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Convolutional Gaussian Processes (1709.01894v1)

Published 6 Sep 2017 in stat.ML and cs.LG

Abstract: We present a practical way of introducing convolutional structure into Gaussian processes, making them more suited to high-dimensional inputs like images. The main contribution of our work is the construction of an inter-domain inducing point approximation that is well-tailored to the convolutional kernel. This allows us to gain the generalisation benefit of a convolutional kernel, together with fast but accurate posterior inference. We investigate several variations of the convolutional kernel, and apply it to MNIST and CIFAR-10, which have both been known to be challenging for Gaussian processes. We also show how the marginal likelihood can be used to find an optimal weighting between convolutional and RBF kernels to further improve performance. We hope that this illustration of the usefulness of a marginal likelihood will help automate discovering architectures in larger models.

Citations (128)
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Summary

  • The paper presents a novel method that embeds convolutional kernels within Gaussian processes to enhance image data analysis.
  • It employs an inter-domain inducing point approximation in patch-space, reducing computational complexity and overfitting risks.
  • Empirical results demonstrate significant error rate improvements on benchmarks like MNIST and CIFAR-10 compared to traditional RBF models.

Insights into "Convolutional Gaussian Processes"

The paper "Convolutional Gaussian Processes" by van der Wilk, Rasmussen, and Hensman explores a significant enhancement of Gaussian Processes (GPs) by incorporating convolutional structures, thus extending their applicability to high-dimensional inputs such as images. This work makes meaningful contributions by devising an inter-domain inducing point approximation method that aligns well with the convolutional kernel, thereby enabling efficient and precise posterior inference for image data. The research further explores various forms of convolutional kernels, applying them to the commonly cited benchmarks MNIST and CIFAR-10, which traditionally pose challenges for Gaussian processes due to their complexity.

Methodological Advancements

The authors put forth a novel framework to merge convolutional structures within Gaussian processes. They propose a model where the process is convolved directly, removing the necessity for additional transformation parameters typically required in convolutional neural networks. This approach retains the process's integrity and avoids overfitting by leveraging variational parameters akin to neural network filters.

A significant methodological leap is the application of an inter-domain inducing point approximation that enables the establishment of convolutional kernels as viable entities within a Gaussian processes framework. This is particularly achieved by placing inducing inputs in the patch-space, rather than the entire image space, thereby facilitating reductions in computational burden from evaluating large kernel matrices typically required for high-dimensional data like images.

Empirical Evaluations

The empirical analysis is thorough, with experiments demonstrating the efficacy of convolutional GPs over traditional methods. For instance, a notable improvement is reported on the rectangles dataset, where the convolutional GP achieves an error rate of 1.4% compared to 5% with the best RBF kernel model. On the MNIST dataset, the weighted convolutional kernel reduces error rates to 1.22% from the 1.9% baseline obtained using an RBF kernel. These improvements underscore the potential of convolutional structures to capture image-specific nuances better.

Furthermore, the paper tackles the CIFAR-10 dataset, showcasing the power of the multi-channel convolutional kernel variant through a reduction in error rates to 35.4%, a significant improvement over a competitive baseline. This result is particularly noteworthy given the complexity of CIFAR-10, highlighting the model's capability to manage colour channel interactions and extract useful patterns from RGB images.

Implications and Future Directions

The introduction of convolutional structures into Gaussian processes not only enhances the representational power of GPs for image-based tasks but also opens new avenues for further research into hybrid models that perhaps combine parametric and nonparametric approaches. The approach posits practical implications for developing models that can automate architectural discovery processes in larger models through the exploitation of marginal likelihoods.

Future research directions could aim to explore the scalability of these models further, particularly in larger, deeper architectures. Additionally, extending these methods to handle temporal or sequential data where convolutional approaches could provide similar benefits as seen in computer vision tasks would be valuable. Exploring novel applications in domains such as natural language processing or audio signal processing might also benefit from such convolutional structures within a stochastic process framework.

Overall, the paper makes a substantive contribution to the field by not only enhancing the theoretical understanding of convolutional structures' role within Gaussian processes but also delivering a robust empirical evaluation framework to spur subsequent research developments.

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