Multi-Output Gaussian Processes for Graph-Structured Data

Ayano Nakai-Kasai; Tadashi Wadayama

REVIEW 3 major objections 5 minor 36 references

Multi-Output Gaussian Processes for Graph-Structured Data

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Graph regression can be unified under a single multi-output Gaussian process formulation.

desk verdict A clean unification of graph-GP methods with a new graph PC kernel and genuinely new inference scenarios, but the printed induced-subgraph formula (Eq. 32) is self-referential and must be fixed before anyone can reproduce the paper's flagship experiment. read the letter →

arxiv 2505.16755 v1 pith:D6N3GUIK submitted 2025-05-22 cs.LG

Ayano Nakai-Kasai , Tadashi Wadayama This is my paper · ORCID

classification cs.LG

keywords multi-outputGaussianprocessesgraph-structureddatagraphkernelsseparableprocessconvolutionmissingnodeestimationheterotopickernelregression

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that graph regression should be built directly on the definition of multi-output Gaussian processes rather than on a specialized graph-filter model. Doing so yields one formulation whose block covariance matrix couples nodes and data freely, so the same method covers data observed on all nodes or only some, symmetric or asymmetric sample counts, and test nodes that appear only after training. The paper shows that two earlier graph-GP methods are special cases of this formulation, and it adds kernel designs—sums of separable kernels and a novel graph process convolution—that let the model express trends, periodicity, and node-specific behavior. A sympathetic reader would care because the result is a single flexible regression tool for graph data instead of a collection of special-purpose models.

What carries the argument

The carrying object is the multi-output covariance block matrix $\mathbf{K}_M(X)$ of Eq. (13), with each block $k_{mm'}(X_m, X_{m'})$ encoding both vertex correlation and input-space correlation. The separable kernel $k_{mm'}(x,x')=k(x,x')k_G(m,m')$ is the workhorse: it decouples graph structure from data structure, reduces to the Kronecker form $K_G \otimes K(\bar{X})$ under isotopic data, and puts the earlier graph-filter methods inside the model via $K_G=CC^T$. Sums of separable kernels add multiple latent correlation channels, and the proposed graph process convolution replaces the shared data kernel with a per-vertex convolution whose vertex coupling is itself drawn from graph kernels. Prediction on an induced subgraph uses the same block covariance machinery, with cross-covariance blocks $k_{mv_o}(X_m, \tilde{X}^*_{v_o})$ carrying information from the original graph into the subgraph.

What would settle it

Generate graph signals from a process with node-specific length scales and non-separable cross-node coupling, e.g., each vertex's function is a convolution of neighbors' latent functions with vertex-dependent smoothness, and test the proposed multi-output GP model using the paper's diffusion and graph Matérn kernels; if its predictive log-likelihood does not exceed ordinary single-output GP on held-out nodes, the claim that the formulation captures a variety of correlations not handled by existing studies is not supported in that regime.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the general MOGP definition—vector-valued functions whose covariance is a block matrix $\mathbf{K}_M(X)$ with entries $k_{mm'}(x,x')$—is already the right object for regression on graph-structured data. The graph enters through the cross-node covariance: with a separable kernel $k_{mm'}(x,x')=k(x,x')k_G(m,m')$, the matrix factorizes as $K_G \otimes K(\bar{X})$ for isotopic data, which makes the earlier methods [15], [16] exact special cases with $K_G = CC^T$ for a graph filter $C$. The same skeleton accepts heterotopic inputs, asymmetric numbers of observations per node, sums of separable kernels, and a new graph process convolution $k_{mm'}(x,x') = v^2 s_m s_{m'} (2\pi)^{-D/2}|P|^{-1/2}\exp(-\tfrac{1}{2}(x-x')^T P^{-1}(x-x'))$ in which $s_m s_{m'}$ and $P^{-1}_m+P^{-1}_{m'}$ are set by graph kernels. It also supports prediction on an induced subgraph, so missing nodes can be estimated from all available data rather than from their own past alone.

Load-bearing premise

The load-bearing premise is that the true correlation between nodes can be expressed by the chosen covariance kernel—either a product of a data kernel and a graph kernel, or the proposed graph convolution—so if the graph kernel does not match the real mechanism linking the nodes, the model falls back toward ordinary single-output GP and can be worse.

