+5chq1ddeylpq0wmp5vleviucje8khlmvst4on26kefuzgyt0fi+thpcvhcuieoyiowbvw16osne+yesdypwxow3o8kg15jf8azflemxbdebvhfr/ekpnusvjzjja2w38bpsznsiy5hm5l1qeupzer7zryewvt/1saj4nj44zkkgbd2iku/bvrkavvrthtlk0+29wmki1thwnwreqwvhm9g4nkqyhql6iji3sghgmlvuzijmuecxmpqjuetskciomheem5cqbyoxwxyfsta1l6oisu4ycxuswqeu8cdxw73wowrecwykecahxcatnaagdcbzcg7x7l96h9zklc8wbp3oa/8r7+gwmo5/n</latexit> sha1_base64="3qhylx9qlp+05qt4pk5toemfbus=">aaacgniczvdltgixfo34rhyhln00ehnmcjlxo+6iblxi4ggjiomudjs000l7x4rm5lomw/0ov8atgz/dp7adlaru0tztc89ttk8qc27adx+ctfwnza3twk5xd2//4lb0dpxovkip86kssrcdypjgefobg2dtwdmia8fawfg2n7eemtzcrq8wivlpkmheq04jwkpfkncdmy6yfsprwgrpadu7z/qlilt3p4vxgtchftsvzr/02x0omkgwarxemi7nxtbliqzobcuk3cswmnaxgbkohrgrzptsqfkmn1lmgeol7ykat9n/gymrrhiywwxezmisz0apywpwbsozt/yz6d1mzfalzc0xaroajsmqxvvshsujsijofisjwkbwhhqecm0oiikfhgpuv4lpighcwczztbl5y4msav+ifl13791k42yevggdofnurr66ra10h5rirxrn0ct6q+/oi/phfdpfm+mam98po4vyvv8aevshtg==</latexit> <latexit sha1_base64="m6axfmcurf5oagaaj1n7fjbneki=">aaacgniczvdlsgmxfm3uv62vapdugkwoumqmcoqu6mzlbwslbs2znnogjpmhusouyt5f3op3ubk3bvwm/8bm24vtl4r7cu654et4keagxpfhya2tb2xu5bclo7t7+wffw6nho2jnwzmqoxtbj4yjhrimcbcshwlgpc9yyx/fzvpwm9ogq/abjhhrstimecapauv1i6wul5nr2k94fev0kal4z2m/whzr7rtwkvdmoizm1egxf7sdrwpjqqccgnpx3ah6cdhaqwbporsbfhe6jkpwstakkplemjwf4lpldhcgtd0h4cn7fymh0kgci6vmmlmyzqwojuzvqmaks5y9m72bifsrvqxmim0cs2qegqtewsmobhbsmy8gfhguzolca64zbtgxgfdn7vcwhrfnkng4czyjbzmrvda8r13x3pulcv1mhlyehamtveeeukr1dicaqikomqbx9ibenrfnw/l0vmbsndpfkagfcr7/aiyqovg=</latexit> = + <latexit sha1_base64="j710iawjldt3kcch2yc+lhtgmcu=">aaace3iczvbnswmxem3wr7p+vt16wsych1k2xtrdsejfywvrc+1astlsg5pslmrwkev/hnhv/4ynetx5fn+g/8bs24ntb8k8vhkzzdw/5kyd6/5yuaxlldw1/lq9sbm1vvpy3bvtmlgenojkurv8rclnew0aa05bsajy+jw2/cfvvm8+ukwzjg5hgfnp4f7eqkywgoq+40se6kewka2ouowiw3bh4sycyhqulz7tavz6bde7hd9oiekiaasey63bftcgl8ukgof0zhcstwnmbrhh2wzgwfdtpeotr86ryqinlmq8cjwx+78jxuildh2jzjkeqwumsml1yaigl7kujrn/s4nkvihliqvdpbcihgdeyqi4arqryr5hwh2qtuaqezbfcfchazgozk5xsb8rtmd4abupkvoolilgsfm87n64xdolmkqehabddiwq6btv0dwqowyisken9ixerefrzxq3pibsndxt2uczyx39asohoxk=</latexit> <latexit sha1_base64="tt7shfqoco2k4yfwnueei7nc9mq=">aaace3iczvdnsgmxej6tf7x+vt16wsych1k2xtrdsejfywvrc20t2ttbhiabjzkvyllwkcsrpocnetz5ad6gb2c29wdbgtbfvvlmmpn8shcdnvftzjawv1bxsuu5jc2t7z387t6dubgmre6vulrpe8med1kdoqrwjdqj0hes4q+v0nrjgwndvxilo4h1jomhpocuokxu274sptosniwvctdf8erejnxfup4dhyvpxouragrd/e+7p2gswyhuegnazs/ctki0ciryoneodysihzi+a1kyeslmj5lspxaplnnza6xtc9gdsp87eiknjdiwyjszmvrkoflcfk0kbzjn6zjjpz2o6mtiktimmholyhdwsxgyxchcot0jiiwlyk0dcntcm4pizaghmtttxdogmlc0puasr+v5rxzb/ar0xvjuvel1eqarhqm4hgmowylu4rpquackgp7hbv6dj+fnexc+ptkm89ezdzphfp0cdochpq==</latexit> xi g... <latexit