Learning about Language with Normalizing Flows Graham Neubig Language Technologies Institute, Carnegie Mellon University

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1 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

2 Learning about Language? 2

3 Learning about Language? Syntactic structure 2

4 Learning about Language? Syntactic structure The cat sat on a green wall 2

5 Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN 2

6 Learning about Language? Syntactic structure The cat sat on a green wall Parts-of-speech: DT NN VBD IN DT JJ NN Dependency: 2

7 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

8 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

9 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

10 Supervised Approaches X Y 3

11 Supervised Approaches X Supervised Learning Y X Y 4

12 Supervised Approaches X Supervised Learning Y X Y 5

13 Unsupervised Approaches X 6

14 Unsupervised Approaches X Learning language models P(X) 6

15 Unsupervised Approaches X Learning language models P(X) Learning continuous features from language models (e.g. word2vec, skipthought, BERT) 6

16 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

17 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

18 Latent Variable Approaches X Unsupervised Y Y???? X 7

19 Latent Variable Approaches X Unsupervised Y Y???? X 8

20 Density Matching for Bilingual Word Embedding Chunting Zhou, Xuezhe Ma, Di Wang, Graham Neubig (NAACL 2019) 9

21 Bilingual Word Embedding 10

22 Bilingual Word Embedding apple pear professor school canine piazza dog cat basketball planet earth 10

23 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

24 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

25 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

26 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

27 Previous Work on Unsupervised BWE

28 Previous Work on Unsupervised BWE Unsupervised methods of minimization some form of distance between distributions of discrete vector sets:

29 Previous Work on Unsupervised BWE Unsupervised methods of minimization some form of distance between distributions of discrete vector sets:

30 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

31 Density Mapping for Bilingual Word Embedding (DeMa-BWE) 12

32 Density Mapping for Bilingual Word Embedding (DeMa-BWE) Japanese Space English Space dog canine mapping function cat bird 12

33 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

34 Normalizing Flows 13

35 <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

36 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

37 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

38 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 (x) p (x) = pz (f (x)) det( <latexit sha1_base64="st0jwp7pyeca65l8lakjgjqile4=">aaacsnicbvdlsinbfk3oxfd8tbyxbooja8kmdouabgtrjuufszrjh3c7cjsprh5qdvsmbx+fg1ezm49w40kr2uzlazpeovbw6pxz63gcveldrvvbkx0qr6yurw9unre2dz5xd7+0tzjpgs2rqetfbgbqyrhbjenhtaorokdhdxb7pvav71abmcq/ajrin4jblempgkzuq0lay30aikfrv2/we273p4t6ok82fiuhpxc/j1t3qw0i91pqjehxhwdxt78vgr6wgye99ko1t+lowd8sb0zqbiblxvwx309efmfmqoexhc9nqzupdxyki4qfguxb3miao5bgekhp5pmqcv7nkn0ejtqumphenz/iitjmfau2gqenzbi3fv/ndtikj7u5jnommbbti8jmcur4uffelxofqzelils0b+viclyrsu1xbane8pc/kvzb0ztshlx9r52ezepyz/vsk6szjx2xu3bbllmlcfbintkre3oenbfn3fkzjzac2cwew0cp/bffwlqq</latexit> 13

39 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 (x) p (x) = pz (f (x)) det( <latexit sha1_base64="st0jwp7pyeca65l8lakjgjqile4=">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</latexit> Intuitively, prevents degenerative mapping of everything to zero vector 13

40 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 (x) p (x) = pz (f (x)) det( <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

41 DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird 14

42 DeMa-BWE: Preliminaries Japanese Space English Space dog canine mapping function cat bird Notations: 14

43 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

44 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

45 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

46 Prior Distribution Assumption on the monolingual word embedding space: Gaussian mixture model 鳥猫犬 15

49 DeMa-BWE: Density Matching 16

50 DeMa-BWE: Density Matching Sampling a continuous vector from the GMM 16

51 DeMa-BWE: Density Matching Sampling a continuous vector from the GMM 鳥猫 x i (x i ) x p(x x i ) 犬 16

52 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

53 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

54 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

57 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

58 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

59 Method Details 17

60 Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal 17

61 Method Details Weak Orthogonality Constraint: Try to make sure that the transformation is close to orthogonal 17

62 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

63 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

64 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

65 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

66 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

67 Main Results on BLI (close languages) 85 Procrustes(R) MUSE (U+R) SL-unsup-ID DeMa-BWE en-de de-en en-es es-en 19

68 Main Results on BLI (distant languages) 65 Procrustes(R) MUSE (U+R) DeMa-BWE en-et et-en en-el el-en en-ja ja-en 20

69 Unsupervised Learning of Syntactic Structure w/ Invertible Neural Projections Junxian He, Graham Neubig, Taylor Berg-Kirkpatrick (EMNLP 2018) 21