Editorial extensions

If this is right

Single-output GP regression and the graph-GP methods of [15] and [16] are recovered as special cases by choosing $k_G(m,m')=\delta_{mm'}$ or $K_G=CC^T$, so the formulation is strictly more general.
Heterotopic data and asymmetric node-wise training sizes are handled directly by Eq. (13), removing the isotopic/symmetric restrictions of earlier methods.
Sum-of-separable kernels let a single model combine multiple data kernels (e.g., trend and periodicity) with multiple graph kernels, adding only a few hyperparameters.
The graph process convolution gives each node its own length-scale-like parameters while still coupling nodes through graph kernels, which the experiments connect to tighter predictive intervals on denser graphs.
Missing node values can be predicted on the induced subgraph using all training data, rather than only the defective node's own observations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The near-tie with ordinary single-output GP on the weather data suggests the advantage is conditional on the graph kernel matching the true coupling mechanism; the paper does not claim a universal win.
The same block-covariance skeleton could extend to multi-fidelity, multi-class, and latent-variable settings the paper lists as future directions, but those extensions are not demonstrated here.
Because the covariance inverse is $O(N^3)$, the practical bottleneck is scaling; a graph-locality sparsification, flagged as future work, would likely determine whether the flexibility matters on large graphs.
A natural next experiment would be to learn the graph kernel's spectral coefficients inside the multi-output GP objective, which this paper does not attempt.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, this is a clean unification paper: the authors take the standard definition of multi-output Gaussian processes, apply it to graph-structured data, and show that the existing graph-GP methods of Venkitaraman et al. and Zhi et al. are special cases with a separable kernel. On top of that they add genuinely new capabilities: heterotopic and asymmetric data configurations, sum-of-separable kernels, a new "graph process convolution" kernel, and an induced-subgraph prediction scenario (predicting missing nodes from observations on the rest of the graph). Second, the induced-subgraph formula as printed is wrong. Eq. (32) states that the predictive mean equals K_MO^T (K_M + Sigma)^{-1} \tilde{Y}^*, where \tilde{Y}^* is the unknown test target. That is circular. It should be the training vector Y, as in Eq. (15). The experiments in Section IV-C report sensible numbers, so the authors almost certainly implemented the corrected formula, but a reader cannot reproduce the method without inferring and fixing this error. That is a real problem, and it sits on the paper's most distinctive inference scenario.

The unification itself is correct and clearly presented. The graph PC kernel in Section III-D is a legitimate contribution: it gives per-vertex flexibility in the data kernel while still injecting graph structure through the s_m s_m' term and the inverse-covariance construction. The experiments are honest. On the weather data, several graph kernels are no better than SOGP in MSE, and the authors do not overclaim; the benefits show up mainly in log-likelihood. The computational complexity discussion is sensible, and the GitHub repository is a plus.

Soft spots, in proportion: the Eq. (32) typo is the main one and should be corrected before publication; it is a typo, not a conceptual error. The empirical evaluation is small-scale — 32-vertex synthetic graphs, 45-node weather data — so practical claims are modest. The overall framework's value depends on the graph kernel matching the actual cross-node mechanism, and the weather results illustrate exactly that fragility. Also, the "extension" to heterotopic data is mathematically just writing down the block covariance that standard MOGP already permits; the novelty is in the packaging and in the graph-specific kernel designs, not in the underlying algebra.

Who this is for: anyone working on Gaussian processes for graph-structured data, or on multi-output GP applications where graph correlation matters. It deserves a serious referee; with the typo fixed, it is a solid contribution to that subfield, though not a breakthrough that reshapes it. Recommendation: send it to review, but require the authors to fix Eq. (32) and ideally to provide a small code snippet or clarified notation so the induced-subgraph predictive formula is unambiguous.

Referee Report

3 major / 5 minor

Summary. This paper proposes a general multi-output Gaussian process (MOGP) regression framework for graph-structured data, built from the standard definition of MOGP rather than from a specific graph-filter model. The formulation allows node-specific inputs and outputs, heterotopic and asymmetric data configurations, flexible kernel choices including sums of separable (SoS) kernels and a newly proposed graph process-convolution (graph PC) kernel, and prediction on induced subgraphs for missing-node estimation. The authors show analytically that SOGP and the graph-MOGP methods of Venkitaraman et al. and Zhi et al. are special cases, and they report experiments on synthetic data, fMRI data, and weather data. The core GP conditioning algebra is standard, and the experimental code is publicly available.

