Kernels Part 2: Affine Quantum Gravity, continued

What is a Kernel?  Really?   What is the actual mathematical motivation?

Most machine learning texts introduce the Kernel  as some horribly abstract , infinite, continuous operator.  It is seemingly disconnected from the problem at hand; one simply need choose any Kernel, cross-validate, cross-validate and cross-validate, and a good solution will emerge.

Or, the Kernel is just some positive definite matrix, constructed from some finite basis set.  So what’s with all the formalism if this is all the Kernel is?

In Quantum Physics, and in particular (Affine) Quantum Gravity (ala Klauder: ), the Kernel representation of the Hilbert Space plays the role of connecting the continuous space of Classical Physics to the discrete representation of Quantum Mechanics.  Let’s take a look.

Let’s start small–at the Quantum level–and define a discrete but infinite basis of functions , say \left\{ \mid n \rangle \right\}_{n=0}^{\infty} , such that we have a complete , orthonormal basis. Then we can write the Resolution of the Identify Operator using Dirac notation as I =\sum_{n=0}^{\infty} |n\rangle\langle n| and \langle n | m \rangle = \delta_{n,m} . In physics, a discrete, orthonormal basis is pretty common, however, we shall see that the Kernel allows us to generalize the Resolution of the Identity Operator to continuous and non-orthogonal basis.  Indeed, frequently we insert Kernels into dot products (the so-called Kernel Trick) \langle n | m \rangle =K(n,m) .  But very rarely do we see the actual Kernel basis functions.  What might these look like?

We now will implement what I call the Actual Kernel Trick.  We can convert our basis of discrete functions \left\{ \mid n \rangle \right\}_{n=0}^{\infty} , to a set of continuous functions , say, \left\{ \mid \xi \rangle \right\} with a parametric resummation:

|\xi\rangle=\exp-\frac{|\xi^{2}|}{2}\sum_{n=0}^{\infty}\frac{\xi^{n}}{\sqrt{n!}}|n\rangle   where \xi is a complex number

(There are many such resummations; this particular form generates the so-called squeezed states, or coherent states, of quantum dynamics. More on this later)

Notice these states are no longer orthonormal: \langle \xi' | \xi \rangle = \exp-\frac{1}{2}(|\xi'^{2}+\xi^{2}-2\xi'\xi|) \neq \delta(\xi'-\xi)

This means that in order to express the Resolution of the Identity Operator , we need to redfine the measure on the space.  Here, we change  \int dxdy to \int \frac{dxdy}{\pi}:

\int|\xi\rangle\langle\xi|\frac{dxdy}{\pi}=\sum_{n,m=0}^{\infty}\frac{|n\rangle\langle m|}{\sqrt{n!m!}}\frac{}{}\int\exp-|\xi^{2}|\xi^{n}\xi^{*m}\frac{dxdy}{\pi}= 1

This is why we need to keep track of the Kernel when computing dot products–the measure on the space has changed.

To satisfy the mathematical conditions for Kernel, we should also check that the new functions { | \xi \rangle } are actually continuous.  I’ll leave this as an exercise.

Again, the actual  intent of the Kernel Trick is to allow us to represent a infinite set of discrete functions with a family of continuous functions.  And in many cases in machine learning, we can express  our dot products \langle \xi' | \xi \rangle with a simple analytic function, with some adjustable parameters.

Notice that this is exactly what we accomplished in our previous post: we converted the Radial Basis Function (RBF) Kernel into an infinite sum of spatial derivative operators.  And this is exactly the point.  We don’t specify  a Kernel to just randomly guess what is going on !  We specify the Kernel to provide a compact, even analytic, representation of our prior knowledge of the problem.

I think that’s enough for tonight. In our next post, I will address some actual physics.


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