Algebraic geometry and statistical learning theory:
"This book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stoch...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge monographs on applied and computational mathematics
25 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "This book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties."--BOOK JACKET. |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke Literaturverz. S. 277 - 283 |
| Beschreibung: | VIII, 286 S. Ill., graph. Darst. |
| ISBN: | 9780521864671 |
Internformat
MARC
| LEADER | 00000nam a2200000zcb4500 | ||
|---|---|---|---|
| 001 | BV035720906 | ||
| 003 | DE-604 | ||
| 005 | 20140507 | ||
| 007 | t | ||
| 008 | 090911s2009 ad|| |||| 00||| eng d | ||
| 010 | |a 2009011366 | ||
| 016 | 7 | |a 015270930 |2 DE-101 | |
| 020 | |a 9780521864671 |c hbk |9 978-0-521-86467-1 | ||
| 035 | |a (OCoLC)299942247 | ||
| 035 | |a (DE-599)HBZHT016053862 | ||
| 040 | |a DE-604 |b ger | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-473 |a DE-29T |a DE-384 |a DE-703 |a DE-11 |a DE-188 |a DE-739 | ||
| 050 | 0 | |a Q325.7 | |
| 082 | 0 | |a 006.3/1 |2 22 | |
| 084 | |a QH 740 |0 (DE-625)141614: |2 rvk | ||
| 084 | |a SK 240 |0 (DE-625)143226: |2 rvk | ||
| 100 | 1 | |a Watanabe, Sumio |d 1959- |e Verfasser |0 (DE-588)139183116 |4 aut | |
| 245 | 1 | 0 | |a Algebraic geometry and statistical learning theory |c Sumio Watanabe |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a VIII, 286 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge monographs on applied and computational mathematics |v 25 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 500 | |a Literaturverz. S. 277 - 283 | ||
| 520 | 1 | |a "This book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties."--BOOK JACKET. | |
| 650 | 4 | |a Computational learning theory |x Statistical methods | |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 0 | 7 | |a Mathematische Lerntheorie |0 (DE-588)4169103-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |D s |
| 689 | 0 | 1 | |a Mathematische Lerntheorie |0 (DE-588)4169103-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Cambridge monographs on applied and computational mathematics |v 25 |w (DE-604)BV011073737 |9 25 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bamberg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017997703&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017997703 | ||
Datensatz im Suchindex
| _version_ | 1804139932021686272 |
|---|---|
| adam_text | Contents
Preface
page vii
Introduction
1
1.1
Basic concepts in statistical learning
1
1.2
Statistical models and learning machines
10
18
26
41
42
48
48
50
53
58
66
72
77
77
80
86
87
91
99
Zeta
function and singular integral
105
4.1
Schwartz distribution
105
4.2
State density function 111
1.3
Statistical estimation methods
1.4
Four main formulas
1.5
Overview of this book
1.6
Probability theory
2
Singularity theory
2.1
Polynomials and analytic functions
2.2
Algebraic set and analytic set
2.3
Singularity
2.4
Resolution of singularities
2.5
Normal crossing singularities
2.6
Manifold
3
Algebraic geometry
3.1
Ring and ideal
3.2
Real algebraic set
3.3
Singularities and dimension
3.4
Real
projective
space
3.5
Blow-up
3.6
Examples
vi
Contents
A3
Mellin
transform
116
4.4 Evaluation
of singular integral
118
4.5
Asymptotic expansion and b-function
128
5
Empirical processes
133
5.1
Convergence in law
133
5.2
Function-valued analytic functions
140
5.3
Empirical process
144
5.4
Fluctuation of Gaussian processes
154
6
Singular learning theory
158
6.1
Standard form of likelihood ratio function
160
6.2
Evidence and stochastic complexity
168
6.3
Bayes
and Gibbs estimation
177
6.4
Maximum likelihood and a posteriori
203
7
Singular learning machines
217
7.1
Learning coefficient
217
7.2
Three-layered neural networks
227
7.3
Mixture models
230
7.4
Bayesian network
233
7.5
Hidden Markov model
234
7.6
Singular learning process
235
7.7
Bias and variance
239
7.8
Non-analytic learning machines
245
8
Singular statistics
249
8.1
Universally optimal learning
249
8.2
Generalized
Bayes
information criterion
252
8.3
Widely applicable information criteria
253
8.4
Singular hypothesis test
258
8.5
Realization of a posteriori distribution
264
8.6
From regular to singular
274
Bibliography 111
Index
284
|
| any_adam_object | 1 |
| author | Watanabe, Sumio 1959- |
| author_GND | (DE-588)139183116 |
| author_facet | Watanabe, Sumio 1959- |
| author_role | aut |
| author_sort | Watanabe, Sumio 1959- |
| author_variant | s w sw |
| building | Verbundindex |
| bvnumber | BV035720906 |
| callnumber-first | Q - Science |
| callnumber-label | Q325 |
| callnumber-raw | Q325.7 |
| callnumber-search | Q325.7 |
| callnumber-sort | Q 3325.7 |
| callnumber-subject | Q - General Science |
| classification_rvk | QH 740 SK 240 |
| ctrlnum | (OCoLC)299942247 (DE-599)HBZHT016053862 |
| dewey-full | 006.3/1 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 006 - Special computer methods |
| dewey-raw | 006.3/1 |
| dewey-search | 006.3/1 |
| dewey-sort | 16.3 11 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik Wirtschaftswissenschaften |
