Efficient factoring of GP likelihoods

October 16, 2020 — October 26, 2020

graphical models
Hilbert space
kernel tricks
machine learning
Figure 1

There are many ways to cleverly slice up GP likelihoods so that inference is cheap.

This page is about some of them, especially the union of sparse and variational tricks. Scalable Gaussian process regressions choose cunning factorisations such that the model collapses down to a lower-dimensional thing than it might have seemed to need, at least approximately. There is a comptilation of tricks to make this go — variational approximations a model, sparse GP models where there are a small number of inducing points (Dezfouli and Bonilla 2015; Edwin V. Bonilla, Krauth, and Dezfouli 2019; Krauth et al. 2016; Hensman, Fusi, and Lawrence 2013; Salimbeni and Deisenroth 2017). You might suspect yourself of using such a method if you find that some important high-dimensional expectation can be evaluated by some function of univariate Gaussians.

This is a related notion to other tricks which factorise a distribution cleverly, such message-passing inference. There are indeed a lot of different factorisations that can be done here; See filtering GPs for one which factorizes over a single input axis. Also Toeplitz and related structures work out nicely for, e.g. lattice-distributed inputs and some other situations I forget right now. I will bet you they can all be used together.

1 Inducing variables

See GP inducing variables.

2 Inducing features

See GP inducing features.

3 Spectral and rank sparsity

Loosely speaking, where the functions can be represented in a small number of basis functions. See, for, example (Adam et al. 2020; Zammit-Mangion and Cressie 2021).

4 SVI for Gaussian processes

As seen in Hensman, Fusi, and Lawrence (2013);Salimbeni and Deisenroth (2017).

5 Low rank methods

Represent the GP in terms of a controlled budget of basis functions. See low-rank Gaussian processes.

6 Vecchia factorisation

Approximate the precision matrix by one with a sparse cholesky factorisation. See Vecchia factorization.

7 Local

Approximate covariance by nearby predictors.

8 Latent Gaussian Process models

The Edwin V. Bonilla, Krauth, and Dezfouli (2019) set up for Latent Gaussian Process models (“LGPMs”) goes as follows:

We are learning a mapping \(\boldsymbol{f}:\mathbb{R}^D\to\mathbb{R}^P\) from data. The dataset looks like \(\mathcal{D}=\left\{\mathbf{x}_{n}, \mathbf{y}_{n}\right\}_{n=1}^{N}\equiv \left\{\mathbf{x}, \mathbf{y}\right\}.\) \(\mathbf{x}_{n}\in \mathbb{R}^D\) is an input vector and \(\mathbf{y}_{n}\in\mathbb{R}^P\) is an output. We decree that the mapping from inputs to outputs, may be expressed by \(Q\) underlying latent functions \(\left\{f_{j}\right\}_{j=1}^{Q}.\) We assume that the \(Q\) latent functions \(\left\{f_{j}\right\}\) are drawn from (a priori) independent zero-mean Gaussian processes.

\[ \begin{aligned} p\left(f_{j} \mid \boldsymbol{\theta}_{j}\right) & \sim \mathcal{G} \mathcal{P}\left(0, \kappa_{j}\left(\cdot, \cdot ; \boldsymbol{\theta}_{j}\right)\right), \quad j=1, \ldots Q, \quad \text { and } \\ p(\mathbf{f} \mid \boldsymbol{\theta}) &=\prod_{j=1}^{Q} p\left(\mathbf{f}_{\cdot j} \mid \boldsymbol{\theta}_{j}\right) \\ &=\prod_{j=1}^{Q} \mathcal{N}\left(\mathbf{f}_{\cdot j} ; \mathbf{0}, \mathbf{K}_{\mathbf{x x}}^{j}\right). \end{aligned} \] Here \(\mathbf{f}\) is the set of all latent function values; \(\mathbf{f}_{\cdot j}=\left\{f_{j}\left(\mathbf{x}_{n}\right)\right\}_{n=1}^{N}\) denotes the values of latent function \(j\). The Gram matrix is \(\mathbf{K}_{\mathrm{xx}}^{j}\), induced by a covariance kernel, \(\kappa_{j}\left(\cdot, \cdot ; \boldsymbol{\theta}_{j}\right)\). The parameters of all kernel functions we call \(\boldsymbol{\theta}=\left\{\boldsymbol{\theta}_{j}\right\}.\) Our observation model can have various likelihoods; We call the corresponding parameter \(\boldsymbol{\phi}\). We assume that our multi-dimensional observations \(\left\{\mathbf{y}_{n}\right\}\) are i.i.d. given the latent functions \(\left\{\mathbf{f}_{n}\right\},\) so that \[ p(\mathbf{y} \mid \mathbf{f}, \boldsymbol{\phi})=\prod_{n=1}^{N} p\left(\mathbf{y}_{n} \mid \mathbf{f}_{n \cdot}, \boldsymbol{\phi}\right) \] \(\mathbf{f}_{n\cdot}=\{f_{j}(\boldsymbol{x}_n)\}_{j=1}^{q}\) is the set of latent \(\boldsymbol{f}\) values upon which \(\mathbf{y}_{n}\) depends.

There are several factorizations to note here

  1. The prior is factored into latent functions per-coordinate
  2. the conditional likelihood is factored over observations (i.e. nosie in independent)

If we further factorise the variational approximation in some way this will work out nicely, e.g. into Gaussian mixtures. This is going to work out well for us when we try to devise a system of inference later to minimise the ELBO. TBC.

For now, though, let us examine exactly tractable inference

9 References

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Bonilla, Edwin V. 2017. “Variational Learning of GP Models.”
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Bonilla, Edwin V., Krauth, and Dezfouli. 2019. Generic Inference in Latent Gaussian Process Models.” Journal of Machine Learning Research.
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