text: We propose a Standing Wave Decomposition (SWD) approximation to Gaussian Process regression (GP). GP involves a costly matrix inversion operation, which limits applicability to large data analysis. For an input space that can be approximated by a grid and when correlations among data are short-ranged, the kernel matrix inversion can be replaced by analytic diagonalization using the SWD. We show that this approach applies to uni- and multi-dimensional input data, extends to include longer-range correlations, and the grid can be in a latent space and used as inducing points. Through simulations, we show that our approximate method outperforms existing methods in predictive accuracy per unit time in the regime where data are plentiful. Our SWD-GP is recommended for regression analyses where there is a relatively large amount of data and or there are constraints on computation time.
  @cite_24: We introduce a kernel approximation strategy that enables computation of the Gaussian process log marginal likelihood and all hyperparameter derivatives in @math time. Our GRIEF kernel consists of @math eigenfunctions found using a Nystrom approximation from a dense Cartesian product grid of inducing points. By exploiting algebraic properties of Kronecker and Khatri-Rao tensor products, computational complexity of the training procedure can be practically independent of the number of inducing points. This allows us to use arbitrarily many inducing points to achieve a globally accurate kernel approximation, even in high-dimensional problems. The fast likelihood evaluation enables type-I or II Bayesian inference on large-scale datasets. We benchmark our algorithms on real-world problems with up to two-million training points and @math inducing points.
  @cite_8: It is important to understand the rich structure of natural sounds in order to solve important tasks, like automatic speech recognition, and to understand auditory processing in the brain. This thesis takes a step in this direction by characterising the statistics of simple natural sounds. We focus on the statistics because perception often appears to depend on them, rather than on the raw waveform. For example the perception of auditory textures, like running water, wind, fire and rain, depends on summary-statistics, like the rate of falling rain droplets, rather than on the exact details of the physical source. In order to analyse the statistics of sounds accurately it is necessary to improve a number of traditional signal processing methods, including those for amplitude demodulation, time-frequency analysis, and sub-band demodulation. These estimation tasks are ill-posed and therefore it is natural to treat them as Bayesian inference problems. The new probabilistic versions of these methods have several advantages. For example, they perform more accurately on natural signals and are more robust to noise, they can also fill-in missing sections of data, and provide error-bars. Furthermore, free-parameters can be learned from the signal. Using these new algorithms we demonstrate that the energy, sparsity, modulation depth and modulation time-scale in each sub-band of a signal are critical statistics, together with the dependencies between the sub-band modulators. In order to validate this claim, a model containing co-modulated coloured noise carriers is shown to be capable of generating a range of realistic sounding auditory textures. Finally, we explored the connection between the statistics of natural sounds and perception. We demonstrate that inference in the model for auditory textures qualitatively replicates the primitive grouping rules that listeners use to understand simple acoustic scenes. This suggests that the auditory system is optimised for the statistics of natural sounds.
  @cite_11: We introduce a new structured kernel interpolation (SKI) framework, which generalises and unifies inducing point methods for scalable Gaussian processes (GPs). SKI methods produce kernel approximations for fast computations through kernel interpolation. The SKI framework clarifies how the quality of an inducing point approach depends on the number of inducing (aka interpolation) points, interpolation strategy, and GP covariance kernel. SKI also provides a mechanism to create new scalable kernel methods, through choosing different kernel interpolation strategies. Using SKI, with local cubic kernel interpolation, we introduce KISS-GP, which is 1) more scalable than inducing point alternatives, 2) naturally enables Kronecker and Toeplitz algebra for substantial additional gains in scalability, without requiring any grid data, and 3) can be used for fast and expressive kernel learning. KISS-GP costs O(n) time and storage for GP inference. We evaluate KISS-GP for kernel matrix approximation, kernel learning, and natural sound modelling.
  @cite_17: Point processes are difficult to analyze because they provide only a sparse and noisy observation of the intensity function driving the process. Gaussian Processes offer an attractive framework within which to infer underlying intensity functions. The result of this inference is a continuous function defined across time that is typically more amenable to analytical efforts. However, a naive implementation will become computationally infeasible in any problem of reasonable size, both in memory and run time requirements. We demonstrate problem specific methods for a class of renewal processes that eliminate the memory burden and reduce the solve time by orders of magnitude.
summary: Previous works have also taken advantage of structured data to increase the computational efficiency of GP. For input data that lie on a grid, @cite_17 @cite_8 sidestep the matrix inversion by solving @math with gradient-based methods. Importantly, the optimization is fast because Toeplitz circulant structure can be exploited for fast matrix multiplication. These fast multiplications take on forms similar to . For input data that do not lie on a grid, @cite_11 (KISS-GP) introduces a latent grid, finds a sparse, approximate representation of @math in terms of @math , and makes inference using the same kind of optimization problem in @cite_17 @cite_8 . The sparsification in KISS-GP and that in our LGSWD-GP share the same flavor in ignoring the diminishing long-range correlations among data points. Recently, @cite_24 further extended this optimization approach to efficiently handle high dimensional data by applying the Nystr " o m approximation for eigen-decomposition and by exploiting properties of Kronecker and Khatri-Rao products for fast matrix multiplication. Overall, our method and these methods all take advantage of the grid structure to form fast matrix multiplication; however, our method differs in that it overcomes the matrix inversion bottleneck via analytic diagonalizaion rather than optimization.