70 <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="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="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="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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76 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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77 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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sha1_base64="wplw8y/0iqfhjrazcsqahuwq07k=">aaacdxiczvdlsgmxfm34rpvvdekmwaqxpcyioo6kblxwcgyhlswtztrqzdikd8qy9b/erx6hk3hrn/gz/ogz6sxseyhck3ppdscniau34lo/zsrq2vrgzmmrvl2zu7dfoth8ncrrlplucaxbatfm8ij5wegwdqwzkyfgrwb8m81bt0wbrqihmmssj8kw4ignbczv7gyyfz72vx6l6tbdvpay8apqruu1+5xf7kdrrliiqcdgddw3hl5knhaq2ltctqylcr2tietygbhjtc/n/u7xqwugoftanghwzv7fsik0ksdikrnm5myza0oju7mqgmmszc/kdzorqs2qtuyktwgwjeb41ut5fcfaijrzesycg8jznnjanamgjhyqqrn9cqyjogkfm2dzzuqtjrim/pp6dd29v6g2boqwsugynaaz5kfl1eb3qil8rjfar+gnvtsvzofz6xznpctosxoe5sr5/gmr950d</latexit> x2 <latexit 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78 Dependency Model with Valence DT NN VBD DT NN IN NN The cat stopped a dog in Paris [Klein and Manning 2004] 25

79 Dependency Model with Valence DT NN VBD DT NN IN NN [Klein and Manning 2004] 26

80 Dependency Parse Induction from POS? DT NN VBD DT NN IN NN 27

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86 Latent Embeddings w/ Neural Projection zi <latexit sha1_base64="a0van9bpw9kdxhs9rkub77rptae=">aaacdniczvdltgixfo3gc/gfunttsexcedjjtnqd0y1lteqwmcgd0oggdjpp75jghi8wbvu7xbm3/okf4r/ygvki3ks5p+ee25yeibbcgov+oiwv1bx1jejmawt7z3evvh/wyfsikwtsjzrub8qwwspwba6ctwpniaweawwjm2zeemlacbxdwzhmvisdiiecerdu43op467hevfkfbfmtgsvay8hfzrxo1f+7fyvtsslgapitmdzy/btooftwsalbmjytoiidfjhwohizvx0anictyztx6hs9ksap+z/jzriiwkmrtjrzm6wmacumfwrgqhmwvbm9g7gmqggspqjtanngheil/2ur3eclkizh2eimcichyp7xdmkymwbozrbr2a6jjpqsbgwbebeyillohlwu6q5d+ev+nuevhedown0ijx0gerofjvqe1ek0st6q+/oi/phfdpfm2nbyxco0vw5339maj0j</latexit> Syntax Model z1 z2 z3 e1 e2 e3 <latexit 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90 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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91 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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92 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="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 ;, ) = p(f 1 (xi ) zi ; ) <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=">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> 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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=">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</latexit> Determinant of Jacobian matrix <latexit sha1_base64="yfyvkxezekmeahc8b85drlg5ulo=">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</latexit> xi <latexit 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93 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="4v3leopic1acf3czlm/0mgv/hee=">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</latexit> p(xi zi ;, ) = p(f 1 (xi ) zi ; ) <latexit 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sha1_base64="virgb/9jv45ewvl+qymhz6smazo=">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</latexit> x f <latexit sha1_base64="0tvux8mzrc5izm1cmvw9px4j8iq=">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</latexit> 1 (e; ) Gaussian distribution Determinant of Jacobian matrix <latexit 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sha1_base64="znkure23lzcjmnesh6crgwekxji=">aaacchiczvdlsgmxfm3uv62vqks3g0vwucpmedrd0y3lio4w2qfk0kwbmkyg5i5qh36bunxvccvu/qs/wz8wm52fbs+ee3luuehkbdfnghznxyqtrk6tb5q3k1vbo7t71f2dby0trahhjjeqe2bnoyuobww47cskyhfw+himr7p54xnvmsnohiyx9querixkbioh7p77zx615jscvoxl4baghopq96u/vyekiaarei617rpodh6kftdc6btsszsnmrnjie0aggfbtz/mvqf2iwegdiivorhyoft/i8vccwwjo8yanptldejjdd2oycsylj2t3/vebpva1dor0qfemalhhz+yke6armtmi0y4ddloyrehtfecfgiajoqzr9hkhbumymkrmizcxuswgddsxdac27na66oiq4yo0de6rs46ry10g9riqwqn0st6q+/wi/vhfvpfm2njknyo0vxz33/v75qx</latexit> 2 z 3 Example of Markov prior log p(x) = log p GHMM (f 1 (x)) + X log i x f 1 (e; ) <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

