Title: Thin and deep Gaussian processes
Venue: NeurIPS 2023
Accepted: yes
Keywords: Gaussian Processes

Deep Gaussian Processes

non-stationary kernels
Abstract: Gaussian processes (GPs) can provide a principled approach to uncertainty quantification with easy-to-interpret kernel hyperparameters, such as the lengthscale, which controls the correlation distance of function values.However, selecting an appropriate kernel can be challenging.

Deep GPs avoid manual kernel engineering by successively parameterizing kernels with GP layers, allowing them to learn low-dimensional embeddings of the inputs that explain the output data.
Following the architecture of deep neural networks, the most common deep GPs warp the input space layer-by-layer but lose all the interpretability of shallow GPs. An alternative construction is to successively parameterize the lengthscale of a kernel, improving the interpretability but ultimately giving away the notion of learning lower-dimensional embeddings. Unfortunately, both methods are susceptible to particular pathologies which may hinder fitting and limit their interpretability.

This work proposes a novel synthesis of both previous approaches: {Thin and Deep GP} (TDGP). Each TDGP layer defines locally linear transformations of the original input data maintaining the concept of latent embeddings while also retaining the interpretation of lengthscales of a kernel. Moreover, unlike the prior solutions, TDGP induces non-pathological manifolds that admit learning lower-dimensional representations.

We show with theoretical and experimental results that i) TDGP is, unlike previous models, tailored to specifically discover lower-dimensional manifolds in the input data, ii) TDGP behaves well when increasing the number of layers, and iii) TDGP performs well in standard benchmark datasets.

Thin and Deep Gaussian Processes

Abstract

Gaussian processes (GPs) can provide a principled approach to uncertainty quantification with easy-to-interpret kernel hyperparameters, such as the lengthscale, which controls the correlation distance of function values. However, selecting an appropriate kernel can be challenging. Deep GPs avoid manual kernel engineering by successively parameterizing kernels with GP layers, allowing them to learn low-dimensional embeddings of the inputs that explain the output data. Following the architecture of deep neural networks, the most common deep GPs warp the input space layer-by-layer but lose all the interpretability of shallow GPs. An alternative construction is to successively parameterize the lengthscale of a kernel, improving the interpretability but ultimately giving away the notion of learning lower-dimensional embeddings. Unfortunately, both methods are susceptible to particular pathologies which may hinder fitting and limit their interpretability. This work proposes a novel synthesis of both previous approaches: Thin and Deep GP (TDGP). Each TDGP layer defines locally linear transformations of the original input data maintaining the concept of latent embeddings while also retaining the interpretation of lengthscales of a kernel. Moreover, unlike the prior methods, TDGP induces non-pathological manifolds that admit learning lower-dimensional representations. We show with theoretical and experimental results that i) TDGP is, unlike previous models, tailored to specifically discover lower-dimensional manifolds in the input data, ii) TDGP behaves well when increasing the number of layers, and iii) TDGP performs well in standard benchmark datasets.

1 Introduction

Gaussian processes (GPs) are probabilistic models whose nonparametric nature and interpretable hyperparameters make them appealing in many applications where uncertainty quantification and data efficiency matter, such as Bayesian optimization [11], spatiotemporal modeling [6], robotics and control [5]. The key modeling choice for a GP prior is its covariance or kernel function, which determines the class of functions it represents. However, due to finite data and a fixed-form kernel function, the GP may not inter- or extrapolate as desired. Stationary kernels, such as the commonly used squared exponential kernel and the Matérn family, assume the existence of a constant characteristic lengthscale, which makes them unsuitable for modeling non-stationary data.

To construct more expressive kernels, we can consider hierarchical GP models (deep GPs or DGP). The most common deep GP construction is a functional composition of GP layers with standard

stationary kernels that results in a non-stationary non-Gaussian process [3, 23]; for clarity, we will refer to this model type as compositional DGPs (CDGPs). However, CDGPs can show pathological behavior, where adding layers leads to a loss of representational ability [9, 7]. Alternatively, it is also possible to extend 'shallow' GPs by making the kernel lengthscales a function of the input [19], resulting in the deeply non-stationary GP [DNSGP, 22]. Although this covariance function approach to DGPs does not degenerate with more layers, care must be taken to guarantee a positive semi-definite kernel matrix. Moreover, the induced space is not a proper inner-product space [18], which hinders the learning of useful manifolds.

