Paper ID: | 8540 |
---|---|

Title: | Function-Space Distributions over Kernels |

The authors have given a novel and expressive manner to learn covariance functions for Gaussian processes. The explanation of the model and the inference procedure are clear and to the point, making it a pleasant read. However, I think section 5 can be improved, perhaps swapping in material in the supplementary paper. Main points 1. A discussion on how one might approach non-stationary kernels may help to initiate future work. 2. A discussion on how one might approach non-axis aligned kernels for multi-dimension input would also be helpful. 3. Line 112. If one need not worry about the normalization factor, does it mean that it is redundant to augment this kernel with a signal-variance factor? 4. Eq. 7. Seems to be missing a few subscript t's for tasks. 5. Section 5. Good to re-iterate how $\Delta$ in equation 2 is obtained, esp for multi-dimensional data. 6. Section 5.1 Please refer to Figure 3. 7. Section 5.2 What is the model predicting for the airline passenger data set? Is it the number of passengers (please indicate)? Why are there negative values in Figure 4a? Would modelling the logarithm of the data make more sense? 8. Figure 4. The lines are too close to prove the point. Please use a zoomed inset. 9. Section 5.3. Are there categorical and nominal data in the data set? If so, how useful/sensible is it to use the spectral method? Also a rough investigation into why the FKL does poorly on the fertility data set will be very helpful. 10. Section 5.4. Please discuss the figure (Figure 5) that you actually place on the main text, rather than go on the figures in the supplementary paper. The divide between the main paper and the supplementary paper is not reasonable. Minor points A. References [22] and [23] are duplicated. B. Lines 118 to 122 reads awkward. Suggest to paraphrase C. Lines 195 to 197. Suggest to use "linear combination" rather than "mixture", since the mixture models have the weights sum to unity.

Review update: Thanks for the response. It addressed my concerns well, and the SM kernel comparison seemed to give consistent results. I'm increasing my score slightly. --- This paper proposes functional kernel learning (FKL) a Bayesian nonparametric framework to infer flexible, stationary kernels for Gaussian process (GP) models, under multitask and multidimensional settings, with its latent GP inferred with elliptical slice sampling. The paper includes extensive empirical evidence to support FKL's superior performance over common kernels. The paper exposites this method in a clear and understandable manner with a step-by-step process. However, from my perspective it can benefit from some improvements on certain theoretical and empirical aspects. Here are my detailed comments on said aspects. In Section 3.1, the description of the Bochner's theorem is crucial to understanding stationary kernels, but it is slightly inaccurate. Bochner's theorem maps stationary kernels to finite measures, instead of Lebesgue measures (line 78), hence not all spectral measures of stationary kernels have valid spectral densities. For a detailed account of this issue, see Samo and Roberts [1]. The trapezoid rule as a variant of the Darboux sum poses another issue: when the density S(omega) is Lebesgue measurable, the Darboux sum is a good approximator only when S(omega) is continuous almost everywhere (Riemann integrable). However, those issues can be resolved with the fact that a mixture of Gaussian measures is dense in the Banach space of finite measures (see Shen et al.[2]), suggesting Riemann integrable densities are dense for spectral measures. In Section 3.2, the logarithm of the spectral density is modeled by a Gaussian process, but the quadratic mean function of this GP is without proper justification. It is slightly confusing why a parametrized, quadratic mean function is necessary for spectral density estimation on a compact subset of the frequency domain (eq. 3). Section 3.3 explores FKL with multidimensional data, which is modeled with a separable product kernel, rendering it inherently restrictive. Theoretically, separable kernels do not support all multidimensional stationary covariances (one example is the multivariate spectral mixture kernel[3]). For complete support over stationary kernels, it is needed to generalize eq. 3 into a multidimensional setting where omega_i are placed on a Cartesian grid. This generalization represents the entirety of stationary functions, but incurs the curse of dimensionality. While product kernels are commonly used in Gaussian process models, it is worth mentioning the capacity of different model specifications. For inference with FKL, the paper proposes an MCMC based inference scheme using elliptical slice sampling. While the inference is suitable for this task, I think it would benefit from formulating the GP regression for f as Bayesian linear regression instead. The trapzoid rule approximation renders the corresponding Gaussian process degenerate, equivalent to a Bayesian linear regression with trigometric basis expansion. In Section 5.4, comparisons for GP interpolation are made between FKL and standard kernels (RBF, RBF w/ ARD, Matérn). Standard kernels generally do not perform well when the dataset exhibits clear periodicities, for their spectral measures converge around frequency 0. It would be prudent to at least consider spectral mixture kernel (with sparse GP regression) as a candidate for parametric, flexible kernels. On a minor note, the placement of Figure 1 can be improved, and the citation style needs to be more consistent. [1] Samo, Y.-L. K., and Roberts, S.. "Generalized spectral kernels." arXiv preprint arXiv:1506.02236 (2015). [2] Shen, Z., Heinonen, M., and Kaski, S.. (2019). Harmonizable mixture kernels with variational Fourier features. Proceedings of Machine Learning Research, in PMLR 89:3273-3282 [3] Yang, Z., Wilson, A. G., Smola, A., and Song, L.. "A la carte–learning fast kernels." Artificial Intelligence and Statistics. 2015

----- Originality ----- The basic idea of this work is not new, but the practical construction is neat and can be of interest to the machine learning community. ----- Quality ----- The technical parts appear correct to me, although I have not been digging into the mathematical details. I enjoy the probabilistic kernel model, since my own impression of highly parametrised approaches for increased expressiveness is that they tend to be very challenging to train. The fact that the proposed approach easily extends to multiple input- and output dimensions is promising, since this is required in many real-world applications. Although the proposed method can describe the entire class of stationary covariance functions (which is a limitation, although of interest for a broad class of problems), it seems to me that the particular interest is functions with periodic behaviour. Therefore, it seems a bit strange to me that no comparison is made to the standard periodic covariance function obtained through the warping u(x)=(cos(x),sin(x)), which have shown promising results in modelling periodic functions elsewhere (see e.g. [1] below). I think that would be the most natural and simplest design choice to capture periodic behaviour. Furthermore, the focus in the one-dimensional comparison with other methods is on extrapolation; apart from the RBF kernel, the interpolation performance seems similar across all methods. In the multiple input case when focus is on interpolation, the performance difference as compared to the Matern kernel decreases - it would be interesting to hear the authors view on whether this improvement is justified by the higher model complexity. Finally, I am wondering about the time scaling properties when solving problems of multiple input dimensions. Intuitively it seems like the numerical integration (Eq. 3) is a bottleneck in these cases, so it would be interesting to know more about it. [1] Ghassemi & Deisenroth, "Analytic long-term forecasting with periodic Gaussian processes", AISTATS 2014. ----- Clarity ----- The paper is very well-written, easy to follow, with clear methodological descriptions. ----- Significance ----- I believe that this paper by itself is significant to the sub-field of GP modelling that focuses on functions with periodic behaviour. Although I am not sure that there are plenty of problems that will benefit from this particular construction, I have a feeling that it has good potential of inspiring more powerful extensions and developments.