Fast Kernel Learning for Multidimensional Pattern Extrapolation
The ability to automatically discover patterns and perform extrapolation is an essential quality of intelligent systems. Kernel methods, such as Gaussian processes, have great potential for pattern extrapolation, since the kernel flexibly and interpretably controls the generalisation properties of these methods. However, automatically extrapolating large scale multidimensional patterns is in general difficult, and developing Gaussian process models for this purpose involves several challenges. A vast majority of kernels, and kernel learning methods, currently only succeed in smoothing and interpolation. This difficulty is compounded by the fact that Gaussian processes are typically only tractable for small datasets, and scaling an expressive kernel learning approach poses different challenges than scaling a standard Gaussian process model. One faces additional computational constraints, and the need to retain significant model structure for expressing the rich information available in a large dataset. In this paper, we propose a Gaussian process approach for large scale multidimensional pattern extrapolation. We recover sophisticated out of class kernels, perform texture extrapolation, inpainting, and video extrapolation, and long range forecasting of land surface temperatures, all on large multidimensional datasets, including a problem with 383,400 training points. The proposed method significantly outperforms alternative scalable and flexible Gaussian process methods, in speed and accuracy. Moreover, we show that a distinct combination of expressive kernels, a fully non-parametric representation, and scalable inference which exploits existing model structure, are critical for large scale multidimensional pattern extrapolation.
The authors combine two recent advances in Gaussian processes, spectral mixture kernels [5], and scalable Gaussian processes for data in grids [14, 22 see below], in order to tackle applications with high amount of data points, like texture extrapolation, inpainting, and video extrapolation.  The paper includes a thorough evaluation of the framework proposed, and comparisons against sparse GP methods, with general purpose covariance functions, and spectral mixture kernels.QualityThe paper is technically sound. The framework proposed by the authors achieves outstanding results in the different applications studied in the paper. The authors claim that they relax the grid assumption, by completing with imaginary observations. It is important to discuss this issue when each of the experiments is being introduced. The imaginary observations are mentioned in section 2.1 and hardly ever mention again. How would the method work in higher dimensional settings (greater than three)? How prone to overfitting is the method? What are the weaknesses of the method?ClarityThe paper is clearly written and well organized. OriginalityAs the authors clearly stated it, the framework proposed is a combination of the spectral mixture kernels, and fast Gaussian process regression on grids. From this point of view, the novelty of the method is incremental. SignificanceI must say that the quality of the results obtained by the proposed method is impressive. The method beats any other of the GP alternatives. I wonder how methods in the image inpainting related literature would compare against the ones presented by the authors.[22] Yuancheng Luo and Ramani Duraiswami (2013): Fast Near-GRID Gaussian process regression, at AISTAST 2013.