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I am just learning about Mercer Kernels, and a question came up. Since using Mercer's theorem, we know that a positive definite kernel matrix can be represented by an inner production of the input vector mapped to new feature space implied by the kernel.

A Gram matrix of $X$ is defined a $K(X;k)\in \mathbb{R}^{m\times m}$ such that $K_{i,j}=k(\hat{x}_i,\hat{x}_j)$. If the matrix $K$ is positive definite, then $k$ is called a Mercer Kernel. By Mercer's Theorem, if we have a Mercer kernel, then there exists a function $\phi: X \to Y $ such that $$k(\hat{x}_i,\hat{x}_j)=\langle \phi(\hat{x}_i),\phi(\hat{x}_j) \rangle $$ The question is, since this is the case, why do we need to use the kernel function at all? Why not just transform the data according to $\phi$ and use the transformed features to train the SVM. Apparently with this approach there should be some difficulty while classifying a new datapoint, but I am not quite finding the issue.

Thanks!

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    $\begingroup$ I'm not totally sure what you're asking, but I will point out that the value of the kernel trick is that you don't need to use the $\phi$ function at all, instead you can work directly with the kernel function. For example, a kernel for a polynomial basis of degree $d$ can be written as $(x^Ty+1)^d$, far simpler to compute than computing $\phi$, which for a three-element $x$ and $d=2$ results in a nine-element $\phi(x)$, which then has to be multiplied by a nine-element $\phi(y)$... and that's a very small example. $\endgroup$ – jbowman Nov 8 '19 at 20:52
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Say our data lives in $\mathbb R$ and we're using the kernel $$k(x, y) = 1 + 2 x y + x^2 y^2,$$ which corresponds to $$\phi(x) = \begin{bmatrix}1 \\ \sqrt 2 x \\ x^2\end{bmatrix}.$$

If we train a linear SVM on the one-dimensional $\phi(x)$ data, or a kernel SVM on the three-dimensional $x$ data, we'll get the same prediction rule for new data out. So, in this case, the kernel trick is "unnecessary."

But say our data lives in $\mathbb R^d$ and we want to use the kernel $$k(x, y) = 1 + 2 x^T y + (x^T y)^2 = (x^T y + 1)^2.$$ Then the corresponding features end up being of dimension $\frac12 d^2 + \frac32 d + 1$. This is suddenly a much larger model, that will take more computation to solve, more memory to store, etc.

Even worse, say we want to use the kernel $$ k(x, y) = \exp\left( - \frac{1}{2 \sigma^2} \lVert x - y \rVert^2 \right) .$$ The $\phi$ features here end up being infinite-dimensional (see here), which means it will take an infinite amount of time and memory to compute with directly. But using the kernel trick, we can do it just fine.

It can also be much easier to choose functions $k$ with certain properties than it is to design feature functions $\phi$. (Kernels are no harder to design than feature functions – you can just write them as $\phi(x)^T \phi(y)$ directly, after all – but the opposite direction can be quite difficult.)

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  • $\begingroup$ This is basically the same as @jbowman's comment, but I was already writing it when the comment appeared, so... :) $\endgroup$ – Dougal Nov 8 '19 at 20:58
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    $\begingroup$ If you know an explicit $\phi$, there is no issue with new data points; you just use $\phi$ of them. What you might be referring to is what happens if you don't know $\phi$ but are trying to work with a known $n \times n$ kernel matrix $\mathbf K$ in a linear way. In that case, you can construct a $\phi$ valid for only those $n$ points by taking a Cholesky decomposition, $\mathbf K = \boldsymbol\Phi \boldsymbol\Phi^T$; then you can use the $i$the row of $\boldsymbol\Phi$ as $\phi(x_i)$, but won't be able to extend that to new data points. $\endgroup$ – Dougal Nov 9 '19 at 0:26
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    $\begingroup$ Because these features are in $\mathbb R^n$, though, you generally don't gain anything computationally with this approach versus just working with the $n\times n$ kernel matrix in the first place (and you lose a lot); it's only in fairly limited circumstances where this Cholesky trick is useful. $\endgroup$ – Dougal Nov 9 '19 at 0:28
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    $\begingroup$ @jeffery_the_wind When we do $\mathbf K = \boldsymbol\Phi \boldsymbol\Phi^T$, we've found points in $\mathbb R^n$ whose inner products agree with $k$ on the $n$ input points; this is the feature embedding for a kernel $$k_X(x, y) = \begin{cases} k(x_i, x_j) & \text{if } x = x_i, y = y_j \\ 0 & \text{otherwise}\end{cases}.$$ But we haven't found the value of a valid feature embedding for $k$ even for those $n$ points: that should take values in an infinite-dimensional $\mathcal H$, not $\mathbb R^n$. In particular, the value of $\phi(x)$ shouldn't depend on the points you're comparing it to. $\endgroup$ – Dougal Nov 9 '19 at 1:12
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    $\begingroup$ @jeffery_the_wind They're sparse, but (for a strictly positive definite kernel like the Gaussian) they still need to be $n$ dimensional; one point needs to use all $n$ dimensions. $\endgroup$ – Dougal Nov 9 '19 at 3:53

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