Significance. If the load-bearing formulas are corrected, the paper provides a useful unifying view of Gaussian process regression on graphs: known graph-filter-based methods become particular kernel choices inside a standard MOGP, and the framework opens new configurations and inference scenarios that earlier methods do not cover. The analytic special-case relations in Section III-F are clear and valuable, and the release of code is a concrete strength. The novelty is primarily integrative rather than a new inference principle, and the empirical evidence is mixed: graph-aware models outperform baselines on some datasets and metrics but not others. The main contribution is therefore the clarification and extension of the design space, not a demonstrated universal performance gain.

major comments (3)

[III-E, Eqs. (28), (32), (33)] Eq. (32) is self-referential: the predictive mean conditions on \tilde{Y}^*, the vector being predicted. The right-hand side should multiply by the training output vector Y, as in Eq. (15). In addition, because Eq. (28) adds observation noise to the test block, the induced-subgraph predictive covariance in Eq. (33) is missing the term +\sigma^2 I_{\tilde{T}} that appears in the analogous SOGP formula (7) and MOGP formula (16). Without these corrections, the induced-subgraph prediction scenario described in Section III-E is not reproducible as written.
[III-D, Eq. (26)] Eq. (26) sets P_m^{-1}+P_{m'}^{-1} = k_{G,2}(m,m')^{-1}I_D. This construction is not valid without additional conditions. If k_{G,2}(m,m') is non-positive for some pair, the right-hand side is not positive definite, so it cannot equal the left-hand side; for example, the Laplacian pseudoinverse listed in Table III has negative off-diagonal entries. Even when all entries are positive, the right-hand side has the additive form u_m+u_{m'}, which is not satisfied by most graph kernels, so the equation cannot generally be realized by choosing per-vertex matrices P_m. The authors should either state explicit feasibility conditions or replace Eq. (26) with a parameterization that is guaranteed to be valid.
[IV-B, Table IV, and IV-D, Table VI] The empirical support for the claim that graph information improves performance is partial. In Table IV, ICM achieves the best MSE for all graph degrees k=6,12,18,24, and the proposed MOGP improves only the log-likelihood for two of the four degrees. In Table VI, on the weather dataset SOGP has the best mean MSE and graph PC underperforms SOGP, while the clearest gains are for SoS kernels in log-likelihood. The conclusions in Sections IV-B and V should be qualified to reflect that the benefit depends strongly on the kernel choice, the graph, and the evaluation metric.

minor comments (5)

[III-A] The phrase "we re-difine N" should be "we redefine N".
[III-B] The word "assymetric" should be "asymmetric".
[III-D] The phrase "One extention is" should be "One extension is".
[IV-B, Eq. (39)] The interval for q_j is written as [5, 5], which is degenerate; this is presumably a typo for a symmetric interval such as [-5, 5].
[IV-A and Table IV] The arrows in the table headers (▼, ▲) are not explicitly defined in the text or caption; please state that lower MSE and higher log-likelihood are better.

Circularity Check

1 steps flagged · score 6.0 of 10

Induced-subgraph predictive mean (Eq. 32) is self-referential: it multiplies by the unknown target \tilde{Y}^* instead of the training outputs Y, so the showcase inference scenario reduces to the quantity it is supposed to predict as written.

self definitional [Section III-E, Eq. (32)]
"The predictive distribution of ˜Y ∗ is given by p( ˜Y ∗| ˜X ∗, D) = N(µIS( ˜X ∗), ΣIS( ˜X ∗)), where µIS( ˜X ∗) = KMO(X, ˜X ∗)T(KM(X) + Σ)−1 ˜Y ∗, (32)"

In this induced-subgraph inference scenario, ˜Y ∗ is defined as the target values to be predicted (the unobserved outputs at the test vertices). Eq. (32) then defines the predictive mean µIS as a linear function of ˜Y ∗ itself, with the training outputs Y absent from the right-hand side. Standard conditioning of the joint Gaussian in Eq. (28) would give KMO(X, ˜X ∗)T(KM(X)+Σ)−1Y, as in Eq. (15). As written, the prediction rule is not computable from the training data; the estimator is defined in terms of the estimand, so the showcased missing-node prediction reduces to the unknown quantity by construction.

