Covariance kernel zoo

Stationary

A popular assumption, more or less implying implies that no region of the process is special. In this case the kernel may be written as a function purely of the distance between observations, i.e.

\[K(s,t)=K(\|s-t\|)\] for some distance \(\|\cdot\|\) between the observation coordinates.

Wiener process kernel

From the naming, we might suspect that a Gaussian process would also describe a standard Wiener process, which after all is a process with Gaussian increments, which is certain type of dependence. It is over a boring index space, time \(t\in \mathbb{R}\), but there is indeed nothing stopping us.

We can read this right off the Wiener process Wikipedia page. For a Gaussian process \(\{W_t\}_{t\in\mathbb{R}},\)

\[ {\displaystyle \operatorname {cov} (W_{s},W_{t})=s \wedge t} \]

Here \(s \wedge t\) here means “the minimum of \(s\) and \(t\)”.

That result is standard. From it we can immediately construct the kernel \(K(s,t):=s \wedge t\).

Causal kernels

Time-indexed processes are more general than a standard Wiener process. 🏗

What constraints make a covariance kernel causal?

Markov kernels

How can we know from inspecting a kernel whether it implies an independence structure of some kind? The Wiener process and causal kernels clearly imply certain independences. Any kernel \(k(s,t)=k(s\wedge t)\) is clearly Markov. Are there more general ones? TODO: relate to kernels of bounded support. 🏗

Squared exponential

A.k.a. exponentiated quadratic. Often radial basis functions mean this also.

The classic, default, analytically convenient, because it is proportional to the Gaussian density and therefore cancels out with it at opportune times.

\[k_{\textrm{SE}}(x, x') = \sigma^2\exp\left(-\frac{(x - x')^2}{2\ell^2}\right)\]

Rational Quadratic

Duvenaud reckons this is everywhere but TBH I have not seen it. Included for completeness.

\[k_{\textrm{RQ}}(x, x') = \sigma^2 \left( 1 + \frac{(x - x')^2}{2 \alpha \ell^2} \right)^{-\alpha}\]

Note that \(\lim_{\alpha\to\infty} k_{\textrm{RQ}}= k_{\textrm{SE}}\).

Matérn

The Matérn stationary (and in the Euclidean case, isotropic) covariance function is one model for covariance. See Carl Edward Rasmussen’s Gaussian Process lecture notes for a readable explanation, or chapter 4 of his textbook (Rasmussen and Williams 2006).

\[ k_{\textrm{Mat}}(x, x')=\sigma^{2} \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2 \nu} \frac{x - x'}{\rho}\right)^{\nu} K_{\nu}\left(\sqrt{2 \nu} \frac{x - x'}{\rho}\right) \]

where \(\Gamma\) is the gamma function, \(\ K_{\nu }\) is the modified Bessel function of the second kind, and \(\rho,\nu\geq 0\).

AFAICT you use this for covariances hypothesised to be less smooth than the squared exponential covariance. And other things?

Periodic

\[ k_{\textrm{Per}}(x, x') = \sigma^2\exp\left(-\frac{2\sin^2(\pi|x - x'|/p)}{\ell^2}\right) \]

Locally periodic

This is an example of a composed kernel, explained below.

\[\begin{aligned} k_{\textrm{LocPer}}(x, x') &= k_{\textrm{Per}}(x, x')k_{\textrm{SE}}(x, x') \\ &= \sigma^2\exp\left(-\frac{2\sin^2(\pi|x - x'|/p)}{\ell^2}\right) \exp\left(-\frac{(x - x')^2}{2\ell^2}\right) \end{aligned}\]

Obviously there are other possible localisations of a periodic kernel. This is a Locally Periodic kernel.

“Integral” kernel

I just noticed the ambiguously named Integral kernel:

I’ve called the kernel the ‘integral kernel’ as we use it when we know observations of the integrals of a function, and want to estimate the function itself.

Examples include:

• Knowing how far a robot has travelled after 2, 4, 6 and 8 seconds, but wanting an estimate of its speed after 5 seconds…
• Wanting to know an estimate of the density of people aged 23, when we only have the total count for binned age ranges…

I would argue that all kernels are naturally defined in terms of integrals, but the author seems to mean something particular. I suspect I would call this a sampling kernel, but that name is also overloaded. Anyway, what is actually going on here? Where is it introduced? Possibly one of (Smith, Alvarez, and Lawrence 2018; O’Callaghan and Ramos 2011; Murray-Smith and Pearlmutter 2005).

