Are all classifiers linear in some high dimensional space?
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Of all possible classifiers (including SVMs, locally weighted regression, softmax regression, lots others I'm sure I don't know about, etc.), are they all linear in some high dimensional space?
E.g. An SVM with a Gaussian kernel is linear in infinite-dimensional space.
classification theory
asked Apr 10 '18 at 18:00
sidranesidrane
161
$endgroup$
add a comment |
$begingroup$
Of all possible classifiers (including SVMs, locally weighted regression, softmax regression, lots others I'm sure I don't know about, etc.), are they all linear in some high dimensional space?
E.g. An SVM with a Gaussian kernel is linear in infinite-dimensional space.
classification theory
asked Apr 10 '18 at 18:00
sidranesidrane
161
$endgroup$
1
$begingroup$
The question really is linear with respect to what? logistic regression is "non-linear", but it is linear in parameters. Polynomial terms are non-linear, but linear in parameters. Neural networks aren't linear in parameters, but could be in terms of variables (potentially). You should think about what you mean by "linear" and try to figure out whether all classifiers would meet that definition of linear.
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– Ryan
Apr 10 '18 at 20:55
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this has also been discussed on stats.se: stats.stackexchange.com/questions/215696
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– jld
Apr 11 '18 at 15:08
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@Chaconne thank you for your comment. That post you linked was very informative.
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– sidrane
Apr 15 '18 at 17:27
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@Ryan what I mean by linear, is if we can project into any higher-dimensional feature space (using a potentially nonlinear mapping), can we draw a single separating hyperplane between the classes of data? And the answer linked by Chaconne indicates that generally, yes you can. The exception is in classifiers that combine multiple "smaller" or "weaker" classifiers together, e.g. K-means, decision trees, boosting, and others like it.
$endgroup$
– sidrane
Apr 15 '18 at 17:31
add a comment |
$begingroup$
Of all possible classifiers (including SVMs, locally weighted regression, softmax regression, lots others I'm sure I don't know about, etc.), are they all linear in some high dimensional space?
E.g. An SVM with a Gaussian kernel is linear in infinite-dimensional space.
classification theory
asked Apr 10 '18 at 18:00
sidranesidrane
161
$endgroup$
Of all possible classifiers (including SVMs, locally weighted regression, softmax regression, lots others I'm sure I don't know about, etc.), are they all linear in some high dimensional space?
E.g. An SVM with a Gaussian kernel is linear in infinite-dimensional space.
classification theory
classification theory
asked Apr 10 '18 at 18:00
sidranesidrane
161
asked Apr 10 '18 at 18:00
sidranesidrane
161
asked Apr 10 '18 at 18:00
sidranesidrane
161
asked Apr 10 '18 at 18:00
sidranesidrane
161
asked Apr 10 '18 at 18:00
sidranesidrane
161
161
1
$begingroup$
The question really is linear with respect to what? logistic regression is "non-linear", but it is linear in parameters. Polynomial terms are non-linear, but linear in parameters. Neural networks aren't linear in parameters, but could be in terms of variables (potentially). You should think about what you mean by "linear" and try to figure out whether all classifiers would meet that definition of linear.
$endgroup$
– Ryan
Apr 10 '18 at 20:55
$begingroup$
this has also been discussed on stats.se: stats.stackexchange.com/questions/215696
$endgroup$
– jld
Apr 11 '18 at 15:08
$begingroup$
@Chaconne thank you for your comment. That post you linked was very informative.
$endgroup$
– sidrane
Apr 15 '18 at 17:27
$begingroup$
@Ryan what I mean by linear, is if we can project into any higher-dimensional feature space (using a potentially nonlinear mapping), can we draw a single separating hyperplane between the classes of data? And the answer linked by Chaconne indicates that generally, yes you can. The exception is in classifiers that combine multiple "smaller" or "weaker" classifiers together, e.g. K-means, decision trees, boosting, and others like it.
