Kernel Trick and Inner Product Preservation

I understand that the point of using the kernel trick is to project the problem onto a higher dimensional space, where the problem is linearly separable. In this explanation, https://www.quora.com/What-is-the-kernel-trick, it states that the inner product $$\langle x,y \rangle$$ will be equal to $$\langle \phi(x),\phi(y)\rangle$$. Understanding this equality seems to be key to understanding how this trick works.

My question is, how do know that our function $$\phi$$ will preserve the inner product and what are the conditions for this to happen?

I have tried google searching this and despite many references to the kernel trick, I do not believe that this has been answered anywhere.

It doesn't mean that $$\langle x, y \rangle = \langle \phi(x), \phi(y)\rangle$$.
Kernel method in general means that for an algorithm that involve $$\langle x, y\rangle$$, we can replace it with a function $$K(x,y)=\langle \phi(x), \phi(y)\rangle$$ where computing $$K(x,y)$$ is easy. It is known that for $$K$$ that satisfies Mercer's theorem, there is a corresponding $$\phi$$ which is the map to the higher dimensional space.
We do not need to know $$\phi$$ explicitly and $$\phi$$ can be very complicated.
• what gives you the right to replace with $K(x,y)$? Also are you saying that the equality in the guys post is by chance? Sep 22 '19 at 8:11
• for some algorithms, we might find that it just involve inner product with some features space, in that case, rather than computing $\langle x, y \rangle$, that is working with $x$ directly, we might like to perform a transformation $\phi(x)$, and then work with it. Since it just involves inner product, what matters is just $\langle \phi(x), \phi(y) \rangle$, and we can compute it via $K(x,y)$ and we don't really care about $\phi$ explicitly sometimes. As for the "by chance" thing, I might need you to be more explicit. Sep 22 '19 at 8:17
• what is the justification for be able to swap to $K(x,y)$? Sep 22 '19 at 9:59
• The justification is rather than working with $x$ directly, we want to work with $\phi(x)$, for example, rather than working with $x$, you might want to work with $(x, x^2)$. Then you want to work with inner product of $(x, x^2)$ and $(y, y^2)$, this is denote as $K(x,y)= \langle (x, x^2), (y, y^2) \rangle$. Sep 22 '19 at 10:05
• Yes but why can you swap $x$ for $\phi(x)$? What are the requirements on $\phi$? Sep 22 '19 at 10:06