Composing kernels

A sum or product (or outer sum, or tensor product) of kernels is still a kernel. For other transforms YMMV.

For example, in the case of Gaussian processes, suppose that, independently,

\[\begin{aligned} f_{1} &\sim \mathcal{GP}\left(\mu_{1}, k_{1}\right)\\ f_{2} &\sim \mathcal{GP}\left(\mu_{2}, k_{2}\right) \end{aligned}\] then

\[ f_{1}+f_{2} \sim \mathcal{GP} \left(\mu_{1}+\mu_{2}, k_{1}+k_{2}\right) \] so \(k_{1}+k_{2}\) is also a kernel.

More generally, if \(k_{1}\) and \(k_{2}\) are two kernels, and \(c_{1}\), and \(c_{2}\) are two positive real numbers, then:

\[ K(x, x')=c_{1} k_{1}(x, x')+c_{2} k_{2}(x, x') \] is again a kernel. What with the multiplication as well, we note that all polynomials of kernels where the coefficients are positive are in turn kernels. (Genton 2001)

Note that the additivity in terms of kernels is not the same as additivity in terms of induced feature spaces. The induced feature map of \(k_{1}+k_{2}\) is their concatenation rather than their sum. Suppose \(\phi_{1}(x)\) gives us the feature map of \(k_{1}\) for \(x\) and likewise \(\phi_{2}(x)\).

\[\begin{aligned} k_{1}(x, x') &=\phi_{1}(x)^{\top} \phi_{1}(x') \\ k_{2}(x, x') &=\phi_{2}(x)^{\top} \phi_{2}(x')\\ k_{1}(x, x')+k_{2}(x, x') &=\phi_{1}(x)^{\top} \phi_{1}(x')+\phi_{2}(x)^{\top} \phi_{2}(x')\\ &=\left[\begin{array}{c}{\phi_{1}(x)} \\ {\phi_{2}(x)}\end{array}\right]^{\top} \left[\begin{array}{c}{\phi_{1}(x')} \\ {\phi_{2}(x')}\end{array}\right] \end{aligned}\]

If \(k:\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}\) is a kernel and \(\psi: \mathcal{X}\to\mathcal{Y}\) this is also a kernel

\[\begin{aligned} k_{\psi}:&\mathcal{X}\times\mathcal{X}\to\mathbb{R}\\ & (x,x')\mapsto k_{y}(\psi(x), \psi(x')) \end{aligned}\]

which apparently is now called a deep kernel. if \(k\) is stationary and \(\psi\) is invertible then this is a stationary reducible kernel.

Also if \(A\) is a positive definite operator, then of course it defines a kernel \(k_A(x,x'):=x^{\top}Ax'\)

(Genton 2001) uses the properties of covariance to construct some other nifty ones:

Let \(h:\mathcal{X}\to\mathbb{R}^{+}\) have minimum at 0. Then, using the identity for RVs

\[ \mathop{\textrm{Cov}}\left(Y_{1}, Y_{2}\right)=\left[\mathop{\textrm{Var}}\left(Y_{1}+Y_{2}\right)-\mathop{\textrm{Var}}\left(Y_{1}-Y_{2}\right)\right] / 4 \]

we find that the following is a kernel

\[ K(x, x')=\frac{1}{4}[h(x+x')-h(x-x')] \]

All these to various cunning combination strategies, which I may return to discuss. 🏗 Some of them are in the references. For example (D. Duvenaud et al. 2013) position their work in the wider field.

There is a large body of work attempting to construct a rich kernel through a weighted sum of base kernels (e.g. (Bach 2008; Christoudias, Urtasun, and Darrell 2009)). While these approaches find the optimal solution in polynomial time, speed comes at a cost: the component kernels, as well as their hyperparameters, must be specified in advance […]

(Hinton and Salakhutdinov 2008) use a deep neural network to learn an embedding; this is a flexible approach to kernel learning but relies upon finding structure in the input density, p(x). Instead we focus on domains where most of the interesting structure is in f(x).