In this paper, we address the shortcomings of previous DGP paradigms by retaining the flexibility provided by learning hierarchical lengthscale fields while also enabling manifold learning. Instead of pursuing neural networks to enhance standard GPs [1, 26], which may lead to overfitting [17], we retain the nonparametric formulation by using additional GPs to model a linear projection of each input onto a latent manifold. The key insight of our approach is that such projections are input-dependent and tailored towards more interpretable lower-dimensional manifolds with corresponding lengthscale fields. The resulting Thin and Deep GP 2 (TDGP) avoids the pitfalls of other DGP constructions while maintaining its hierarchical composition and modeling capacity beyond shallow GPs.

Our contributions are three-fold:

We propose TDGP, a new hierarchical architecture for DGPs that is highly interpretable and does not degenerate as the number of layers increases. Notably, TDGP is the only deep architecture that induces both a lengthscale field and data embeddings.

We prove that TDGPs and compositional DGPs are the limits of a more general DGP construction. Thus, we establish a new perspective on standard CDGPs while reaping the benefits of inducing a lengthscale field.

We demonstrate that TDGPs perform as well as or better than previous approaches. Our experiments also show that TDGP leans towards inducing low-dimensional embeddings.

2 Background

Gaussian processes (GPs) are distributions over functions and fully characterized by a mean function m and a kernel (covariance function) k [20]. If a function f is GP distributed, we write f ∼ GP ( m,k ) . If not stated otherwise, we assume that the prior mean function is 0 everywhere, i.e., m ( · ) ≡ 0 . Typically, the kernel possesses a few interpretable hyperparameters, such as lengthscales or signal variances, estimated by the standard maximization of the marginal likelihood [20].

The squared exponential (SE) kernel is arguably the most commonly used covariance function in the GP literature and, in its most general form [25], it can be written as

where a , b ∈ R D , the constant σ 2 > 0 defines the signal variance, and ∆ ∈ R D × D is a lengthscale matrix. Notably, the SE kernel is stationary , i.e., k SE ( a + c , b + c ) = k SE ( a , b ) for any c ∈ R D . Furthermore, when the lengthscale matrix ∆ = λ 2 I , the kernel is isotropic , meaning that it can be written as a function π SE ( d 2 ) of the squared distance d 2 = ∥ a -b ∥ 2 2 . For more details on the significance of the lengthscale parameter, we refer the reader to Appendix A.

Stationary kernels enforce invariances, which may not always be desirable. However, stationary kernels can be used as building blocks to derive broader families of kernels (including non-stationary kernels), either by composing them with deformation functions or through mixtures of lengthscales.

Deformation kernels result from applying a deformation function τ : R D → R Q to a and b before feeding them to a stationary kernel k in R Q . Thus, a deformation kernel k τ follows

For a linear transformation τ ( x ) = Wx , we can interpret k τ as a stationary kernel with lengthscale matrix ∆ = [ W ⊺ W ] -1 . However, for more intricate τ ( · ) , interpreting or analyzing k τ can be very challenging. For instance, this is the case for deep kernel learning [DKL, 26] models, in which τ ( · ) is an arbitrary neural network.

Compositional deep GPs [CDGPs, 3] also rely on deformation kernels, but they use a GP prior to model τ ( · ) . The kernel of the τ ( · ) process can also (recursively) be considered to be a deformation kernel, thereby extending DGPs to arbitrary depths. However, stacking GP layers reduces the interpretability of DGPs, and their non-injective nature makes them susceptible to pathologies [9].

Lengthscale mixture kernels [19] are a generalization of the process of convolving stationary kernels with different lengthscales from Higdon et al. [14]. For arbitrary isotropic kernels k , i.e., k ( a , b ) = π k ( ∥ a -b ∥ 2 2 ) , Paciorek and Schervish [19] construct a non-stationary k lmx as a function of a field of lengthscale matrices ∆ : R D → R D × D , such that

where δ = 2( a -b ) ⊺ ( ∆ ( a ) + ∆ ( b )) -1 ( a -b ) . The explicit lengthscale field ∆ makes this model more interpretable than general deformation kernels. However, Paciorek [18] notes that δ ( a , b ) may violate the triangle inequality and therefore does not induce a manifold over the input space. This departure from the properties of stationary kernels is due to the matrix inside the quadratic form being a function of both a and b . Another caveat is that the scale of k lmx is also controlled by a pre-factor term that depends on ∆ , allowing for unintended effects (such as unwanted long-range correlations), especially in rapidly varying lengthscale fields.