full rationale

The general MOGP formulation is obtained from the standard conditional-Gaussian equations (Eqs. 12-16); hyperparameters are fitted by marginal likelihood (Eq. 17) and evaluated on held-out outputs, so the main regression derivation is not circular. The claim that SOGP and [15],[16] are special cases is analytic (Sect. III-F reduces their model to KM(X)=CC^T⊗K(\bar X), which is the separable kernel of Eq. (20)), not a fitted equivalence or a self-citation chain. The one load-bearing circular step is Eq. (32) in Section III-E: the predictive mean for the induced-subgraph target ˜Y ∗ is written as KMO(X, ˜X ∗)T(KM(X)+Σ)−1 ˜Y ∗, so the unknown target appears on the right-hand side and the training outputs Y do not. This makes the showcased missing-node inference scenario self-referential as written; standard conditioning would use Y. The reported experiments suggest the implemented formula may have been the corrected one, but the paper itself does not provide a reproducible prediction rule for this scenario. Because the rest of the framework is independent, the circularity is partial rather than total.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The MOGP framework itself uses standard Gaussian process hyperparameters fitted by marginal likelihood. The graph-specific ingredients are the graph kernels borrowed from prior work and the ad hoc insertion of graph kernels into the process-convolution covariance in graph PC. No hidden or unexplained constants are needed for the central claim.

free parameters (4)

Noise variance sigma^2 = Shared across vertices (sigma^2_1=...=sigma^2_M) in all experiments
Set equal in Section IV-A to reduce hyperparameter count; part of the observation model in Eq. (9).
SE kernel hyperparameters (v^2, length scale l) = Determined by maximizing log-marginal likelihood (Eq. 17)
Used as k(x,x') in separable, SoS, and graph PC kernels; standard GP hyperparameters.
Graph kernel hyperparameters (alpha, beta_i, etc.) = Fit by maximum marginal likelihood, e.g., alpha in global filtering, diffusion, Matérn
Appears in the graph kernels listed in Table III; these parameters are optimized per experiment.
Graph PC hyperparameters (s_m s_m', Lambda = l I) = Set via graph kernels kG,1 and kG,2 with length scale l fit to data
In Eqs. (25)-(26) and Section IV-C/D, graph PC uses graph kernels to define the scalar and precision terms; l is a free parameter.

assumptions (5)

standard math A vector-valued Gaussian process with cross-covariance matrix k_M(x,x') is a valid prior for M correlated functions
Foundation of MOGP in Section III-A; follows from the definition of a Gaussian process.
domain assumption The graph kernels used (Laplacian, diffusion, Matérn, etc.) are valid positive semi-definite kernels on the graph
Relied on in Sections III-C/III-D and Table III; taken from prior literature, validity for arbitrary graphs assumed.
domain assumption The joint distribution of training outputs Y and test outputs Y_tilde* is the Gaussian given in Eq. (28) with block covariance
Required for the induced-subgraph inference in Section III-E; standard GP conditioning if the prior is correctly specified.
standard math The sum of kernels is a valid kernel
Used to justify the sum-of-separable (SoS) kernel in Section III-C.
ad hoc to paper Substituting graph kernels into the process-convolution terms via s_m s_m' = kG,1(m,m') and P_m^{-1}+P_{m'}^{-1}=kG,2(m,m')^{-1}I preserves convolution semantics and kernel validity
Proposed in Section III-D as a heuristic design; no derivation proves this is a well-defined convolution over graph processes.

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Cite this review

Pith. "Pith review of Multi-Output Gaussian Processes for Graph-Structured Data." pith.science (2026). https://pith.science/paper/D6N3GUIK

@misc{pith2026250516755,
  author       = {Pith},
  title        = {Pith review of: Multi-Output Gaussian Processes for Graph-Structured Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D6N3GUIK}},
  note         = {Machine review of arXiv:2505.16755}
}

read the original abstract

Graph-structured data is a type of data to be obtained associated with a graph structure where vertices and edges describe some kind of data correlation. This paper proposes a regression method on graph-structured data, which is based on multi-output Gaussian processes (MOGP), to capture both the correlation between vertices and the correlation between associated data. The proposed formulation is built on the definition of MOGP. This allows it to be applied to a wide range of data configurations and scenarios. Moreover, it has high expressive capability due to its flexibility in kernel design. It includes existing methods of Gaussian processes for graph-structured data as special cases and is possible to remove restrictions on data configurations, model selection, and inference scenarios in the existing methods. The performance of extensions achievable by the proposed formulation is evaluated through computer experiments with synthetic and real data.

