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Gaussian Processes: A Quick Introduction

Mark Ebden

arXiv:1505.02965v2math.ST

TL;DR

The paper introduces Gaussian processes as a supervised, less-parametric approach to estimating dependent variables from noisy observations and extends the discussion to broader applications. It explains covariance-based prediction and parameter selection, while illustrating uses in regression, dimensionality reduction, and other models.

  • Problem

    Given noisy observations of a dependent variable, the central problem is estimating that variable at a new input value.

  • Method

    Gaussian process regression represents f(x) without committing to a specific functional model, using covariance functions and conditional Gaussian prediction with parameters selected by maximizing log p(y|x, θ).

  • Results

    Gaussian processes support regression, dimensionality reduction, integration, global optimization, mixture-of-experts models, and unsupervised learning, while covariance structure informs predictions from observed data.

  • Takeaways & Limitations

    Gaussian processes provide a flexible framework in which data can inform predictions and latent representations, while retaining explicit structural assumptions through covariance functions.

  • Takeaways & Limitations

    Gaussian processes still require structural assumptions, including a carefully chosen covariance function, and are not completely free-form.

Abstract

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A gentle introduction to Gaussian processes (GPs). The three parts of the document consider GPs for regression, classification, and dimensionality reduction.

1 MOTIVATION

The motivation is to estimate a dependent variable at a new input from noisy observations. Gaussian process regression offers a less parametric supervised-learning approach that avoids committing to a specific functional form while retaining assumptions about f(x).

  • Prediction asks for the best estimate of a dependent variable at a new input x∗ from noisy observations.
  • Linear regression fits a straight line when linearity is plausible, while model selection can compare quadratic, cubic, and nonpolynomial possibilities.
  • GPR represents f(x) without claiming a specific model form, allowing the training data to be used more subtly.
  • GPR is less parametric but still requires basic assumptions about f(x); otherwise, more general techniques should be considered.

2 DEFINITION OF A GAUSSIAN PROCESS

Gaussian processes extend multivariate Gaussian distributions to functions: every finite subset of function values follows a multivariate Gaussian distribution. Covariance determines how observations relate, including smoothness, noise handling, and predictive influence.

  • A Gaussian process generates data over a domain such that every finite subset follows a multivariate Gaussian distribution.
  • With a commonly assumed zero mean, the covariance function k(x, x′) specifies relationships between observations.
  • Nearby inputs receive high covariance for smooth functions, whereas distant observations can have negligible influence during interpolation.
  • Observations are related to the underlying function f(x) through a Gaussian noise model to represent measurement errors.
  • The exposition folds observational noise into k(x, x′) using the Kronecker delta, while distinguishing predictions y∗ from the noiseless f∗.
  • The covariance function is evaluated across point combinations and summarized in covariance matrices for Gaussian-process regression.

3 HOW TO REGRESS USING GAUSSIAN PROCESSES

GP regression conditions a joint Gaussian model on observed data to obtain a Gaussian distribution for a new prediction. Its mean supplies the estimate, while its variance quantifies uncertainty and supports confidence intervals.

  • The key modelling assumption is that the data can be represented as a sample from a multivariate Gaussian distribution.
  • Conditioning on the observed data produces a Gaussian conditional distribution p(y∗|y) for the prediction.
  • The mean of the conditional distribution is the best estimate for y∗.
  • The variance of the conditional distribution captures uncertainty in the estimate.
  • Figure 1 applies the procedure to estimate the dependent variable at x∗= 0.2 from the example data.
  • For 1,000 test points, the method plots y∗ ± 1.96 var(y∗), giving a 95% confidence interval.

4 GPR IN THE REAL WORLD

Real-world GPR depends on selecting covariance functions and their parameters appropriately. Covariance functions can encode short- and long-term trends, periodicity, and other expected dynamics, but choosing them remains a form of structural prior knowledge.