Deeply non-stationary GPs [DNSGPs, 7, 22] use the lengthscale mixture kernel and parameterize the function ∆ ( x ) with a warped GP prior to obtain a deep GP model. Similar to CDGPs, this model can be extended in depth [22, 7] by considering the kernel of ∆ ( x ) to be non-stationary with its lengthscales stemming from another GP. A practical issue in these models is guaranteeing that ∆ ( x ) is positive semi-definite. Therefore, in practice, ∆ ( x ) is usually restricted to be diagonal.

3 Thin and Deep GPs (TDGPs)

As discussed in the previous section, deep GP constructions follow naturally from hierarchical extensions of a base kernel. Therefore, we arrange the presentation of TDGPs in three parts. First, we propose a kernel that admits interpretations both in terms of its lengthscale and of its induced manifold. Second, we use this kernel to build a novel type of deep GP model (TDGP). Third, we describe how to carry out inference for TDGPs. Finally, we discuss limitations of our approach.

Kernel. We address the drawbacks of the approaches based on deformation and lengthscale mixture kernels - i.e., the lack of interpretability and failure to induce a manifold, respectively - by proposing a synthesis of both methods, retaining their positives whilst mitigating some of their issues. Starting from the discussion in Section 2, a squared exponential kernel with lengthscale matrix [ W ⊺ W ] -1 corresponds to a deformation of an isotropic kernel:

Therefore, this is the same as applying a linear deformation τ ( x ) = Wx .

We propose extending this to a non-linear transformation that is locally linear by letting W vary as a function of that input, so τ ( x ) = W ( x ) x . This results in the TDGP kernel:

Equation (5) cannot be written as a Mahalanobis distance like in Eq. (4), but in the neighborhood of x , [ W ⊺ ( x ) W ( x )] -1 is a lengthscale matrix, thus allowing k TDGP to be implicitly parametrized by a lengthscale field just like the lengthscale mixture kernels of Paciorek and Schervish [19]. Hence, it allows for better interpretability than the deformation kernels considered in compositional DGPs.

However, unlike the lengthscale mixture approach of Eq. (3), our kernel in Eq. (5) does not introduce an input-dependent pre-factor, thereby avoiding pathologies when the lengthscale varies rapidly. Moreover, since the distance induced by the quadratic form obeys the triangle inequality, it induces a manifold in the input space. Hence, it addresses the two issues of lengthscale mixture kernels.

Model. In an analogous manner to the compositional DGP and DNSGP, we present how to use this kernel to build a hierarchical GP model. Our L -layers deep model is described as follows:

where

Therefore, this model can be extended arbitrarily by deforming the kernel of the entries of W ℓ with another locally linear transformation with its own matrix W ℓ -1 .

Figure 1 compares the graphical models of TDGP and compositional DGP. We notice that TDGP learns hierarchical input-dependent lengthscale fields, instead of a straight composition of processes. We call this model Thin and Deep GP (TDGP) by the fact that our graphical model always has cycles of bounded length due to the connection of every hidden layer with the inputs X . Therefore, it has finite girth, in contrast to the unbounded girth of the compositional DGP graphical model. Importantly, however, TDGP are related to CDGPs as both can be seen as locally affine deformations of the input space (proof in Appendix B).

Theorem 3.1 ( Relationship between TDGP and CDGP). Any L -layer CDGP prior over a function f ( x ) = h L ( h L -1 ( · · · h 1 ( x ) · · · )) is a special case of a generalized TDGP prior with equal depth defined over the augmented input space ˜ x = [ x , 1] ⊺ . Since linear deformations ˜ W ˜ x in the augmented space correspond to affine transformations Wx + d in the original space, the special case of the CDGP model corresponds to a TDGP where the prior variance of W ℓ approaches zero.

Unlike DNSGP, which makes the positive semi-definite matrix ∆ a function and requires the use of warped GP priors, our hidden layers W are arbitrary matrices that admit a regular Gaussian prior. By placing zero-mean priors on the entries of W , we encourage the MLE estimate to maximally reduce the latent dimensionality of h . This is because the latent dimensionality becomes linked to the number of rows with non-zero variance in the prior, e.g., if the prior kernel variance i -th row of W ℓ tends to zero, the posterior over that row is highly concentrated at zero, eliminating the i -th latent dimension. In a compositional DGP, this inductive bias also corresponds to a zero-mean prior in the hidden layers. However, as discussed by Duvenaud et al. [9], and reproduced in Fig. 2, this choice introduces a pathology that makes the derivatives of the process almost zero.