Figures

Figures reproduced from arXiv: 2505.16755 by the authors.

**Figure 1.** Training data for the experiment in Sect. IV-B. Lines and points are underlying functions and training data for each vertex, respectively. The log-likelihood provides the goodness of the model including the predictive covariance matrix and is defined as Log-likelihood = − T 2 log(2π) − 1 2 log det[Σ∗ ] − 1 2 (y ∗ − µ ∗ ) TΣ∗−1 (y ∗ − µ ∗ ). (38) Lower MSE and higher log-likelihood indicate better performance. B. V… view at source ↗

**Figure 2.** Predicted mean and 95% confidence interval of 10 test points for the case of k = 24. The x coordinates of the predicted means of SOGP and MOGP for graph are shifted by a small amount for better visibility [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Graph and data used in the experiments on the synthetic data in Sect. IV-C. The red dots represent the observed data. The shaded area represents the region to be predicted. vertical line on the right in the figure. The data yi are generated from the following function: yi = sin(xi) xi + ϵi , (40) where ϵi ∼ N (0, 0.01) is the additive Gaussian noise. The connected vertices in the graph have adjacent input regions an… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Predicted means (red lines) and 95% confidence intervals (shaded area) on the synthetic data in Sect. IV-C for four representative methods. global filtering and graph Matern-2 with unnormalized graph ´ Laplacian were used, respectively. We set Λ = ℓI. These kernels wer…

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Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

[1]

C. E. Rasmussen and C. K. I. Williams, Gaussian processes for machine learning, The MIT Press, 2006

work page 2006
[2]

N. A. C. Cressie, Statistics for Spatial Data, Revised Edition. New York, NY: Wiley, 1993

work page 1993
[3]

M. L. Stein, Interpolation of Spatial Data: Some Theory for Kriging. New York, NY: Springer, 1999

work page 1999
[4]

M. P. Deisenroth, D. Fox, and C. E. Rasmussen, ”Gaussian processes for data-efficient learning in robotics and control,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 37, no. 2, pp. 408–423, Feb. 2015

work page 2015
[5]

Liu, Y .-S

H. Liu, Y .-S. Ong, X. Shen, and J. Cai, ”When Gaussian process meets big data: A review of scalable GPs,” IEEE Trans. Neural Netw. Learn. Syst., vol. 31, no. 11, pp. 4405–4423, Nov. 2020

work page 2020
[6]

Gaussian process optimization with matual information,

E. Contal, V . Perchet, and N. Vayatis, “Gaussian process optimization with matual information,” in Proc. International Conference on Machine Learning, PMLR, vol. 32, no. 2, pp. 253–261, Jun. 2014

work page 2014
[7]

Gaus- sian process nonparametric tensor estimator and its minimax optimality,

H. Kanagawa, T. Suzuki, H. Kobayashi, N. Shimizu, Y . Tagami, “Gaus- sian process nonparametric tensor estimator and its minimax optimality,” in Proc. International Conference on Machine Learning, PMLR vol. 48, pp. 1632–1641, Jun. 2016

work page 2016
[8]

Neal, Bayesian Learning for Neural Networks, Springer, 2012

M. Neal, Bayesian Learning for Neural Networks, Springer, 2012

work page 2012

Show all 36 references

[9]

X. Dong, D. Thanou, L. Toni, M. Bronstein and P. Frossard, ”Graph signal processing for machine learning: A review and new perspectives,” IEEE Signal Processing Magazine, vol. 37, no. 6, pp. 117–127, Nov. 2020

2020
[10]

A. J. Smola and R. Kondor, ”Kernels and regularization on graphs,” in Proc. Ann. Conf. Comp. Learn. Theory, ser. Lect. Notes Comp. Sci., B. Sch¨olkopf and M. Warmuth, Eds. Springer, pp. 144–158, Aug. 2003

2003
[11]

