Learning theory: an approximation theory viewpoint
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ. |
| Series: | Cambridge monographs on applied and computational mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Beschreibung für Leser Inhaltsverzeichnis |
| Physical Description: | XII, 224 S. graph. Darst. |
| ISBN: | 052186559X 9780521865593 |
Staff View
MARC
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| 245 | 1 | 0 | |a Learning theory |b an approximation theory viewpoint |c Felipe Cucker ; Ding-Xuan Zhou |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2007 | |
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Record in the Search Index
| _version_ | 1804136201027846144 |
|---|---|
| adam_text | Contents
Foreword
ix
Preface
xi
1
The framework of learning
1
1.1
Introduction
1
1.2
A formal setting
5
1.3
Hypothesis spaces and target functions
9
1.4
Sample, approximation, and generalization errors
11
1.5
The bias-variance problem
13
1.6
The remainder of this book
14
1.7
References and additional remarks
15
2
Basic hypothesis spaces
17
2.1
First examples of hypothesis space
17
2.2
Reminders I
18
2.3
Hypothesis spaces associated with Sobolev spaces
21
2.4
Reproducing Kernel Hubert Spaces
22
2.5
Some Mercer kernels
24
2.6
Hypothesis spaces associated with an RKHS
31
2.7
Reminders II
33
2.8
On the computation of empirical target functions
34
2.9
References and additional remarks
35
3
Estimating the sample error
37
3.1
Exponential inequalities in probability
37
3.2
Uniform estimates on the defect
43
3.3
Estimating the sample error
44
3.4
Convex hypothesis spaces
46
3.5
References and additional remarks
49
vi
Contents
Polynomial decay of the approximation error
54
4.1
Reminders III
55
4.2
Operators defined by a kernel
56
4.3
Mercer s theorem
59
4.4
RKHSs revisited
61
4.5
Characterizing the approximation error in RKHSs
63
4.6
An example
68
4.7
References and additional remarks
69
Estimating covering numbers
72
5.1
Reminders IV
73
5.2
Covering numbers for Sobolev smooth kernels
76
5.3
Covering numbers for analytic kernels
83
5.4
Lower bounds for covering numbers
101
5.5
On the smoothness of box spline kernels
106
5.6
References and additional remarks
108
Logarithmic decay of the approximation error
109
6.1
Polynomial decay of the approximation error for if00
kernels
110
6.2
Measuring the regularity of the kernel
112
6.3
Estimating the approximation error in RKHSs
117
6.4
Proof of Theorem
6.1 125
6.5
References and additional remarks
125
On the bias—variance problem
127
7.1
A useful lemma
128
7.2
Proof of Theorem
7.1 129
7.3
A concrete example of bias-variance
132
7.4
References and additional remarks
133
Least squares regularization
134
8.1
Bounds for the regularized error
135
8.2
On the existence of target functions
139
8.3
A first estimate for the excess generalization error
140
8.4
Proof of Theorem
8.1 148
8.5
Reminders V
151
8.6
Compactness and regularization
151
8.7
References and additional remarks
155
Support vector machines for classification
157
9.1
Binary classifiers
159
Contents
vii
9.2
Regularized classifiers
161
9.3
Optimal
hyperplanes:
the separable case
166
9.4
Support vector machines
169
9.5
Optimal
hyperplanes:
the nonseparable case
171
9.6
Error analysis for separable measures
173
9.7
Weakly separable measures
182
9.8
References and additional remarks
185
10
General regularized classifiers
187
10.1
Bounding the misclassification error in terms of the
generalization error
189
10.2
Projection and error decomposition
194
10.3
Bounds for the regularized error
Ί)(γ ,φ)
of fY
196
10.4
Bounds for the sample error term involving fy
198
10.5
Bounds for the sample error term involving/, j,
201
10.6
Stronger error bounds
204
10.7
Improving learning rates by imposing noise conditions
210
10.8
References and additional remarks
211
References
214
Index
222
|
| adam_txt |
Contents
Foreword
ix
Preface
xi
1
The framework of learning
1
1.1
Introduction
1
1.2
A formal setting
5
1.3
Hypothesis spaces and target functions
9
1.4
Sample, approximation, and generalization errors
11
1.5
The bias-variance problem
13
1.6
The remainder of this book
14
1.7
References and additional remarks
15
2
Basic hypothesis spaces
17
2.1
First examples of hypothesis space
17
2.2
Reminders I
18
2.3
Hypothesis spaces associated with Sobolev spaces
21
2.4
Reproducing Kernel Hubert Spaces
22
2.5
Some Mercer kernels
24
2.6
Hypothesis spaces associated with an RKHS
31
2.7
Reminders II
33
2.8
On the computation of empirical target functions
34
2.9
References and additional remarks
35
3
Estimating the sample error
37
3.1
Exponential inequalities in probability
37
3.2
Uniform estimates on the defect
43
3.3
Estimating the sample error
44
3.4
Convex hypothesis spaces
46
3.5
References and additional remarks
49
vi
Contents
Polynomial decay of the approximation error
54
4.1
Reminders III
55
4.2
Operators defined by a kernel
56
4.3
Mercer's theorem
59
4.4
RKHSs revisited
61
4.5
Characterizing the approximation error in RKHSs
63
4.6
An example
68
4.7
References and additional remarks
69
Estimating covering numbers
72
5.1
Reminders IV
73
5.2
Covering numbers for Sobolev smooth kernels