Inference. To estimate the posterior distributions and the hyperparameters of our model, we perform variational inference (VI). We introduce inducing points u for the last-layer GP f ( · ) and inducing points V for the W ( · ) processes. Similar to how VI methods for DGPs [3] were based on the nondeep GPLVM model [24], our method builds upon VI for the hyperparameters of square exponential kernels in shallow GPs as discussed by Damianou et al. [4]. However, we replace the Gaussian prior p ( W ) with a GP prior p ( W ( · )) . For instance, the variational distribution for a two-layer TDGP is:

where for the final layer the parameters ˇ µ u , ˇ Σ u are not estimated but replaced with their closed-form optimal solutions. Appendix C contains a derivation of the ELBO and more details.

Limitations. For large numbers of layers, our VI scheme for TDGP may become computationally intensive. By stacking additional hidden layers W ( x ) , we add D × Q GPs into the model; consequently, the number of variational parameters increases, which can slow down optimization. More specifically, inference uses O ( L × D × Q × m 2 ) parameters and takes time O ( L × D × Q × ( m 2 + m 2 n ) ) to compute the ELBO. Additional details on runtime as a function of data size and width can be found in the Appendix E. Furthermore, without the addition of a bias to the input data x , TDGP performs locally linear transformations W ( x ) x , meaning that the neighborhood of x = 0 is kept unchanged for each layer.

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4 Experiments

To assess whether TDGP accurately captures non-smooth behavior and has competitive results with other GP models, we performed a regression experiment on a synthetic dataset with input-dependent linear projections and another experiment in varied empirical datasets. In this section, we examine the two-layer TDGP against a two-layer CDGP, two-layer DNSGP, DKL, and the shallow sparse GP (SGP). These models are evaluated on average negative log-predictive density (NLPD) and mean relative absolute error (MRAE). All metrics are 'the lower, the better'. Importantly, throughout the experiments, we also emphasize (i) the interpretability of our model compared to the prior art and (ii) TDGP's inductive bias towards learning low-dimensional embeddings. In all experiments, inputs and targets are normalized so that the training set has zero mean and unit variance. Appendix D contains more details of architecture, training, and initialization. We implemented the experiments in Python using GPflow [10], GPflux [8], and Keras [2]. Code is available as supplementary material at https://github.com/spectraldani/thindeepgps .

4.1 Synthetic experiment

Data. To assess our intuition that TDGP leans towards inducing low-dimensional manifolds, we show how well TDGP and competitors can fit a composite function f = g ◦ h , where g : R → R and h : R 2 → R are non-linear functions. In this context, h acts as a 'funnel' inducing a 1D manifold. For more details on f and how we sample from the input space, we refer the reader to Appendix D.

Results. Figure 3 shows the posterior mean of TDGP and competing methods for our synthetic dataset. Note TDGP accurately fits the target function (leftmost panel), while the other methods fail to capture the shape of several of the local maxima. Consequently, Table 1 shows that TDGP performs significantly better in terms of both NLPD and MRAE in the test data.

Examining the latent space. TDGP outperformed competitors in predictive performance, but how do their latent spaces compare? To further understand the nuances that distinguish each model, Fig. 4

plots their respective latent spaces and lengthscale fields - along with the true latent space induced by h as a baseline. While TDGP induces both a latent space and a lengthscale field, it is important to highlight that the same does not hold for CDGP and DNSGP. Thus, Fig. 4 does not show a latent space for DNSGP or a lengthscale field for CDGP. Notably, TDGP's latent space perfectly captures

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the shape of h , while DKL and CDGP fail to do so. Analyzing the lengthscale fields, we conclude that TDGP successfully learns a non-stationary kernel, unlike DNSGP.

Finally, Fig. 5 shows that TDGP learns to give much higher importance to one of its latent dimensions, supporting our intuition that TDGPs lean towards learning low-dimensional manifolds. While the shallow GP and CDGP also weigh one dimension higher than the other, this discrepancy is less accentuated. It is worth mentioning that measuring the relevance of latent dimensions depends on the architecture we evaluate. For DGPs and DKL, we can use the inverses of the output layer's kernel lengthscales. Naturally, the same rationale applies to shallow GPs. For TDGP, the analogous variable is the kernel variance of each hidden layer's row, since the

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larger this value is, the more distant from zero the values of that row are. Overall, these results suggest that TDGP leans towards learning sparser representations than its competitors.