Signal-adapted tight frames on graphs,

H. Behjat, U. Richter, D. Van De Ville, and L. S ¨ornmo, “Signal-adapted tight frames on graphs,” IEEE Transactions on Signal Processing, vol. 64, no. 22, pp. 6017–6029, Nov. 2016

2016
[12]

Learning parametric dictio- naries for signals on graphs,

D. Thanou, D. I. Shuman, and P. Frossard, “Learning parametric dictio- naries for signals on graphs,” IEEE Transactions on Signal Processing, vol. 62, no. 15, pp. 3849–3862, Aug. 2014

2014
[13]

Spatial gaussian process regression with mobile sensor networks,

D. Gu and H. Hu, “Spatial gaussian process regression with mobile sensor networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 8, pp. 1279–1290, Aug. 2012

2012
[14]

M. A. ´Alvarez, L. Rosasco, and N. D. Lawrence, ”Kernels for vector- valued functions: A review,” Found. Trends ® Mach. Learn., vol. 4, no. 3, pp. 195–266, Jun. 2012

2012
[15]

Gaussian processes over graphs,

A. Venkitaraman, S. Chatterjee, and P. Handel, “Gaussian processes over graphs,” in Proc. ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5640–5644, May 2020

2020
[16]

Y .-C. Zhi, Y . C. Ng, and X. Dong, ”Gaussian processes on graphs via spectral kernel learning,” IEEE Trans. Signal Inf. Process. Netw., vol. 9, pp. 304–314, Apr. 2023

2023
[17]

Data Reconstruction Coverage Based on Graph Signal Processing for Wireless Sensor Networks,

J. Feng, F. Chen and H. Chen, “Data Reconstruction Coverage Based on Graph Signal Processing for Wireless Sensor Networks,” IEEE Wireless Communications Letters, vol. 11, no. 1, pp. 48–52, Jan. 2022

2022
[18]

A Time-Vertex Signal Processing Framework: Scalable Processing and Meaningful Rep- resentations for Time-Series on Graphs,

F. Grassi, A. Loukas, N. Perraudin and B. Ricaud, “A Time-Vertex Signal Processing Framework: Scalable Processing and Meaningful Rep- resentations for Time-Series on Graphs,” IEEE Transactions on Signal Processing, vol. 66, no. 3, pp. 817–829, Feb, 2018

2018
[19]

Mat ´ern Gaussian processes on graphs,

V . Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. Deisenroth, and N. Durrande, “Mat ´ern Gaussian processes on graphs,” in Proc. Inter- national Conference on Artificial Intelligence and Statistics (AISTATS), San Diego, California, USA, PMLR vol. 130, Apr. 2021

2021
[20]

Geometric deep learning: going beyond euclidean data,

M. M. Bronstein, J. Bruna, Y . LeCun, A. Szlam, and P. Vandergheynst, “Geometric deep learning: going beyond euclidean data,” IEEE Signal Processing Magazine, vol. 34, no.4, pp. 18–42, Jul.2017

2017
[21]

J. M. Phillips, Graph-Structured Data. In: Mathematical Foundations for Data Analysis, Springer Series in the Data Sciences, Springer, Cham., 2021

2021
[22]

Deep convolutional networks on graph- structured data,

M. Henaff, J. Bruna, Y . LeCun, “Deep convolutional networks on graph- structured data,” arXiv preprint, arXiv:1506.05163 [cs.LG], Jun. 2015

2015 arXiv
[23]

Gaussian process kernels for pattern discovery and extrapolation,

A. G. Wilson and R. P. Adams, “Gaussian process kernels for pattern discovery and extrapolation,” in Proc. International Conference in Ma- chine Learning, Jun. 2013

2013
[24]

Y .-C. Zhi, F. L. Opolka, Y . C. Ng, P. Li `o, and X. Dong, ”Transductive kernels for Gaussian processes on graphs,” arXiv [cs.LG], Nov. 2022

2022
[25]

Goovaerts

P. Goovaerts. Geostatistics For Natural Resources Evaluation, Oxford University Press, USA, 1997. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 11

1997
[26]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. Academic Press, 2014

2014
[27]

Borchani, G

H. Borchani, G. Varando, C. Bielza, and P. Larra ˜naga, ”A survey on multi-output regression,” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, vol. 5, pp. 216–233, 2015

2015
[28]

