A visual introduction to kernel functions
kelvinpaschal.com
kelvinpaschal.com
>> ... A GP works by constructing an infinite amount of guesses or functions of the true process you want to approximate. As you accumulate more observations, it changes the shape of these functions to match the data, and hence the true process (just like the way you change your mind after getting new information)
Why is the 'true process' changing here? I understand our best guess or model is changing with new observations, but the true process should not be changing. If it actually is, then the formulation should be changed to isolate the parameters that is feeding back to it.
>> ... A GP works by constructing an infinite amount of guesses or functions of the true process you want to approximate. As you accumulate more observations, it changes the shape of these functions to match the data, ... A GP is simply a distribution over functions (or guesses). Because we have an infinite amount of guesses, the expected true guess (or best model) is the mean of all plausible guesses.
So is the shape of each function changing? OK. What is the 'distribution' over the functions doing? Is that also changing? Is the said 'distribution' just flat mean of these functions?
>> GP(m(x), k(x, x'))
What is 'x' here? (Sigh! We need to learn to define the variables before using.) I can infer that x' is not derivative of x.
>> In the context of GPs, a kernel or covariance function k(x, x') = Cov(f(x), f(x')), encodes which function values should vary together.
It does not seem the 'f' here is intended to be the specific 'f' introduced at the beginning of the article.
>> I will use the rest of this post to go over different kernel representations and their visualizations.
The plots now have y and x, and x1 and x2. How are these related?
And with k(x, x') = Cov(f(x), f(x')), what is 'f' for the various kernel functions being plotted.
The rest of the post looks fine as plots of the various functions given. But given the above, I have not understood their importance as kernel functions or use for GP.
If the author has a CV attached to their blog, the purpose is to signal competence and the target audience is future employers .
> What is 'x' here?
So as I understand it, a Gaussian Process is defined in terms of a set of random variables which are indexed, typically by either time (t), or space (x). So in the concrete example, x here would be the amount of cheese inserted into the magical machine. In general the "index" can be a vector. Say if the magical machine instead required inserting both cheese and milk to produce some amount of gold, the index x would be two-dimensional, to represents the various amounts of cheese and milk you inserted.
> It does not seem the 'f' here is intended to be the specific 'f' introduced at the beginning of the article.
Right, it's general, and it's kinda confusing to use f when everything else seems to use X_t or similar. Here f is actually a random variable index by x, so one example could be
f(x) = r_1 + x * r_2
> The plots now have y and x, and x1 and x2. How are these related?
The left plot shows three realizations of y = f(x), ie for three different choices (samples) of the random variables that goes into f(x). The right-hand plot shows the output of the kernel function for two indices x and x'. In the first example, the kernel function was the dot product between the two inputs, but given the indices are 1-dimensional that reduces to just k(x, x') = x * x'.
Back to the example, you can feed the machine various amounts of cheese and record the various amounts of gold you get back. The amount of cheese are the indices which you use with the kernel function you picked, which you run through the Gaussian Process regression math, and you get a new function which takes an index (amount of cheese) and returns a normal distribution that predicts the amount of gold for that index (amount of cheese).
The process spits out the mean and the variance of the normal distribution, so you can look at the variance to determine how certain you can be about the prediction which will be centered around the mean.
As I understand it, the point of the left plot is that you can use it to get an idea for which kernel function to use for your measured data. And as mentioned you can easily make new kernel functions by adding (OR-like) and multiplying (AND-like) other kernel functions.
Also the author made a mistake, he mentioned kernel functions are parameterless, but he meant non-parametric. The kernel functions he shows like the periodic kernel has hyperparameters l and p for example.
At least that's my current understanding.
>> f(x) = r_1 + x r_2*
>> The right-hand plot shows the output of the kernel function for two indices x and x'.
The kernel function here would be k(x, x') = Cov[f(x), f(x')] = Cov[f(r_1 + x * r_2), f(r_1 + x' * r_2)].
In this case, I am guessing we should be able to figure what k(x, x') would be, but perhaps would not be x * x' for this case. x * x' sounds to be a very special case.
As I understand it, it would instead be
k(x, x') = Cov[f(x), f(x')] = Cov[r_1 + x * r_2, r_1 + x' * r_2]
He's not saying the true process is changing, just the functions that are being sampled from the GP. The true process refers to the true, underlying function so it's deterministic if you have correctly identified all its inputs.
> So is the shape of each function changing?
Yes, the function changes shape as you get more data because the parameters governing that function (that we define in the kernel) are updated with new observational data, so that over time it converges to the 'true' process/function we are trying to discover.