$endgroup$
– sidrane
Apr 15 '18 at 17:31
add a comment |
1
$begingroup$
The question really is linear with respect to what? logistic regression is "non-linear", but it is linear in parameters. Polynomial terms are non-linear, but linear in parameters. Neural networks aren't linear in parameters, but could be in terms of variables (potentially). You should think about what you mean by "linear" and try to figure out whether all classifiers would meet that definition of linear.
$endgroup$
– Ryan
Apr 10 '18 at 20:55
$begingroup$
this has also been discussed on stats.se: stats.stackexchange.com/questions/215696
$endgroup$
– jld
Apr 11 '18 at 15:08
$begingroup$
@Chaconne thank you for your comment. That post you linked was very informative.
$endgroup$
– sidrane
Apr 15 '18 at 17:27
$begingroup$
@Ryan what I mean by linear, is if we can project into any higher-dimensional feature space (using a potentially nonlinear mapping), can we draw a single separating hyperplane between the classes of data? And the answer linked by Chaconne indicates that generally, yes you can. The exception is in classifiers that combine multiple "smaller" or "weaker" classifiers together, e.g. K-means, decision trees, boosting, and others like it.
$endgroup$
– sidrane
Apr 15 '18 at 17:31
1
1
$begingroup$
The question really is linear with respect to what? logistic regression is "non-linear", but it is linear in parameters. Polynomial terms are non-linear, but linear in parameters. Neural networks aren't linear in parameters, but could be in terms of variables (potentially). You should think about what you mean by "linear" and try to figure out whether all classifiers would meet that definition of linear.
$endgroup$
– Ryan
Apr 10 '18 at 20:55
$begingroup$
The question really is linear with respect to what? logistic regression is "non-linear", but it is linear in parameters. Polynomial terms are non-linear, but linear in parameters. Neural networks aren't linear in parameters, but could be in terms of variables (potentially). You should think about what you mean by "linear" and try to figure out whether all classifiers would meet that definition of linear.
$endgroup$
– Ryan
Apr 10 '18 at 20:55
$begingroup$
this has also been discussed on stats.se: stats.stackexchange.com/questions/215696
$endgroup$
– jld
Apr 11 '18 at 15:08
$begingroup$
this has also been discussed on stats.se: stats.stackexchange.com/questions/215696
$endgroup$
– jld
Apr 11 '18 at 15:08
$begingroup$
@Chaconne thank you for your comment. That post you linked was very informative.
$endgroup$
– sidrane
Apr 15 '18 at 17:27
$begingroup$
@Chaconne thank you for your comment. That post you linked was very informative.
$endgroup$
– sidrane
Apr 15 '18 at 17:27
$begingroup$
@Ryan what I mean by linear, is if we can project into any higher-dimensional feature space (using a potentially nonlinear mapping), can we draw a single separating hyperplane between the classes of data? And the answer linked by Chaconne indicates that generally, yes you can. The exception is in classifiers that combine multiple "smaller" or "weaker" classifiers together, e.g. K-means, decision trees, boosting, and others like it.
$endgroup$
– sidrane
Apr 15 '18 at 17:31
$begingroup$
@Ryan what I mean by linear, is if we can project into any higher-dimensional feature space (using a potentially nonlinear mapping), can we draw a single separating hyperplane between the classes of data? And the answer linked by Chaconne indicates that generally, yes you can. The exception is in classifiers that combine multiple "smaller" or "weaker" classifiers together, e.g. K-means, decision trees, boosting, and others like it.
$endgroup$
– sidrane
Apr 15 '18 at 17:31
add a comment |
1 Answer
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The answer is yes!
Remark 1: by "they all linear" I assume, for any classifier, an existence of an equivalent linear classifier in higher dimensions (that is, a decision hyperplane that that yields identical results to the original classifier for any given input).
Remark 2: it is definitely true for any classifier on a countable domain (e.g., all possible 8bit RGB 256x256 images is quite large but still a countable domain).
For classifiers on uncountable domains (like fields of real numbers), the answer would depend if Cover's theorem holds for infinitely large number of points.
However, I'd rather not be worried about that, since all real-world classifiers implemented on classical computers are discrete :-)
answered 12 mins ago
m0nzderrm0nzderr
11
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1 Answer
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$begingroup$
The answer is yes!
