How is the optimal hyperplane computed?
Let’s introduce the notation used to define formally a hyperplane:
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where
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is known as the
weight vector and
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as the
bias.
See also
A more in depth description of this and hyperplanes you can find in the section 4.5 (Seperating Hyperplanes) of the book: Elements of Statistical Learning by T. Hastie, R. Tibshirani and J. H. Friedman.
The optimal hyperplane can be represented in an infinite number of different ways by scaling of
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and
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. As a matter of convention, among all the possible representations of the hyperplane, the one chosen is
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where
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symbolizes the training examples closest to the hyperplane. In general, the training examples that are closest to the hyperplane are called
support vectors. This representation is known as the
canonical hyperplane.
Now, we use the result of geometry that gives the distance between a point
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and a hyperplane
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In particular, for the canonical hyperplane, the numerator is equal to one and the distance to the support vectors is
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[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
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(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
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(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
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(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
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(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
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or enter new name. (Default extension: sty)
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l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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Transcript written on math.log.
Recall that the margin introduced in the previous section, here denoted as
System Message: WARNING/2 (M)
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
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Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
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(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
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l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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, is twice the distance to the closest examples:
System Message: WARNING/2 (M = \frac{2}{||\beta||}
)
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
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Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
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(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
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l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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Transcript written on math.log.
Finally, the problem of maximizing
System Message: WARNING/2 (M)
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This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
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Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
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(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
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l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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is equivalent to the problem of minimizing a function
System Message: WARNING/2 (L(\beta))
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
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(/usr/share/texmf-texlive/tex/latex/euler/euler.sty
Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
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(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
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or enter new name. (Default extension: sty)
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l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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Transcript written on math.log.
subject to some constraints. The constraints model the requirement for the hyperplane to classify correctly all the training examples
System Message: WARNING/2 (x_{i})
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
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(/usr/share/texmf-texlive/tex/latex/euler/euler.sty
Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
(/etc/texmf/tex/latex/config/color.cfg)
(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
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l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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Transcript written on math.log.
. Formally,
System Message: WARNING/2 (\min_{\beta, \beta_{0}} L(\beta) = \frac{1}{2}||\beta||^{2} \text{ subject to } y_{i}(\beta^{T} x_{i} + \beta_{0}) \geq 1 \text{ } \forall i,
)
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf-texlive/tex/latex/tools/bm.sty)
(/usr/share/texmf-texlive/tex/latex/euler/euler.sty
Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
(/etc/texmf/tex/latex/config/color.cfg)
(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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Transcript written on math.log.
where
System Message: WARNING/2 (y_{i})
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf-texlive/tex/latex/tools/bm.sty)
(/usr/share/texmf-texlive/tex/latex/euler/euler.sty
Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
(/etc/texmf/tex/latex/config/color.cfg)
(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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No pages of output.
Transcript written on math.log.
represents each of the labels of the training examples.
This is a problem of Lagrangian optimization that can be solved using Lagrange multipliers to obtain the weight vector
System Message: WARNING/2 (\beta)
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf-texlive/tex/latex/tools/bm.sty)
(/usr/share/texmf-texlive/tex/latex/euler/euler.sty
Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
(/etc/texmf/tex/latex/config/color.cfg)
(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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Transcript written on math.log.
and the bias
System Message: WARNING/2 (\beta_{0})
latex exited with error:
[stderr]
[stdout]
This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./math.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?’ option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf-texlive/tex/latex/tools/bm.sty)
(/usr/share/texmf-texlive/tex/latex/euler/euler.sty
Package: `euler’ v2.5 <1995/03/05> (FJ and FMi)
) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty
(/etc/texmf/tex/latex/config/color.cfg)
(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def))
! LaTeX Error: File `bbm.sty’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage
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of the optimal hyperplane.