sha1_base64="ytthjcjf7thl3c1aric7iyn4ixu=">aaacdxiczvdlsgmxfm34rpvvdekmwaqxpuxfuhdfny4rolbqdiwtztrqzdikd8qy9b/erx6hk3hrn/gz/ogz6sxseyhck3ppdscniau34lo/zsrq2vrgzmmrvl2zu7dfoth8ncrrlhlucau7atfm8ih5wegwtqwzkyfg7wb8m83bt0wbrqihmmtml2qy8zbtapbq9akzpk/7vf+punu3l7wmggwooqja/cpvb6boilkevbbjug03bj8lgjgvbfrujybfhi7jkhutjihkxk9zv1n8apkbdpw2jwkcs/83uiknjdcyyqyzuvnggflc1kwkrjjr2tp53uxkuatklrnpe5ofixbe+smp4gryrgc+wkrgudjlbg+4zhtexajcnbdfwxrenkfgeyzbjbqliswd77x+xxfvl6rnmyksejpgj+gmndalaqi71eieokigv/sg3p0x58p5dl5m0hwn2dlcc+v8/wghz507</latexit> xi,l <latexit 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Experiments Dataset: English Penn Treebank POS tagging Trained and tested on whole PTB Grammar induction Trained on sentences of length <= 10 in section 2-21 Tested on sentences in section 23 41
Part-of-speech Induction 100 100 Many-to-1 % 75 50 Feature-based 80.2 80.8 75.4 62.7 HM Gaussia Neural VM 75 50 Feature-based 72.1 74.1 68.5 53.8 HM Gaussia Neural 42
Part-of-speech Induction 100 100 Many-to-1 % 75 50 Feature-based 80.2 80.8 75.4 62.7 HM Gaussia Neural VM 75 50 Feature-based 72.1 74.1 68.5 53.8 HM Gaussia Neural Outperform feature-based SOTA 42
Dependency Parse Induction Directed % on len <= 10 75 50 25 Oracle POS 49.6 DM 55.4 Gaussi 60.2 Neural Directed % on all 75 50 25 Oracle POS 35.8 DM 43.1 Gaussi 47.9 Neural 43
Original Embedding Space adjective adverb noun singular noun proper noun plural verb base verb gerund verb past tense verb past participle verb 3rd singular cardinal number 44
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sha1_base64="znhkautewogmapwollzuede8uty=">aaacqhiczve9swnbej3z2/gzw5vdifieclfro8hgmojrwqthbznnfvfj2j1tw3f/wvz/lb3/w0os3etsgdowzns3m8vbn0kmhamo+gwwfpewv1bx1msbm1vbo7v1zrtncsuxy4009i5hdqxq2cvbeu8yi0wlem+tx4uqfvue1gmjr2muyu+xby1swrlvvix90d9trk1ohoe8ae9ba6br6ded43hgek5qe5fmuft2lfgvyjyel1jw4txhxvgje8b7dzvt6hrfwgwzhnpmekbg+qmphln/jwqmngi09j1vcjo1iifjpgv6lhqqklxpjo9upjjmoppvk3wp+yee0tneixswe2o+0zhmmiqtvsaea2grkxx5wlgv/ishhzllohn7zkqkqqzvyo3p3cjf9kcix8oitltxupazpe549lx3uf3f03nqpw6dtakrcnzghw7gcnpwaudwcr3oaochvmibvacfwvfwpvnhqjddyx7mrpdzcypztby=</latexit> <latexit 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sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">aaacs3iczvbnswmxem1wq7v+txr0slged6vsrvbvohepclyl3vky6awn5mnjztwy7f/wqr/ku//dk3gw2/zg60cylzdvwsulesetbsgnv1palq+svtaq6xubw9u1+s6d1alh0gfaanonqaxbfxsqo4buyodksmb99hhzzo+fwfiu1s2oe+hl+qb4zbnfggphwae1rtakjux/b+0zajbzxq/q3le41cyvojajam2vhstyz6hbzgtk1tc1kfd2sb+g56ciemw/m5jn/qphdp1yg3cu+hp270zgpzuur05znds3kxjuwtimu+fifq14znk3yxk1i9ksrmbgdseixqf9jkskrvbs6inohy/al4lxh9waqzf2gdld3vd8nqkgmntxzzmizf6thgqemzasqmewvurzgmnsjc/nezjy4hixcnsx0/+gc9q6awu3x43zi1ncfbjh9skhazmtck6uydxpeezg5jw8kxfvw/vyvr2fqbtkzxz2yvyvyr/6pltn</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> Projected Embedding Space w/ Markov Prior z i e i x f 1 (e; ) <latexit 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sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> x i 45
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sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> Projected Embedding Space w/ Markov Prior z i e i x f 1 (e; ) <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> x i 45