4.2 Bathymetry case study

Data. As a case-study, we also apply TDGP to the bathymetry dataset GEBCO. This dataset contains a global terrain model (elevation data) for ocean and land. As an example of a non-stationary task, we selected an especially challenging subset of the data covering the Andes mountain range, ocean, and land (see Appendix D.2 for details). We subsample 1,000 points from this region and compare the methods via five-fold cross-validation.

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Results. The NLPD and MRAE results are listed in Table 2, where we observed our method to be slightly better than others. However, more importantly, this dataset can be thoroughly examined and interpreted. Figure 8 shows correlation plots for a point located on the lower slope of the Andes. We observe more sensible correlations for TDGP and CDGP compared to other methods - the correlation is high in the locations across the slope. This plot also highlights a recurring problem with DNSGP: Despite a high inverse lengthscale barrier in the Andes, there is still a correlation between the slope of the moun-

tain and the sea level. Additionally, Fig. 7 shows the domain coloring of learned latent spaces and the sum of the eigenvalues of the lengthscale fields. Once again, we note that only TDGP can be analyzed in both ways. This is an advantage in settings where expert priors on the smoothness exist, as these can be placed directly on the lengthscale field instead of the less accessible latent space mapping. As expected, methods that learn the lengthscale field place high inverse lengthscale values around the mountain range and low values in the smooth oceans and plains.

4.3 Benchmark datasets

Data. We also compare the methods on four well-known regression datasets from the UCI repository. To assess each model fairly, we adopt a ten-fold separation of the datasets into training and testing.

Results. Figure 9 shows the average NLPD for each method in the UCI datasets along with their respective standard deviations. TDGP performs either on par with other methods or outperforms them. Figure 10 also shows the relative relevance of each method's latent dimensions. Similarly

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to what we observed in the synthetic experiments, TDGP's inductive bias leads to a sharp split between relevant and irrelevant dimensions compared to the other approaches. Even for the cases with similar generalization errors, TDGP shows better interpretability and bias towards learning lower-dimensional representations.

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5 Related work

Expressive non-stationary GPs can be constructed in a variety of ways. We give a brief overview of some contributions available in the literature below.

Convolution of SE and scale mixture kernels. Non-stationary covariance learning can be done by convolving two SE kernels with different lengthscales. This well-studied approach was later extended to arbitrary kernels by Paciorek and Schervish [19]. Gibbs [12] introduced this formulation,

representing the lengthscale as the sum of squares of an arbitrary function applied to each covariance input. Later, Higdon et al. [14] introduced this approach in the spatial context, varying the focus points of the ellipse representing the 2 × 2 lengthscale matrix of a SE covariance with independent GPs, which constitutes a two-layer DGP in 2D. Heinonen et al. [13] introduced a method to jointly vary the covariance amplitude and lengthscale, and the observation noise over the domain, using Hamiltonian Monte Carlo for inference. As discussed in Section 2, all these methods inherit the limitations of their kernel, i.e. issues with the pre-factor and violation of the triangle inequality, which hinders manifold learning.

Compositional deep GPs. Compositional DGPs [15, 3] construct priors over complex functions by repeatedly applying a sequence of non-parametric transformations to the input, with each transformation being represented by a GP with a stationary covariance. One of the the most significant issues with CDGPs is that, unlike neural networks which compose linear functions through nonlinear scalar activations, every latent layer is a non-linear function. Therefore, several authors have suggested improvements to DGPs by adding skip connections that directly concatenate the hidden layer with the input space [9] or the use of linear mean functions [16, 23] for improved performance and capability. CDGP research also explores improvements in inference, such as the use of mini-batching and more efficient approximations to intractable integrals [23], and the inclusion of auxiliary latent variables to allow for more flexible posteriors [21]. Since TDGP is based on deformation kernels and is related to CDGPs, we believe it can leverage many of these advances in inference, which is left to future investigations.

Deep kernel learning. An alternative to CDGPs is to use layers of parametric transformations to learn arbitrarily complex kernels. Deep kernel learning [1, 26] parameterizes such transformations with a deterministic neural network. Since these deep kernels have many parameters and are not constructed in a Bayesian way, they are prone to overfitting and miscalibrated uncertainties, and can perform poorly in practice [17]. Nevertheless, by using Bayesian injective warping functions that are well-suited for spatial applications, Zammit-Mangion et al. [27] were able to achieve good results in low-dimensional datasets.