Remark 1: by "they all linear" I assume, for any classifier, an existence of an equivalent linear classifier in higher dimensions (that is, a decision hyperplane that that yields identical results to the original classifier for any given input).
Remark 2: it is definitely true for any classifier on a countable domain (e.g., all possible 8bit RGB 256x256 images is quite large but still a countable domain).
For classifiers on uncountable domains (like fields of real numbers), the answer would depend if Cover's theorem holds for infinitely large number of points.
However, I'd rather not be worried about that, since all real-world classifiers implemented on classical computers are discrete :-)
answered 12 mins ago
m0nzderrm0nzderr
11
New contributor
m0nzderr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
The answer is yes!
Remark 1: by "they all linear" I assume, for any classifier, an existence of an equivalent linear classifier in higher dimensions (that is, a decision hyperplane that that yields identical results to the original classifier for any given input).
Remark 2: it is definitely true for any classifier on a countable domain (e.g., all possible 8bit RGB 256x256 images is quite large but still a countable domain).
For classifiers on uncountable domains (like fields of real numbers), the answer would depend if Cover's theorem holds for infinitely large number of points.
However, I'd rather not be worried about that, since all real-world classifiers implemented on classical computers are discrete :-)
answered 12 mins ago
m0nzderrm0nzderr
11
New contributor
m0nzderr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
The answer is yes!
Remark 1: by "they all linear" I assume, for any classifier, an existence of an equivalent linear classifier in higher dimensions (that is, a decision hyperplane that that yields identical results to the original classifier for any given input).
Remark 2: it is definitely true for any classifier on a countable domain (e.g., all possible 8bit RGB 256x256 images is quite large but still a countable domain).
For classifiers on uncountable domains (like fields of real numbers), the answer would depend if Cover's theorem holds for infinitely large number of points.
However, I'd rather not be worried about that, since all real-world classifiers implemented on classical computers are discrete :-)
answered 12 mins ago
m0nzderrm0nzderr
11
New contributor
m0nzderr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
The answer is yes!
Remark 1: by "they all linear" I assume, for any classifier, an existence of an equivalent linear classifier in higher dimensions (that is, a decision hyperplane that that yields identical results to the original classifier for any given input).
Remark 2: it is definitely true for any classifier on a countable domain (e.g., all possible 8bit RGB 256x256 images is quite large but still a countable domain).
For classifiers on uncountable domains (like fields of real numbers), the answer would depend if Cover's theorem holds for infinitely large number of points.
However, I'd rather not be worried about that, since all real-world classifiers implemented on classical computers are discrete :-)
answered 12 mins ago
m0nzderrm0nzderr
11
New contributor
m0nzderr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 12 mins ago
m0nzderrm0nzderr
11
New contributor
m0nzderr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 12 mins ago
m0nzderrm0nzderr
11
answered 12 mins ago
m0nzderrm0nzderr
11
11
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m0nzderr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$begingroup$
The question really is linear with respect to what? logistic regression is "non-linear", but it is linear in parameters. Polynomial terms are non-linear, but linear in parameters. Neural networks aren't linear in parameters, but could be in terms of variables (potentially). You should think about what you mean by "linear" and try to figure out whether all classifiers would meet that definition of linear.
$endgroup$
– Ryan
Apr 10 '18 at 20:55
$begingroup$
this has also been discussed on stats.se: stats.stackexchange.com/questions/215696
$endgroup$
– jld
Apr 11 '18 at 15:08
$begingroup$
@Chaconne thank you for your comment. That post you linked was very informative.
$endgroup$
– sidrane
Apr 15 '18 at 17:27
$begingroup$
@Ryan what I mean by linear, is if we can project into any higher-dimensional feature space (using a potentially nonlinear mapping), can we draw a single separating hyperplane between the classes of data? And the answer linked by Chaconne indicates that generally, yes you can. The exception is in classifiers that combine multiple "smaller" or "weaker" classifiers together, e.g. K-means, decision trees, boosting, and others like it.
$endgroup$
– sidrane
Apr 15 '18 at 17:31