76
5.3
Covering numbers for analytic kernels
83
5.4
Lower bounds for covering numbers
101
5.5
On the smoothness of box spline kernels
106
5.6
References and additional remarks
108
Logarithmic decay of the approximation error
109
6.1
Polynomial decay of the approximation error for "if00
kernels
110
6.2
Measuring the regularity of the kernel
112
6.3
Estimating the approximation error in RKHSs
117
6.4
Proof of Theorem
6.1 125
6.5
References and additional remarks
125
On the bias—variance problem
127
7.1
A useful lemma
128
7.2
Proof of Theorem
7.1 129
7.3
A concrete example of bias-variance
132
7.4
References and additional remarks
133
Least squares regularization
134
8.1
Bounds for the regularized error
135
8.2
On the existence of target functions
139
8.3
A first estimate for the excess generalization error
140
8.4
Proof of Theorem
8.1 148
8.5
Reminders V
151
8.6
Compactness and regularization
151
8.7
References and additional remarks
155
Support vector machines for classification
157
9.1
Binary classifiers
159
Contents
vii
9.2
Regularized classifiers
161
9.3
Optimal
hyperplanes:
the separable case
166
9.4
Support vector machines
169
9.5
Optimal
hyperplanes:
the nonseparable case
171
9.6
Error analysis for separable measures
173
9.7
Weakly separable measures
182
9.8
References and additional remarks
185
10
General regularized classifiers
187
10.1
Bounding the misclassification error in terms of the
generalization error
189
10.2
Projection and error decomposition
194
10.3
Bounds for the regularized error
Ί)(γ ,φ)
of fY
196
10.4
Bounds for the sample error term involving fy
198
10.5
Bounds for the sample error term involving/, j,
201
10.6
Stronger error bounds
204
10.7
Improving learning rates by imposing noise conditions
210
10.8
References and additional remarks
211
References
214
Index
222 |
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| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| edition | 1. publ. |
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| id | DE-604.BV022218587 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:28:21Z |
| indexdate | 2024-07-09T20:52:38Z |
| institution | BVB |
| isbn | 052186559X 9780521865593 |
| language | English |
| lccn | 2006037012 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015429838 |
| oclc_num | 76141644 |
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| owner_facet | DE-703 DE-634 DE-91G DE-BY-TUM DE-739 DE-83 DE-188 DE-384 |
| physical | XII, 224 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge monographs on applied and computational mathematics |
| spelling | Cucker, Felipe 1958- Verfasser (DE-588)133832783 aut Learning theory an approximation theory viewpoint Felipe Cucker ; Ding-Xuan Zhou 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2007 XII, 224 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on applied and computational mathematics Computational learning theory Approximation theory Maschinelles Lernen (DE-588)4193754-5 gnd rswk-swf Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Maschinelles Lernen (DE-588)4193754-5 s Approximationstheorie (DE-588)4120913-8 s DE-604 Zhou, Ding-Xuan Verfasser (DE-588)113762224 aut http://www.loc.gov/catdir/toc/ecip074/2006037012.html Inhaltsverzeichnis http://www.loc.gov/catdir/enhancements/fy0703/2006037012-d.html Beschreibung für Leser Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015429838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cucker, Felipe 1958- Zhou, Ding-Xuan Learning theory an approximation theory viewpoint Computational learning theory Approximation theory Maschinelles Lernen (DE-588)4193754-5 gnd Approximationstheorie (DE-588)4120913-8 gnd |
| subject_GND | (DE-588)4193754-5 (DE-588)4120913-8 |
| title | Learning theory an approximation theory viewpoint |
| title_auth | Learning theory an approximation theory viewpoint |
| title_exact_search | Learning theory an approximation theory viewpoint |
| title_exact_search_txtP | Learning theory an approximation theory viewpoint |
| title_full | Learning theory an approximation theory viewpoint Felipe Cucker ; Ding-Xuan Zhou |
| title_fullStr | Learning theory an approximation theory viewpoint Felipe Cucker ; Ding-Xuan Zhou |
| title_full_unstemmed | Learning theory an approximation theory viewpoint Felipe Cucker ; Ding-Xuan Zhou |
| title_short | Learning theory |
| title_sort | learning theory an approximation theory viewpoint |
| title_sub | an approximation theory viewpoint |
| topic | Computational learning theory Approximation theory Maschinelles Lernen (DE-588)4193754-5 gnd Approximationstheorie (DE-588)4120913-8 gnd |
| topic_facet | Computational learning theory Approximation theory Maschinelles Lernen Approximationstheorie |
| url | http://www.loc.gov/catdir/toc/ecip074/2006037012.html http://www.loc.gov/catdir/enhancements/fy0703/2006037012-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015429838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cuckerfelipe learningtheoryanapproximationtheoryviewpoint AT zhoudingxuan learningtheoryanapproximationtheoryviewpoint |
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