6 Conclusion

This work presented Thin and Deep GP (TDGP), a new hierarchical architecture for DGPs. TDGP can recover non-stationary functions through a locally linear deformation of stationary kernels. Importantly, this construction allows us to interpret TDGP in terms of the latent embeddings and the lengthscale field it induces. Additionally, while TDGP shares a connection with CDGP, our prior does not concentrate on 'flat' samples when the number of layers increases - as is the case with CDGP. Furthermore, our experiments show that TDGP performs as well as or better than previous approaches. Moreover, TDGP has a robust inductive bias toward learning low-dimensional embeddings, which is instrumental for better interpretability.

That being said, we expect TDGP will be especially useful for geospatial modeling in cases where we are modeling non-stationary functions and when we have expert knowledge on how this function should vary over space/time, which could be inserted as priors over the locally linear transformations. We also believe that recent improvements in inference available in the literature could greatly enhance the expressivity and ease of training of TDGP.

Acknowledgments

This work was supported in part by the CONFAP-CNPq-THE UK Academies program (grant UKA00160-00003.01.00/19). Diego Mesquita was supported by the Silicon Valley Community Foundation (SVCF) through the Ripple impact fund, the Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) through the Jovem Cientista do Nosso Estado program, and the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) through the grant 2023/00815-6.

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Hugh Salimbeni. 'Deep Gaussian Processes: Advances in Models and Inference'. PhD thesis. Imperial College London, 2019.

Hugh Salimbeni and Marc Peter Deisenroth. 'Deeply non-stationary Gaussian processes'. In: 2nd Workshop on Bayesian Deep Learning (NeurIPS) . 2017.

Hugh Salimbeni and Marc Peter Deisenroth. 'Doubly Stochastic Variational Inference for Deep Gaussian Processes'. In: Advances in Neural Information Processing Systems (NeurIPS) . 2017.

Michalis K. Titsias and Neil D. Lawrence. 'Bayesian Gaussian Process Latent Variable Model'. In: Artificial Intelligence and Statistics (AISTATS) . 2010.

Michalis K. Titsias and Miguel Lázaro-Gredilla. 'Variational Inference for Mahalanobis Distance Metrics in Gaussian Process Regression'. In: Advances in Neural Information Processing Systems (NeurIPS) . 2013.

Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric P. Xing. 'Deep Kernel Learning'. In: Artificial Intelligence and Statistics (AISTATS) . 2016.

Andrew Zammit-Mangion, Tin Lok James Ng, Quan Vu, and Maurizio Filippone. 'Deep Compositional Spatial Models'. In: Journal of the American Statistical Association 117.540 (2022).

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nTechniques such as Deep Kernel Learning and extensions such as Deep GPs have been proposed in order to overcome the flexibility constraints associated with standard shallow GPs. However, although these techniques are better suited to model non-stationary data, they are still susceptible to issues such as pathologies when extended to more than a few layers, as well as overfitting. These techniques also lack the interpretability of the latent space and associated length-scales that make standard GPs so appealing. In order to mitigate these limitations, the authors propose a novel approach that generalises over the more widely-used compositional DGPs via a hierarchical model that incorporates locally linear deformations of stationary kernels. A synthetic example is crafted in order to showcase how TDGP improves over competing methods for problems exhibiting heavy non-stationarity. This is further complemented by an analysis on a real-world dataset, as well as 4 benchmark datasets from the UCI repository.\n\nSTRENGTHS:\n- The paper is very well-written and a pleasure to read. I commend the authors for structuring the paper in a manner that clearly shows connections to related models, while also showcasing the key contributions of this work.\n- I particularly appreciated Sections 4.1 and 4.2, which capture the various dimensions along which TDGP can lead to improvements over other methods (predictive performance, uncertainty quantification, and interpretability).\n- The problem statement tackled in this paper is well-motivated, and the limitations of other techniques are clearly pointed out both verbally and visually in the experiments section. One possible improvement to this could be highlighting application domains where TDGP is expected to be most impactful – i.e., which application domains tend to non-stationary data where well-calibrated uncertainty is especially useful. While fairly minor, this could help clarify the significance of this work earlier on the paper.\n\nWEAKNESSES:\n- The authors comment in the *Limitations* section that the increase in the number of variational parameters could slow down optimisation – it could be worth supporting this statement by specifying the computational complexity, and maybe an analysis reporting wall clock time for one of the experiments in Section 4. \n- I would also be interested in whether the increased parametrisation of the model results in models that are less stable or difficult to consistently converge. This has implications on the practical utility of the model.\n- The paper focuses heavily on using the SE kernel throughout – can this be extended to other kernels or is the value-add of this extension not considered to be worthwhile for the paper?\n\nQUESTIONS:\nI do not have additional questions beyond the points raised in the *Weaknesses* section. I would appreciate if the authors could read through this and indicate whether they can improve upon these aspects in a future revision.\n\nThe paper is a solid addition to the literature on hierarchical GP models – besides proposing a novel method, it also gather and illustrates potential deficiencies in existing methods that are widely-used in the community. I also believe that the paper is already in a state that is fit for publication, and can only be improved further with additional minor iterations.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-02"}

