Hello! I can't understand the passages associated to the topic "Classifying with the kernel K" in lecture 7b notes.
We start saying that
y = φ(x)^T w
and considering the representer theorem we arrive at the expression
φ(x)^T φ(X)^T α
but I would have written
φ(x)^T X^T α
Is that right? Are the two expressions equivalent? Because, if they are, I can't see why.
Thanks for your answer!
I think that when you do feature augmentation, you have always to consider φ(X), the augmented matrix, and not X, even for the prediction. It wouldn't make sense to augment a datapoint and then use the matrix X to predict, and the dimensions in most of the cases wouldn't even match.
I agree, I would say the same, if you learn parameters in the augmented space, since the kernel is non-linear, you cannot relate them to the original vector in a linear relationship (here matrix multiplication)
@Anonymous said:
Can you explain a bit more why you would have written φ(x)^T X^T α?
I was just applying the result of the representer theorem, saying that w = X^T α.
Hence, φ(x)^T w = φ(x)^T X^T α.
But what the others said completely makes sense. Thanks everyone!
Classifying with the kernel K
Hello! I can't understand the passages associated to the topic "Classifying with the kernel K" in lecture 7b notes.
We start saying that
y = φ(x)^T w
and considering the representer theorem we arrive at the expression
φ(x)^T φ(X)^T α
but I would have written
φ(x)^T X^T α
Is that right? Are the two expressions equivalent? Because, if they are, I can't see why.
Thanks for your answer!
I think that when you do feature augmentation, you have always to consider φ(X), the augmented matrix, and not X, even for the prediction. It wouldn't make sense to augment a datapoint and then use the matrix X to predict, and the dimensions in most of the cases wouldn't even match.
2
I agree, I would say the same, if you learn parameters in the augmented space, since the kernel is non-linear, you cannot relate them to the original vector in a linear relationship (here matrix multiplication)
Can you explain a bit more why you would have written φ(x)^T X^T α?
I was just applying the result of the representer theorem, saying that w = X^T α.
Hence, φ(x)^T w = φ(x)^T X^T α.
But what the others said completely makes sense. Thanks everyone!
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