(Wilson and Adams 2013) derive kernels of the form \(SE × \cos(x − x_0\)), forming a basis for stationary kernels. These kernels share similarities with \(SE × Per\) but can express negative prior correlation, and could usefully be included in our grammar.

See (Grosse et al. 2012) for a mind-melting compositional matrix factorization diagram, constructing a search over hierarchical kernel decompositions.

Examples of existing machine learning models which fall under our framework. Arrows represent models reachable using a single production rule. Only a small fraction of the 2496 models reachable within 3 steps are shown, and not all possible arrows are shown.

Stationary spectral kernels

(Sun et al. 2018; Bochner 1959; Kom Samo and Roberts 2015; Yaglom 1987) construct spectral kernels in the sense that they use the spectral representation to design the kernel and guarantee it is positive definite and stationary. See Bochner’s theorem.

Non-stationary spectral kernels

(Sun et al. 2018; Remes, Heinonen, and Kaski 2017; Kom Samo and Roberts 2015) use a generalised Bochner Theorem (Yaglom 1987) often called Yaglom’s Theorem, which does not presume stationarity. See Yaglom’s theorem.

It is not immediately clear how to use this; spectral representations are not an intuitive way of constructing things.

Locally stationary kernels

(Genton 2001) defines these are kernels that have a particular structure, specifically, a kernel that can be factored into a stationary kernel (_2) and a non negative function \(K_1\) in the following way:

\[ K(\mathbf{s}, \mathbf{t})=K_{1}\left(\frac{\mathbf{s}+\mathbf{t}}{2}\right) K_{2}(\mathbf{s}-\mathbf{t}) \]

Global structure then depends on the mean location \(\frac{\mathbf{s}+\mathbf{t}}{2}\). (Genton 2001) describes some nifty spectral properties of these kernels.

Other constructions might vie for the title of “locally stationary”. To check. 🏗

Compactly supported

We usually think about compactly supported kernels in the stationary isotropic case, where we mean kernels that vanish whenever the distance between two observation \(s,t\) is larger than a certain cut-off distance \(L,\) i.e. \(\|s-t\|>L\Rightarrow K(s,t)=0\). These are great because they make the Gram matrix sparse (for example, if the cut-off is much smaller than the diameter of the observations and most observations have few covariance neighbours) and so can lead to computational efficiency even for exact inference without any special tricks. They don’t seem to be popular? Statisticians are generally nervous around inferring the support of a parameter, or assigning zero weight to any region of a prior without good reason, so maybe it is that?

(Genton 2001) mentions

\[ \max \left\{\left(1-\frac{\|\mathbf{s}-\mathbf{t}\|}{\tilde{\theta}}\right)^{\tilde{\nu}}, 0\right\} \] and handballs us to (Gneiting 2002b) for a bigger smörgåsbord of stationary compactly supported kernels. Gneiting (2002b) article has a couple of methods designed to produce certain smoothness properties at boundary and origin, but mostly concerns producing compactly supported kernels via clever integral transforms.

Genton kernels

That’s my name for them because they seem to originate in (Genton 2001).

For any non-negative function \(h:\mathcal{T}\to\mathbb{R}^+\) with \(h(\mathbf{0})=0,\) the following is a kernel:

\[ K(\mathbf{s}, \mathbf{t})=\frac{1}{4}[h(\mathbf{s}+\mathbf{t})-h(\mathbf{s}-\mathbf{t}) \] Genton gives the example of \(h:s,t\mapsto s^\top t.\)

The motivation is the identity

\[ \operatorname { Covariance }\left(Y_{1}, Y_{2}\right)= \left[\operatorname { Variance }\left(Y_{1}+Y_{2}\right)-\operatorname { Variance }\left(Y_{1}-Y_{2}\right)\right] / 4. \]

Kernels with desired symmetry

(D. Duvenaud 2014, chap. 2) summarises Ginsbourger et al’s work on kernels with desired symmetries / invariances. 🏗 This produces for example, the periodic kernel above, but also such cute tricks as priors over Möbius strips.

Stationary reducible kernels

See kernel warping.

References