Remarks on multi-output Gaussian process regression,

H. Liu, J. Cai, and Y .-S. Ong, “Remarks on multi-output Gaussian process regression,” Knowl. Based Syst., vol. 144, pp. 102–121, Mar. 2018

2018
[29]

D. M. Higdon, Space and space-time modelling using p rocess con- volutions. In C. Anderson, V . Barnett, P. Chatwin, and A. El-Shaarawi, editors, Quantitative methods for current environmental issues, pp. 37–56. Springer-Verlag, 2002

2002
[30]

Large scale multi-output multi-class classifica- tion using Gaussian processes,

C. Ma, M. A. ´Alvarez, “Large scale multi-output multi-class classifica- tion using Gaussian processes,” Mach. Learn., vol. 112, no. 1, pp. 1077– 1106, 2023

2023
[31]

Predicting the output from a complex computer code when fast approximations are available,

M. C. Kennedy, A. O’Hagan, “Predicting the output from a complex computer code when fast approximations are available,” Biometrika, vol. 87, no. 1, pp. 1–13, Mar. 2000

2000
[32]

Matrix formulation of co-kriging,

D. E. Myers, “Matrix formulation of co-kriging,” Mathematical Geology, vol. 14, pp. 249–257, Jun. 1982

1982
[33]

Focused multi-task learning in a gaussian process framework,

G. Leen, J. Peltonen, S. Kaski, “Focused multi-task learning in a gaussian process framework,” Machine Learning vol. 89, pp. 157–182, Jun. 2012

2012
[34]

Gaussian process latent variable model-based multi-output modeling of incomplete data,

Z. Hu, C. Wang, J. Wu, and D. Du, “Gaussian process latent variable model-based multi-output modeling of incomplete data,” IEEE Trans. Autom. Sci. Eng., vol. 21, no. 2, pp. 1941–1951, Apr. 2024

1941
[35]

DGP-LVM: Derivative Gaussian process latent variable models,

S. Mukherjee, M. Claassen, and P. C. B ¨urkner, “DGP-LVM: Derivative Gaussian process latent variable models,” arXiv preprint arXiv:2404.04074, Apr. 2024

2024 arXiv
[36]

[Online]

Swedish meteorological and hydrological institute (smhi). [Online]. Available: http://opendata-download-metobs.smhi.se/ Ayano Nakai-Kasai received the bachelor’s degree in engineering, the master’s degree in informatics, and Ph.D. degree in informatics from Kyoto Uni- versity,...

2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

C. E. Rasmussen and C. K. I. Williams, Gaussian processes for machine learning, The MIT Press, 2006

work page 2006

[2] [2]

N. A. C. Cressie, Statistics for Spatial Data, Revised Edition. New York, NY: Wiley, 1993

work page 1993

[3] [3]

M. L. Stein, Interpolation of Spatial Data: Some Theory for Kriging. New York, NY: Springer, 1999

work page 1999

[4] [4]

M. P. Deisenroth, D. Fox, and C. E. Rasmussen, ”Gaussian processes for data-efficient learning in robotics and control,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 37, no. 2, pp. 408–423, Feb. 2015

work page 2015

[5] [5]

Liu, Y .-S

H. Liu, Y .-S. Ong, X. Shen, and J. Cai, ”When Gaussian process meets big data: A review of scalable GPs,” IEEE Trans. Neural Netw. Learn. Syst., vol. 31, no. 11, pp. 4405–4423, Nov. 2020

work page 2020

[6] [6]

Gaussian process optimization with matual information,

E. Contal, V . Perchet, and N. Vayatis, “Gaussian process optimization with matual information,” in Proc. International Conference on Machine Learning, PMLR, vol. 32, no. 2, pp. 253–261, Jun. 2014

work page 2014

[7] [7]

Gaus- sian process nonparametric tensor estimator and its minimax optimality,

H. Kanagawa, T. Suzuki, H. Kobayashi, N. Shimizu, Y . Tagami, “Gaus- sian process nonparametric tensor estimator and its minimax optimality,” in Proc. International Conference on Machine Learning, PMLR vol. 48, pp. 1632–1641, Jun. 2016

work page 2016

[8] [8]

Neal, Bayesian Learning for Neural Networks, Springer, 2012

M. Neal, Bayesian Learning for Neural Networks, Springer, 2012

work page 2012