| edition | 1. publ. |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>03038nam a2200493zcb4500</leader><controlfield tag="001">BV035720906</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20140507 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">090911s2009 ad|| |||| 00||| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">2009011366</subfield></datafield><datafield tag="016" ind1="7" ind2=" "><subfield code="a">015270930</subfield><subfield code="2">DE-101</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780521864671</subfield><subfield code="c">hbk</subfield><subfield code="9">978-0-521-86467-1</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)299942247</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)HBZHT016053862</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-473</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-384</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-188</subfield><subfield code="a">DE-739</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">Q325.7</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">006.3/1</subfield><subfield code="2">22</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">QH 740</subfield><subfield code="0">(DE-625)141614:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 240</subfield><subfield code="0">(DE-625)143226:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Watanabe, Sumio</subfield><subfield code="d">1959-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)139183116</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Algebraic geometry and statistical learning theory</subfield><subfield code="c">Sumio Watanabe</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">1. publ.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cambridge [u.a.]</subfield><subfield code="b">Cambridge Univ. Press</subfield><subfield code="c">2009</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">VIII, 286 S.</subfield><subfield code="b">Ill., graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Cambridge monographs on applied and computational mathematics</subfield><subfield code="v">25</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Hier auch später erschienene, unveränderte Nachdrucke</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Literaturverz. S. 277 - 283</subfield></datafield><datafield tag="520" ind1="1" ind2=" "><subfield code="a">"This book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties."--BOOK JACKET.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Computational learning theory</subfield><subfield code="x">Statistical methods</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Geometry, Algebraic</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Mathematische Lerntheorie</subfield><subfield code="0">(DE-588)4169103-9</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Algebraische Geometrie</subfield><subfield code="0">(DE-588)4001161-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Algebraische Geometrie</subfield><subfield code="0">(DE-588)4001161-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Mathematische Lerntheorie</subfield><subfield code="0">(DE-588)4169103-9</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Cambridge monographs on applied and computational mathematics</subfield><subfield code="v">25</subfield><subfield code="w">(DE-604)BV011073737</subfield><subfield code="9">25</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Bamberg</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017997703&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-017997703</subfield></datafield></record></collection> |
| id | DE-604.BV035720906 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:51:56Z |
| institution | BVB |
| isbn | 9780521864671 |
| language | English |
| lccn | 2009011366 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017997703 |
| oclc_num | 299942247 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG DE-29T DE-384 DE-703 DE-11 DE-188 DE-739 |
| owner_facet | DE-473 DE-BY-UBG DE-29T DE-384 DE-703 DE-11 DE-188 DE-739 |
| physical | VIII, 286 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge monographs on applied and computational mathematics |
| series2 | Cambridge monographs on applied and computational mathematics |
| spelling | Watanabe, Sumio 1959- Verfasser (DE-588)139183116 aut Algebraic geometry and statistical learning theory Sumio Watanabe 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2009 VIII, 286 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on applied and computational mathematics 25 Hier auch später erschienene, unveränderte Nachdrucke Literaturverz. S. 277 - 283 "This book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties."--BOOK JACKET. Computational learning theory Statistical methods Geometry, Algebraic Mathematische Lerntheorie (DE-588)4169103-9 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Mathematische Lerntheorie (DE-588)4169103-9 s DE-604 Cambridge monographs on applied and computational mathematics 25 (DE-604)BV011073737 25 Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017997703&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Watanabe, Sumio 1959- Algebraic geometry and statistical learning theory Cambridge monographs on applied and computational mathematics Computational learning theory Statistical methods Geometry, Algebraic Mathematische Lerntheorie (DE-588)4169103-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4169103-9 (DE-588)4001161-6 |
| title | Algebraic geometry and statistical learning theory |
| title_auth | Algebraic geometry and statistical learning theory |
| title_exact_search | Algebraic geometry and statistical learning theory |
| title_full | Algebraic geometry and statistical learning theory Sumio Watanabe |
| title_fullStr | Algebraic geometry and statistical learning theory Sumio Watanabe |
| title_full_unstemmed | Algebraic geometry and statistical learning theory Sumio Watanabe |
| title_short | Algebraic geometry and statistical learning theory |
| title_sort | algebraic geometry and statistical learning theory |
| topic | Computational learning theory Statistical methods Geometry, Algebraic Mathematische Lerntheorie (DE-588)4169103-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Computational learning theory Statistical methods Geometry, Algebraic Mathematische Lerntheorie Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017997703&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011073737 |
| work_keys_str_mv | AT watanabesumio algebraicgeometryandstatisticallearningtheory |