Mixing
Choosing a kernel embedding for a linear separator
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The following is a slide from a presentation on machine learning, the context is using the kernel trick for non-linear separation. 
Does it make any difference that the mapping is $xmapsto (x,x^2)$ rather then $xmapsto (x^2,x)$ or will the data be separable under one mapping iff it is separable under the other? In general, in the context of embedding samples to some feature space, does the order of the coordinates matter? I may be missing something quite fundamental here but it would seem that this can have an effect on the separability of the embedded vectors.
linear-algebra machine-learning
asked Jul 16 at 14:39
D.M.
183
add a comment |Â
up vote
1
down vote
favorite
The following is a slide from a presentation on machine learning, the context is using the kernel trick for non-linear separation.
Does it make any difference that the mapping is $xmapsto (x,x^2)$ rather then $xmapsto (x^2,x)$ or will the data be separable under one mapping iff it is separable under the other? In general, in the context of embedding samples to some feature space, does the order of the coordinates matter? I may be missing something quite fundamental here but it would seem that this can have an effect on the separability of the embedded vectors.
linear-algebra machine-learning
asked Jul 16 at 14:39
D.M.
183
Do you see why the given diagram is separable by half-spaces? Can you construct a similar diagram for the map $x to (x^2,x)$?
– Daniel Littlewood
Jul 16 at 14:47
For the points in the picture, sure. But who's to say, a priori, that there does not exist a collection of points for which first map yields a separation while the second map does not?
– D.M.
Jul 16 at 14:55
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The following is a slide from a presentation on machine learning, the context is using the kernel trick for non-linear separation.
Does it make any difference that the mapping is $xmapsto (x,x^2)$ rather then $xmapsto (x^2,x)$ or will the data be separable under one mapping iff it is separable under the other? In general, in the context of embedding samples to some feature space, does the order of the coordinates matter? I may be missing something quite fundamental here but it would seem that this can have an effect on the separability of the embedded vectors.
linear-algebra machine-learning
asked Jul 16 at 14:39
D.M.
183
The following is a slide from a presentation on machine learning, the context is using the kernel trick for non-linear separation.
Does it make any difference that the mapping is $xmapsto (x,x^2)$ rather then $xmapsto (x^2,x)$ or will the data be separable under one mapping iff it is separable under the other? In general, in the context of embedding samples to some feature space, does the order of the coordinates matter? I may be missing something quite fundamental here but it would seem that this can have an effect on the separability of the embedded vectors.
linear-algebra machine-learning
asked Jul 16 at 14:39
D.M.
183
asked Jul 16 at 14:39
D.M.
183
asked Jul 16 at 14:39
D.M.
183
asked Jul 16 at 14:39
D.M.
183
183
Do you see why the given diagram is separable by half-spaces? Can you construct a similar diagram for the map $x to (x^2,x)$?
– Daniel Littlewood
Jul 16 at 14:47
For the points in the picture, sure. But who's to say, a priori, that there does not exist a collection of points for which first map yields a separation while the second map does not?
– D.M.
Jul 16 at 14:55
add a comment |Â
Do you see why the given diagram is separable by half-spaces? Can you construct a similar diagram for the map $x to (x^2,x)$?
– Daniel Littlewood
Jul 16 at 14:47
For the points in the picture, sure. But who's to say, a priori, that there does not exist a collection of points for which first map yields a separation while the second map does not?
– D.M.
Jul 16 at 14:55
Do you see why the given diagram is separable by half-spaces? Can you construct a similar diagram for the map $x to (x^2,x)$?
– Daniel Littlewood
Jul 16 at 14:47
Do you see why the given diagram is separable by half-spaces? Can you construct a similar diagram for the map $x to (x^2,x)$?
– Daniel Littlewood
Jul 16 at 14:47
For the points in the picture, sure. But who's to say, a priori, that there does not exist a collection of points for which first map yields a separation while the second map does not?
– D.M.
Jul 16 at 14:55
For the points in the picture, sure. But who's to say, a priori, that there does not exist a collection of points for which first map yields a separation while the second map does not?
– D.M.
Jul 16 at 14:55
add a comment |Â
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Do you see why the given diagram is separable by half-spaces? Can you construct a similar diagram for the map $x to (x^2,x)$?
– Daniel Littlewood
Jul 16 at 14:47
For the points in the picture, sure. But who's to say, a priori, that there does not exist a collection of points for which first map yields a separation while the second map does not?
– D.M.
Jul 16 at 14:55