comments: SUMMARY:
This paper extends the CDGP to address two weaknesses in deep Gaussian processes. The solution is similar intuitively to a residual connection. Instead of letting each layer depend only on the last layer, they let each layer depend on the last layer and the input. This allows the model to induce a manifold and use a lengthscale field.

STRENGTHS:
This seems like an important problem, and the solution seems principled.

WEAKNESSES:
There should be more discussion of what it means to not induce a manifold, why violating the triangle inequality leads to that, and why not inducing a manifold is a drawback. I think inducing a manifold implies that one can make the latent space plots: why are these valuable? Why do they require the triangle inequality?

There should be more discussion of how a lengthscale field leads to interpretability.

In both the latent space and the inverse lengthscale plots, there should be some discussion about why we want to see these plots: for instance, in figure 7, we have the two sets of plots: what can we glean from them about the data/problem/etc? Figure 8 has a little bit (the zone of high correlation extends into the mountains), but this is not obvious for a non-expert application reader. You could have a short tutorial perhaps of how to read these plots in the supplement.

QUESTIONS:
Most of my questions are in the weaknesses.

LIMITATIONS:
They have a good discussion of limitations: mainly that it is computationally expensive and leads to requiring more parameters.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of non-stationary kernel design for Gaussian processes. Two main approaches for this problem are deformation kernels and length-scale mixture kernels. On one hand, deformation kernels, eg deep GPs, have found great success while trading off expressivity and the ability to learn latent manifolds in intermediate layers, for lack of interpretability and being prone to pathologies. On the other, length-scale mixture kernels retain interpretability but typically break the triangle inequality in the induced latent space, precluding a manifold structure. The authors propose thin and deep GPs, which aim to combine the benefits of both these approaches, while mitigating their drawbacks. The approach learns a position-dependent matrix, parameterized by a GP, which learns a pointwise linear transformation of the data. The output of such a layer can then be composed to learn a new matrix acting on the original data, enabling a deep learning structure. Numerical examples are provided which appear competitive with existing techniques.\n\nSTRENGTHS:\nThis paper is clearly presented and combines existing ideas in a novel way. The numerical examples seem well selected, with a mixture of synthetic/analytic examples that demonstrate the utility of the method, with standard benchmark datasets allowing for meaningful comparisons on real data. A case study is also provided that is geared towards the strengths of the technique.\n\nWEAKNESSES:\nThe main weakness of the paper in my view is Theorem 3.1. It appears that the proof that the CDGP is a generalization of deep GPs hinges upon the introduction of a new variable that plays the role of the deep GP, and then turning off their method's parameters by concentrating their priors around zero. This seems highly misleading unless the method is introduced in a way that uses the augmented space as default, or some numerical evidence is given that it should be turned off if appropriate. In this case, presumably the deep GP should also be tested with and without an augmented space for fair comparison. It seems that the augmented space is not used in any of the experiments in any of the models presented, since the local linearity leaving the data unchanged around zero is listed as a limitation.\n\nQUESTIONS:\nHave you tested the method with and without the data being augmented with a bias term? Has this been tested in prior methods as well?\n\nSince the matrices are position dependent, the dimension of their range can vary across the domain as a different number of their singular values. How does this influence the latent manifold interpretation that motivated your work?\n\nLIMITATIONS:\nBesides those listed above, the computational expense is listed as a major limitation. This is a common drawback of related work, and it may be helpful to list the memory and computational complexity of this method with L layers, as is commonly done in the support literature.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new deep Gaussian process (DGP) model, the thin and deep GP (TDGP), which does not suffer from a diminishing signal as the number of layers increases. The crux of the TDGP model is the covariance function that, for each layer $\\ell$, acts on a linear combination of a (non-linear) transformation of the outputs of the previous layer and the inputs $\\mathbf{x}$, $\\mathbf{h}^\\ell(\\mathbf{x}) = \\mathbf{W}^\\ell(\\mathbf{h}^{\\ell-1}(\\mathbf{x})) \\mathbf{x}$. This hierarchical construction is shown to cover the usual DGP models while not suffering from diminishing signals as the depth increases. To perform inference in this model, the authors adopt a mean-field VI scheme, which is demonstrated to be effective through a number of experiments on both synthetic and real-world data.