101 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=">aaaczhiczvdbsinbeo3mett4iyv7jmhgebrcmbfb92fbdl/20ywnck4mpz0a09ixobvgtwj6yb/gv/dz9gv8dxuspgy0okltp041vscvbbeyjp8a0yel5zxvty/n9y3nre3wzqcrqyvdome00oympxyev9bdjgjusgnu5gku84fvdf/6eyzlwv3csql9se8vlzijgkhbay/lprv5gemd4e/cuxrsv2bjwtlotznumo34pujnoe3mctnyazxkq80qcqqzonbepkmjfucnciban7pkqknza72h2wavlwd7bnqfjw8dm4wlbcjtge/z/ycclvzshavlnexcr2zqa2e7qyujwaf6m2ltjzlv5ljti4wt7jtfsdjvo67kckgx2r5fjwluce1ypoqggipjajqzhk6j2ygayjd4urbeln2zmsn4zbsuva1dnvaunnjs/sipmx4ch1xo33r6hvroul+6yc/t9sw3ud1rzi8ckcoskjnyqx6qs9ijjpwht+sz/g28rjvrbvr5jo0a85ldshdr/is9mluc</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=">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</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

102 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 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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 <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

103 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> 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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=">aaaci3iczva9t8mwehx4lourwagseqgjpfisfmcrygeseqwv2hi5rkot2nfkxxbvliufgxibn8dmhfgygfjz+ac4lqmfj1n37t076+4fmwcaxpfvghkdg5+ylewvp2dm5+bthcuzlrnfai1illujwjpyfteamoc0esukrcbppegd5f36jvwayegu+jftc3wrszardiby7zxqt1ubsftxl2xzebrtzzt5fzx5bmu3192yowjnp/c+wxpl5311+/huo+rbn62ojimgercotw56bgztfctghnos2eo0jthp4qvanddcgup2orgiczym03fcqcylwbmwpydsllta0dxkpolfvzwbkbkugrv0rz7ybwa17ougfihslli61h8wgxc/nbiotobgzlhhmhahpjm75nsyogr43wbmfdonokslfszgfc0aj7y/jvwhtd3yqdk9mv4domeu0djaq5viq3uogo5rfduqqdfoft2jb+vgerkerzehdmt6nllcv8j6+wkduqmq</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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sha1_base64="cabdscof/szjsejxwej+zulgkro=">aaacdxiczvdltgixfl2dl8qxaulgzuri4okqwy26i7pxcykicrdsla40tnnje8dijvydcavf4cq49rvc+a/+gr1gixct5p6ee25zevxicioe9+1kvlbx1jeym7mt7z3dvfz+wb1rsaastpvquuktwwqpwr05ctamncpsf6zhd2/seeoracnveiejihuk6yc84jsgpzptxyzp4y7v5gteyzuuuwzkm1cohnv+hgcg2s3/tnukxpkfsauxplx2iuwkrcongo1z7diwinah6bowhsgrzhssid+xe2qznhsobu+i7ot9v5eqastbgvwmzcznugaveqzovtiqauufmdznsppfxxztktabwtccwwun4weuiwvp1ecqcxevm2bj9rhmfmxiaki1t19x6yboqtemmlmzlrctwqb189jvyavzrk5hwlk4hhm4gzjcqavuoqp1ocdgbv7hzxl23p0p53mqztiznuoyk+frd02knxw=</latexit> [Dinh et al. 2014] 40

104 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

105 Part-of-speech Induction Many-to-1 % Feature-based HM Gaussia Neural VM Feature-based HM Gaussia Neural 42

106 Part-of-speech Induction Many-to-1 % Feature-based HM Gaussia Neural VM Feature-based HM Gaussia Neural Outperform feature-based SOTA 42

