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Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP) (1503.01057v1)

Published 3 Mar 2015 in cs.LG and stat.ML

Abstract: 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.

Citations (496)

Summary

  • The paper introduces SKI, a framework that unifies and extends inducing point methods via local cubic kernel interpolation to enable scalable Gaussian Process models.
  • It reduces the standard O(n^3) complexity to O(n) while allowing the use of more inducing points than training points, enhancing kernel expressiveness.
  • KISS-GP leverages structured algebra with Kronecker and Toeplitz methods to achieve efficient kernel learning, outperforming traditional methods like FITC in both runtime and accuracy.

Structured Kernel Interpolation for Scalable Gaussian Processes (KISS-GP)

The paper "Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)" introduces a novel framework called Structured Kernel Interpolation (SKI) that aims to address the computational challenges associated with Gaussian Processes (GPs) on large datasets. This research, presented by Wilson and Nickisch, offers a significant contribution to the field by unifying and extending the scalability of Gaussian Process models through an innovative approximation technique.

Summary

Gaussian Processes (GPs) are powerful non-parametric models widely used for their flexibility in learning complex functions. However, the standard GPs carry a prohibitive computational burden, approximately O(n3)\mathcal{O}(n^3), limiting their applicability to smaller datasets. Previous attempts to scale GPs have largely focused on inducing point methods, which reduce computational costs but often at the expense of model accuracy and kernel expressiveness.

SKI offers a fresh perspective by interpreting inducing point methods as performing global Gaussian Process interpolation on kernel functions. This insight allows the authors to propose KISS-GP, a specific variant within the SKI framework that employs local cubic kernel interpolation. KISS-GP retains accuracy while naturally integrating with Kronecker and Toeplitz algebra to improve scalability, with computational costs reduced to O(n)\mathcal{O}(n).

Key Contributions

  1. Unified Framework: SKI generalizes existing inducing point methods by framing them under a common interpolation strategy. This new understanding enables the development of more scalable Gaussian Process models.
  2. Scalable Algorithms: The introduction of KISS-GP represents a major stride in scalability. By leveraging structured kernel interpolation, KISS-GP can handle inducing points mm greater than the number of training points nn, expanding the applicability of GPs to large datasets.
  3. Efficient Kernel Learning: The ability to use a large number of inducing points without significant computational overhead enables expressive kernel learning, overcoming limitations previously observed in sparse methods.
  4. Implementation Insights: By utilizing cubic and inverse distance weighting strategies, SKI creates sparse approximations that allow significant computational savings, making high-dimensional and large-scale problems more tractable.

Numerical Results

The empirical evaluations underscore the efficacy of KISS-GP. The framework demonstrates strong performance in kernel matrix reconstruction and sound modeling tasks. Notably, KISS-GP consistently outperforms FITC and other similar algorithms in runtime efficiency and accuracy, particularly when the inducing point numbers exceed the dataset size.

Theoretical and Practical Implications

The theoretical underpinnings of SKI suggest that the framework can yield accurate approximations of complex kernels, engaging larger datasets without succumbing to computational bottlenecks. Practically, this means that practitioners can leverage GPs in new domains, such as large-scale time series and spatial data applications, previously prohibitive with standard GP methods.

Future Directions

The potential avenues for further research stem from the versatility of the SKI framework. Future work could explore alternative interpolation strategies, integrate probabilistic programming frameworks, or extend the approach to deep Gaussian Process models. SKI also lays the groundwork for combining with stochastic variational methods, enhancing their performance on even more massive datasets.

In conclusion, KISS-GP and the broader SKI framework represent a meaningful advancement in the scalable modeling of Gaussian Processes, promising to unlock new possibilities in both research and application domains.