Projected Embedding Space w/ DMV Prior adjective adverb noun singular noun proper noun plural verb base verb gerund verb past tense verb past participle verb 3rd singular cardinal number 46
Projected Embedding Space w/ DMV Prior adjective adverb noun singular noun proper noun plural verb base verb gerund verb past tense verb past participle verb 3rd singular cardinal number 46
Projected Embedding Space w/ DMV Prior adjective adverb noun singular noun proper noun plural verb base verb gerund verb past tense verb past participle verb 3rd singular cardinal number 46
Projected Embedding Space w/ DMV Prior adjective adverb noun singular noun proper noun plural verb base verb gerund verb past tense verb past participle verb 3rd singular cardinal number 47
Projected Embedding Space w/ DMV Prior adjective adverb noun singular noun proper noun plural verb base verb gerund verb past tense verb past participle verb 3rd singular cardinal number 48
Conclusion 49
Conclusion 50
Conclusion Normalizing flows for unsupervised learning X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> 50
Conclusion Normalizing flows for unsupervised learning X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> Space Learning ofjapanese bilingual lexicons English Space dog canine mapping function cat bird 50
Conclusion Normalizing flows for unsupervised learning X = f 1 (Z) <latexit sha1_base64="/fc4xhzsnc/ymh6xxat25pjhrca=">aaab/3icbvdlsgnbejynrxhfq4ixl4nbiafdbht0igs9eixghpisyxyymwyzftdtk4r1d/6kfw+kepu3vpk3tpi9agjbq1hvtxexgwmuwlk+jdzc4tlysn61sla+sbllbu80vbhlyuo0fkfsuuqxwqnwbw6ctsljio8k1nshv2o/+cck4mfwc6oiot7pb9zjlicwuuzec19g7z45ttnu0oeba5kw7o66zteqwxpgewjnpigy1lrmv6cx0thnavbblgrbvgroqirwklha6mskryqosz+1nq2iz5stto5p8afwetglpa4a8et9pzeqx6mr7+pon8bazxpj8t+vhyn37iq8igjgaz0u8mkbictjmhcps0zbjdqhvhj9k6ydigkfhvlbh2dpvjxpgpwyfvku3jwwq5dzhhm0jw5qcdnodfxrnaqhoqloet2jv/rmpbkvxrvxmw3ngdnmlvod4/mhl4gu7g==</latexit> Z = f (X) <latexit sha1_base64="idq8cdh7ny313/u9s3rzypq+wfi=">aaab+nicbvbns8naen3ur1q/uj16wsxcvzskcnoril48vraf2iay2w7apztn2j0ojfanepggifd/itf/jds2b219mpb4b4azex4suabh+bzyk6tr6xv5zclw9s7unl3cb+oouzq1acqi1fajzojl1gaogrvjxujoc9byr9dtv/xaloarvinxzlyqdcqpocvgpj5dvmexooilxrgyijny+6rnl5ykmwnejm5gsihdvwd/dfsrtuimgqqidcd1yvbsoobtwsafbqjztoiidfjhuelcpr10dvoehxulj4nimzkaz+rvizsewo9d33sgbiz60zuk/3mdbiill+uytobjol8ujajdhkc54d5xjiiyg0ko4uzwtideeqomryijwv18ezk0qxx3tfk9psvvrri48ugqhaeycte5qqebvecnrnejekav6m16sl6sd+tj3pqzspkd9afw5w/cxzmi</latexit> Space Learning ofjapanese bilingual lexicons English Space dog canine mapping function cat bird Learning of syntactic structure The cat sat on a green wall 50
Thank You! Questions? https://github.com/violet-zct/dema-bwe The cat sat on a green wall https://github.com/jxhe/struct-learning-with-flow 51