> What is the 'distribution' over the functions doing? Is that also changing? Is the said 'distribution' just flat mean of these functions?
I think you're confused because the example given with cheese is really confusing when we're trying to understand the functions as arising from a multivariate distribution. So, I'll try to clarify that part. GPs are typically used to represent some function where the input is time or distance. This is why its called a 'process' - because the variables in a random process are indexed by space or time. So in this 1D example, in the X domain, [x1, x2, x3] represents something like fixed increments of increasing cheese. f(X) represents the gold amount. Now imagine gold can take any value from 0-100. Now plot all possible values of f(x1) on the x axis of a grid, f(x2) on the y-axis of the grid, and f(x3) on the z-axis of the grid. We have 100^3 points in this 3D grid. If we select one point, it's x,y,z coordinates correspond to the f(x1), f(x2) and f(x3) gold amounts. The dimension index, corresponds (typically) to something like time, or distance. In this example it's cheese.
In a GP, we're modeling the sampled f(X) point as if its from a 3D multivariate normal distribution. So sampling one point gives us the gold amount for cheese amount 1, 2, and 3. This is the 'function', and as we sample more points, we get more 'functions' that give us varying gold amounts for cheese amount 1, 2, and 3. And because it's a multivariate distribution, we can capture correlations between dimensions, so the amount of gold you get for cheese-1, should influence how much gold you get at cheese-2 because its close by. This relationship is defined by the covariance function of the gaussian.
> GP(m(x), k(x, x')) What is 'x' here? (Sigh! We need to learn to define the variables before using.) I can infer that x' is not derivative of x.
x refers to some amount of gold, and k(x, x') just means that the kernel consumes any two values in our X vector (i.e. [x1, x3] or [x1, x2]).
> "In the context of GPs, a kernel or covariance function k(x, x') = Cov(f(x), f(x')), encodes which function values should vary together." It does not seem the 'f' here is intended to be the specific 'f' introduced at the beginning of the article.
I believe it is the same f actually. He's saying the kernel function takes in two values of x (cheese), and outputs the covariance between their output gold amounts. This illustrates his previous point that the "closeness" between x values should be reflected in the gold amounts.
> The plots now have y and x, and x1 and x2. How are these related?
y is gold. x is cheese. x1, x2 correspond to the first two x-values in the linear plot.
> And with k(x, x') = Cov(f(x), f(x')), what is 'f' for the various kernel functions being plotted.
f(X) is the approximation of the "true" process we're trying to learn from observational data. The observations are tuples of cheese and gold amoutns, so f(x), f(x') is just the corresponding gold amount, we don't actually model that function explicitly. The gaussian distribution we are sampling from for functions just models correlations between our variables, so it represents the function implicitly.
>> So sampling one point gives us the gold amount for cheese amount 1, 2, and 3. This is the 'function', and ...
I get this part, so each point in this N-dimensional space yields a function f of the index, and this is the function.
>> Yes, the function changes shape as you get more data because the parameters governing that function
Getting more data should now get more such points (in N-dimensional space), but with each such point being the 'function' how is it changing shape.
Nevertheless, I think I have much better glimpses after reading your and other other responses here than from the original article, which I still find confusing even on reading again.
So how does it change shape? Well this part is actually something I don't fully grasp myself yet. But I can sketch a crude bayesian interpretation, which is how I think of it. Not completely correct but works as a placeholder until I fully work out the math of updating the parameters.
Basically, from a bayesian perspective we can condition the joint distribution of function outputs as a likelihood conditioned on the kernel parameters theta: p(f(x1), f(x2), ... | theta).
Then we can derive the posterior distribution over theta p(theta | f(x1), f(x2), ...) like so:
p(theta | f(x1), f(x2), ...) ≈ p(f(x1), f(x2), ... | theta) p(theta).
So we fit the theta parameters based on how well it fits the observed data we feed our bayesian model.
FWIW, I recommend chapter 14 of Richard McElreath's Statistical Rethinking for a better introduction of GPs. This article kind of glosses over a lot of the intuition and introductory concepts that you need to really grok it.
This is about inner product functions in a specific kind of Hilbert spaces, a notion that is very useful in many branches of applied mathematics. Machine learning and functional analysis included.
The name collision is unfortunate.
One unfortunate difficulty is that these kernels don't map so well to GPU kernels unless explicitly embedded in high dimensional spaces. This is one of the reasons why kernel methods has recently fallen out of favour in machine learning - lack of mechanical sympathy. Note this just one reason, there are others.