\n\nSTRENGTHS:\n1. The paper presents an interesting and original approach to DGPs, an important field that should interest the NeurIPS community.\n2. The paper is of high overall quality and very well-written.\n3. The presented model has the potential to be of considerable significance, given that it generalises previous DGP models while not suffering from the same pathologies.\n\nWEAKNESSES:\n1. While I think the paper presents a great idea with lots of potential, my main concern is the lack of a deeper analysis of the model. Given that the proposed model is essentially just a particular transformation of the inputs at each layer and a rather straightforward mean-field VI scheme, I would expect some insights into the behaviour of the model, even if these were just empirical. The current experiments do present some interesting insights, but, say, what happens to the model's performance as we increase the depth of the model? What if we change the width of the layers? What if we use different covariance functions than the squared exponential? How does the induced covariance matrix look at different layers of the model compared to, say, DNSGP? Why does the model encourage low-dimensional latent spaces, and does this happen too for models deeper than just two layers? A Gaussian prior shouldn't encourage sparsity, so this aspect of the model is particularly puzzling.\n2. As mentioned, a standard mean-field VI scheme seems very rough given that this is known to perform poorly for many models, including DGPs. To my knowledge, the current state-of-the-art is still the doubly stochastic VI scheme by Salimbeni & Deisenroth (2017), so it would make sense to try something similar (and unless I misunderstand something, it seems fairly straightforward to do this for the proposed model). \n3. One clarity issue (perhaps the only one) I find with the paper is that it is unclear to me which DGP model the authors refer to as \"CDGP\". Deep GPs have evolved dramatically from the original formulation of Damianou & Lawrence (2013), and the mentioned pathologies are not as pronounced anymore (to my knowledge). It is great that the authors compare against DNSGP, but comparing against the doubly stochastic DGP rather than the original formulation makes much more sense. As it is unclear if the authors actually compare to the current state-of-the-art, it is also unclear if the proposed model addresses actual problems.\n4. Experimentally, the model seems to work well, but the authors use only two layers for all models. This is particularly strange as a key selling point of the proposed model is that it doesn't degenerate as the number of layers increases. But it is also an odd choice since deeper models should work better (and DNSGP demonstrably does for two of the UCI datasets). It is also strange that MRAE statistics are only reported for the GEBCO dataset; these should also be reported for the UCI experiments.\n5. The model is claimed to be \"interpretable\", but this only seems to be the case for a two-layer model (i.e., one with a single latent space). For deeper models, the distance matrix is still a highly non-linear function of the inputs, which I don't think will be particularly interpretable.\n\n\nReferences:\n\n- Andreas Damianou & Neil Lawrence, \"Deep Gaussian processes\", AISTATS 2013.\n- Hugh Salimbeni & Marc Peter Deisenroth. \"Doubly Stochastic Variational Inference for Deep Gaussian Processes\", NeurIPS 2017.\n\nQUESTIONS:\n1. Which specific variation of the DGP model is referred to as \"CDGP\"? \n2. In lines 138-139, you say that \"By placing zero-mean priors on the entries of $\\mathbf{W}$, we encode an inductive bias to maximally reduce the latent dimensionality of $\\mathbf{h}$.\" Why is this? Intuitively, since the prior is Gaussian, it shouldn't encourage sparsity.\n3. Your contribution seems to be an input transformation, which isn't tied to a specific covariance function, so is there a reason that you chose the squared exponential (or exponentiated quadratic) covariance function? The Matérn class is more often used in practice, I believe.\n4. Is a model with more than two layers still interpretable? If so, how would you interpret the latent space?\n5. In Figures 4 and 7, how is the inverse length scale computed for TDGP? Also, why can we not compute the inverse length scales for CDGP or DKL? And why can we not visualise the latent space for DNSGP?\n6. In lines 51-52, you mention that TDGP is the only deep architecture that induces data embeddings. What do you mean by \"data embedding\" here, and why do other models not have this property? For instance, even for the original DGP model, we can easily visualise the distribution of a particular sample at a particular layer.\n\nLIMITATIONS:\nThe authors have adequately addressed limitations in a dedicated paragraph.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-13"}