  • GPR reliability depends on sensible covariance-function parameters θ = {l, σf, σn}, estimated by maximizing log p(y|x, θ).
  • An optimizer can find a useful parameter choice; in the example, l = 1 and σf = 1.27.
  • Integrating over multiple possible covariance-function parameters is preferable to endorsing only one parameter value.
  • Figure 3(a): For short-term fluctuations plus a long-term trend, additive covariance terms can use different length parameters, with l2 ≈ 6l1 in the example.
  • Figure 3(b): For a long-term hill-like trend with periodicity, additive covariance terms represent the trend and frequency ν.
  • Covariance functions can encode arbitrarily complex expected dynamics provided K is positive definite.
  • Choosing among covariance functions resembles choosing model types and reflects prior knowledge about the physical process under study.

5 DISCUSSION

The discussion highlights Gaussian processes as a powerful, extensible technique whose practical use involves algorithmic implementation challenges.

  • 5 DISCUSSION: Practical implementation requires solving algorithmic hurdles beyond the data-analysis mathematics.The document points readers to code repositories for implementation support.
  • 5 DISCUSSION: Gaussian processes can handle higher-dimensional inputs with little change to the overall mathematics.The input x becomes a vector while k(x, x′) remains scalar.
  • 5 DISCUSSION: The GP mean can be generalized from the zero vector to functions of x.

APPENDIX

The appendix explains how conditioning a multivariate Gaussian uses covariance structure to estimate one subvector from another.

  • APPENDIX: A multivariate Gaussian sample can be partitioned into consecutive subvectors a and b, with covariance blocks A, B, and C.
  • APPENDIX: When the cross-covariance block C is nonzero, the conditional distribution of b given a remains Gaussian and depends on a.If D is diagonal or block diagonal, knowing a provides no information about b.
  • APPENDIX: The conditional mean is CA^-1a, called the matrix of regression coefficients, and the conditional variance is B − CA^-1C^T, the Schur complement.
  • APPENDIX: Off-diagonal covariance elements allow known components of d to inform estimates of its remaining components.

A Quick Introduction

The document is a continuation focused on Gaussian processes for classification, following prerequisite material on Gaussian-process regression.

  • A Quick Introduction: The classification document follows prerequisite reading on Gaussian Processes for Regression.

1 OVERVIEW

Gaussian-process classification uses a latent GP function followed by a squashing function to produce class probabilities for discrete outputs. The overview introduces binary classification, training, multiclass extension, and the cumulative Gaussian choice.

  • 1 OVERVIEW: Gaussian processes can represent class probabilities by squashing their outputs onto [0, 1].The overview focuses on assigning data points to one of two types.
  • 1 OVERVIEW: Unlike regression, classification links discrete outputs y to latent GP outputs f through a squashing function rather than only a noise process.
  • 1 OVERVIEW: The classifier first evaluates a latent function f describing how relative class likelihood changes over x.
  • 1 OVERVIEW: It then maps f onto [0, 1] using a sigmoidal function π(f) = prob(y = 1|f).
  • 1 OVERVIEW: The document previews slower operational explanations, classifier training, and classification with more than two classes.
  • 1 OVERVIEW: The prescribed squashing function is the cumulative Gaussian Φ(f), mapping high f toward 1 and low f toward 0.
  • 1 OVERVIEW: In the absence of noise, the regression equations can be rewritten in a simplified form.

2 USING THE CLASSIFIER

The classifier first predicts a latent function for a new input, then transforms it into a class-membership probability. This probability is obtained by integrating over uncertainty in the latent prediction.

  • Using the classifier: Given a new input x∗, the trained classifier uses GP outputs to determine its class-membership probability π∗.The training outputs have uncertain distributions with mean values ˆf; the new input is processed to obtain π∗.
  • Using the classifier: The first classification step finds the predictive distribution p(f∗|f) using an adaptation of the GP regression prediction.The adapted regression calculation accounts for the classifier’s latent-function uncertainty.
  • Using the classifier: The second step squashes f∗ through π(f∗) = Φ(f∗), converting latent-function values into class probabilities.The cumulative Gaussian provides the fixed squashing function used for classification.
  • Using the classifier: The expected class probability is computed analytically as an integral of a cumulative Gaussian multiplied by a Gaussian distribution.This analytic solution supplies the probability after averaging over uncertainty in f∗.

