The generalised linear model (GLM) is the standard approach in classical statistics for regression tasks where it is appropriate to measure the data misfit using a likelihood drawn from the exponential family of distributions. In this paper, we apply the kernel trick to give a non-linear variant of the GLM, the generalised kernel machine (GKM), in which a regularised GLM is constructed in a fixed feature space implicitly defined by a Mercer kernel. The MATLAB symbolic maths toolbox is used to automatically create a suite of generalised kernel machines, including methods for automated model selection based on approximate leave-one-out cross-validation. In doing so, we provide a common framework encompassing a wide range of existing and novel kernel learning methods, and highlight their connections with earlier techniques from classical statistics. Examples including kernel ridge regression, kernel logistic regression and kernel Poisson regression are given to demonstrate the flexibility and utility of the generalised kernel machine.

The optimal model parameters of a kernel machine are typically given by the solution of a convex optimisation problem with a single global optimum. Obtaining the best possible performance is therefore largely a matter of the design of a good kernel for the problem at hand, exploiting any underlying structure and optimisation of the regularisation and kernel parameters, i.e. model selection. Fortunately, analytic bounds on, or approximations to, the leave-one-out cross-validation error are often available, providing an efficient and generally reliable means to guide model selection. However, the degree to which the incorporation of prior knowledge improves performance over that which can be obtained using "standard" kernels with automated model selection (i.e. agnostic learning), is an open question. In this paper, we compare approaches using example solutions for all of the benchmark tasks on both tracks of the IJCNN-2007 Agnostic Learning versus Prior Knowledge Challenge.

Multinomial logistic regression provides the standard penalised maximum- likelihood solution to multi-class pattern recognition problems. More recently, the development of sparse multinomial logistic regression models has found ap- plication in text processing and microarray classification, where explicit identifi- cation of the most informative features is of value. In this paper, we propose a sparse multinomial logistic regression method, in which the sparsity arises from the use of a Laplace prior, but where the usual regularisation parameter is inte- grated out analytically. Evaluation over a range of benchmark datasets reveals this approach results in similar generalisation performance to that obtained using cross-validation, but at greatly reduced computational expense.

MOTIVATION: Gene selection algorithms for cancer classification, based on the expression of a small number of biomarker genes, have been the subject of considerable research in recent years. Shevade and Keerthi propose a gene selection algorithm based on sparse logistic regression (SLogReg) incorporating a Laplace prior to promote sparsity in the model parameters, and provide a simple but efficient training procedure. The degree of sparsity obtained is determined by the value of a regularization parameter, which must be carefully tuned in order to optimize performance. This normally involves a model selection stage, based on a computationally intensive search for the minimizer of the cross-validation error. In this paper, we demonstrate that a simple Bayesian approach can be taken to eliminate this regularization parameter entirely, by integrating it out analytically using an uninformative Jeffrey's prior. The improved algorithm (BLogReg) is then typically two or three orders of magnitude faster than the original algorithm, as there is no longer a need for a model selection step. The BLogReg algorithm is also free from selection bias in performance estimation, a common pitfall in the application of machine learning algorithms in cancer classification. RESULTS: The SLogReg, BLogReg and Relevance Vector Machine (RVM) gene selection algorithms are evaluated over the well-studied colon cancer and leukaemia benchmark datasets. The leave-one-out estimates of the probability of test error and cross-entropy of the BLogReg and SLogReg algorithms are very similar, however the BlogReg algorithm is found to be considerably faster than the original SLogReg algorithm. Using nested cross-validation to avoid selection bias, performance estimation for SLogReg on the leukaemia dataset takes almost 48 h, whereas the corresponding result for BLogReg is obtained in only 1 min 24 s, making BLogReg by far the more practical algorithm. BLogReg also demonstrates better estimates of conditional probability than the RVM, which are of great importance in medical applications, with similar computational expense. AVAILABILITY: A MATLAB implementation of the sparse logistic regression algorithm with Bayesian regularization (BLogReg) is available from http://theoval.cmp.uea.ac.uk/~gcc/cbl/blogreg/

Survival analysis is a branch of statistics concerned with the time elapsing before "failure," with diverse applications in medical statistics and the analysis of the reliability of electrical or mechanical components. We introduce a parametric accelerated life survival analysis model based on kernel learning methods that, at least in principal, is able to learn arbitrary dependencies between a vector of explanatory variables and the scale of the distribution of survival times. The proposed kernel survival analysis method is then used to model the growth domain of Clostridium botulinum, the food processing and storage conditions permitting the growth of this foodborne microbial pathogen, leading to the production of the neurotoxin responsible for botulism. A Bayesian training procedure, based on the evidence framework, is used for model selection and to provide a credible interval on model predictions. The kernel survival analysis models are found to be more accurate than models based on more traditional survival analysis techniques but also suggest a risk assessment of the foodborne botulism hazard would benefit from the collection of additional data.

We introduce a simple framework for sparse Bayesian learning with multi-layer perceptron (IMLP) networks, inspired by Tipping's relevance vector machine (RVM). Like the RVM, a Bayesian prior is adopted that includes separate hyperparameters for each weight, allowing redundant weights and hidden layer units to be identified and subsequently pruned from the network, whilst also providing a means to avoid over-fitting the training data. This approach is also more easily implemented, as only the diagonal elements of the Hessian matrix are used in the update formula for the regularisation parameters, rather than the traces of square sub-matrices of the Hessian corresponding to the weights associated with each regularisation parameter. The proposed relevance multi-layer perceptron (RMLP) is evaluated over several publicly available benchmark datasets, demonstrating the viability of the approach, giving rise to similar generalisation performance, but with far fewer weights.

In this paper, we present a simple mathematical trick that simplifies the derivation of Bayesian treatments of a variety of sparse kernel learning methods. The incomplete Cholesky factorisation due to (Fine and Scheinberg, 2001) is used to transform the dual parameter space, such that the covariance matrix of the Gaussian prior over model parameters becomes the identity matrix. The regularisation term is then the familiar weight-decay regulariser, allowing the Bayesian analysis to proceed straight-forwardly via the methods developed by MacKay (1992). As a bye-product, the incomplete Cholesky factorisation algorithm also identifies a subset of the training data forming an approximate basis for the remaining data in feature space, resulting in a sparse model. Bayesian treatments of the kernel ridge regression algorithm (Saunders et al., 1998), with both constant and input dependent variance structures, arc given as illustrative examples of the proposed technique, which we hope will be more widely applicable.

Vector quantisation is a widely used technique in low-bit rate coding of speech and image data, but is highly sensitive to noise in the transmission channel. If the reference vector recalled by a corrupted index differs greatly from the intended reference vector, image quality can be degraded quite dramatically. The index assignment (IA) process attempts to re-order the code book so as to minimise the effects of errors introduced in the transmission channel, by assigning indices with similar binary patterns to similar reference vectors, usually at considerable computational expense. This paper describes a fast, novel index assignment algorithm based on Hall's solution (1970) to the quadratic assignment problem.