107 Dependency Parse Induction Directed % on len <= Oracle POS 49.6 DM 55.4 Gaussi 60.2 Neural Directed % on all Oracle POS 35.8 DM 43.1 Gaussi 47.9 Neural 43

108 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="fx1htpoik2me1ropxl5ys2c3zr4=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit sha1_base64="86pgxmdce2xvlzkoeifesb42x04=">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</latexit> <latexit 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sha1_base64="jbazhnaj26j1i9dtzz4so1plzqy=">aaacoxiczvhltsmwehtkq4ty6jllbeliufvpl8aniqthkahuihvyna216kdkb6bvlb/gylfxcxwfj8qvp+2blpwshc/uwupzjbfcyhh+eo219y3nrea2v9pyd/f2d1r3vhegqcs00gaquauck4iqo4bbbodkrmbdmr6q6w8vyczx6g6noqwlfvy844yio26edo7dbjil4d/olcaxwcrt2+vhqwafbivmugsfe2gow5ia5exa5cefhzyymx2grwcvlwch5uxnfzw4jg0ybdxrgmzyvxmllvzshlnootmlws2g1sj2xbeozj3qz2z3o5vjj5gdusnyzk4iwex8whkvfwikzxvkhqhqb7unqconmbrtbygz3h0lycnqkepn3jkirfa+hyt4zvpkqtiynlrlnmhyulxlpmx5clwzubdq6x8q9bsx3fa2je1ysi7ikemrm3jjrskniqgjkxkj796h9+v9z3fr8bzlazol8h5+ay4ysxk=</latexit> <latexit sha1_base64="znhkautewogmapwollzuede8uty=">aaacqhiczve9swnbej3z2/gzw5vdifieclfro8hgmojrwqthbznnfvfj2j1tw3f/wvz/lb3/w0os3etsgdowzns3m8vbn0kmhamo+gwwfpewv1bx1msbm1vbo7v1zrtncsuxy4009i5hdqxq2cvbeu8yi0wlem+tx4uqfvue1gmjr2muyu+xby1swrlvvix90d9trk1ohoe8ae9ba6br6ded43hgek5qe5fmuft2lfgvyjyel1jw4txhxvgje8b7dzvt6hrfwgwzhnpmekbg+qmphln/jwqmngi09j1vcjo1iifjpgv6lhqqklxpjo9upjjmoppvk3wp+yee0tneixswe2o+0zhmmiqtvsaea2grkxx5wlgv/ishhzllohn7zkqkqqzvyo3p3cjf9kcix8oitltxupazpe549lx3uf3f03nqpw6dtakrcnzghw7gcnpwaudwcr3oaochvmibvacfwvfwpvnhqjddyx7mrpdzcypztby=</latexit> <latexit sha1_base64="znhkautewogmapwollzuede8uty=">aaacqhiczve9swnbej3z2/gzw5vdifieclfro8hgmojrwqthbznnfvfj2j1tw3f/wvz/lb3/w0os3etsgdowzns3m8vbn0kmhamo+gwwfpewv1bx1msbm1vbo7v1zrtncsuxy4009i5hdqxq2cvbeu8yi0wlem+tx4uqfvue1gmjr2muyu+xby1swrlvvix90d9trk1ohoe8ae9ba6br6ded43hgek5qe5fmuft2lfgvyjyel1jw4txhxvgje8b7dzvt6hrfwgwzhnpmekbg+qmphln/jwqmngi09j1vcjo1iifjpgv6lhqqklxpjo9upjjmoppvk3wp+yee0tneixswe2o+0zhmmiqtvsaea2grkxx5wlgv/ishhzllohn7zkqkqqzvyo3p3cjf9kcix8oitltxupazpe549lx3uf3f03nqpw6dtakrcnzghw7gcnpwaudwcr3oaochvmibvacfwvfwpvnhqjddyx7mrpdzcypztby=</latexit> <latexit sha1_base64="virgb/9jv45ewvl+qymhz6smazo=">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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> <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

110 <latexit sha1_base64="4gmlgsvwrggpxtovfy6uyjcrnuq=">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</latexit> <latexit sha1_base64="4gmlgsvwrggpxtovfy6uyjcrnuq=">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</latexit> <latexit 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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

111 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

116 Conclusion 49

117 Conclusion 50

118 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

119 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

120 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

121 Thank You! Questions? The cat sat on a green wall 51

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