3 TRAINING THE GP IN THE CLASSIFIER

Training the GP classifier estimates the latent function and covariance structure from labelled data, using a Laplace approximation and likelihood-based parameter optimization. The resulting GP is then used by the fixed squashing step for classification.

  • Training the GP in the classifier: The classifier’s second step is fixed, so training focuses on estimating the latent values ˆf and the modified covariance matrix K′.Once these quantities are found, the GP producing the predictive distribution is specified.
  • Training the GP in the classifier: Bayes’ theorem decomposes the posterior over latent outputs into likelihood and prior factors, enabling quantitative comparison of candidate GPs.The likelihood factor is determined from the sigmoid-based class model, while the prior is p(f|x).
  • Training the GP in the classifier: The latent-function estimate ˆf is found by setting the posterior-logarithm derivative to zero and iterating because ˆf appears on both sides.An initial guess, such as zero, is updated for several iterations.
  • Training the GP in the classifier: A Laplace approximation represents p(f|x,y) as Gaussian with covariance (K−1 + W)−1, where W is derived from the likelihood curvature.The variance comes from the negative second derivative of the posterior logarithm.
  • Training the GP in the classifier: The Laplace assumption can occasionally be inaccurate, motivating alternative uncertainty approximations such as expectation propagation when classifications are poor.This is an author-stated limitation of the Gaussian approximation used during training.
  • Training the GP in the classifier: Because f varies, the regression prediction must be adapted with K′ = K + W−1 so its variance reflects latent-function uncertainty.The modified covariance accounts for uncertainty propagated through the latent outputs.
  • Training the GP in the classifier: The covariance parameters θ = {l, σf} are optimized by maximizing the marginal likelihood p(y|x, θ), using a Laplace approximation.The resulting expression is intended for numerical optimization, as in GP regression.

4 MULTI-CLASS GPC

Multi-class GPC extends the binary construction by fitting class-specific latent functions, using block-structured covariance matrices and a softmax squashing function. The same GP-LVM framework separately learns low-dimensional representations by maximizing a GP likelihood.

  • 4 MULTI-CLASS GPC: For C > 2 classes, the classifier fits a latent function for each class and concatenates the resulting GP values.The latent representation lengthens as class-specific functions are added.
  • 4 MULTI-CLASS GPC: Multi-class labels use a vector with one 1 for the observed class and 0 for each of the other C −1 classes.The covariance matrix becomes block diagonal across the class-specific matrices K1, …, KC.
  • 4 MULTI-CLASS GPC: The binary cumulative Gaussian is replaced by a softmax function to produce probabilities across multiple classes.The class-specific latent values are grouped into subvectors for the softmax calculation.
  • 4 MULTI-CLASS GPC: The multi-class variance retains the form (K−1+W)−1, with W = diag(π)−ΠΠT encoding class-probability structure.Π is formed by stacking class-specific diagonal probability matrices.
  • 4 MULTI-CLASS GPC: GP classification also extends to multidimensional inputs and can use expectation propagation, confidence intervals, derivative calculations, or variational classifiers.These are listed as possible extensions of the basic classifier.
  • Dimensionality reduction: The GP-LVM learns a q-dimensional representation X of n observations originally described by d dimensions, with q < d.It is introduced as a low-dimensional representation for easier interpretation.
  • Dimensionality reduction: Each observed dimension is modelled as an independent zero-mean GP over X with a squared exponential kernel, encouraging nearby X points to remain nearby in Y.The likelihood of Y is therefore the product of d independent GP likelihoods.
  • Dimensionality reduction: The GP-LVM adjusts both X and kernel parameters θ to maximize the likelihood of the observed data.The kernel parameters are θ = {σ, l, β}.
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