Practical Non-Stationary Graph Gaussian Processes

K. S. Sesh Kumar; Viacheslav Borovitskiy

cs.LGNov 6, 2026

Practical Non-Stationary Graph Gaussian Processes

Authors: K. S. Sesh Kumar, Viacheslav Borovitskiy

Organizations: Brevan Howard Centre for Financial Analysis, Imperial Business School · School of Informatics, University of Edinburgh

Abstract

Gaussian processes on graphs provide a principled framework for prediction and uncertainty estimation, but modelling heterogeneous networks often requires balancing flexibility against parameter complexity. Variability at a node may reflect useful signal that should propagate through the network, unreliable measurements whose influence should be suppressed, or a mixture of both. We introduce Variance-Coupled Gaussian Processes (VCGPs), which use a positive auxiliary signal to modulate graph interactions through a single learned coupling exponent. The exponent controls how strongly high-variability nodes propagate or attenuate information, while a cluster-wise extension accommodates different regimes within the same graph. Under stated structural assumptions, we characterize the unique stationary-rescaling point of the basic family, where correlations match those of a stationary graph kernel while node-dependent variances remain. Experiments on traffic, air-quality and weather datasets show competitive prediction and uncertainty estimation relative to stationary and per-node non-stationary baselines, with the largest gains on networks requiring multiple coupling regimes. Stationary correlations after variance rescaling remain sufficient on some data, and performance depends on the auxiliary signal and model selection.

Figures & tables

Figure 1: One learned

\gamma

selects propagation or isolation. Orange nodes have high

d_{i}

; blue nodes are baseline. Edge thickness shows

|\mathbf{Q}_{ij}|

(top) and correlation deviation from the stationary baseline (bottom). For high-

d_{i}

nodes,

\gamma<0

amplifies coupling and

\gamma>0

suppresses it. Highlighted:

\gamma=0

, uniform coupling;

\gamma=1

, stationary correlations with node-dependent variances.

Method	Specialisation	Precision $\mathbf{Q}$ / Kernel $\mathbf{K}$	Free parameters
stat	$\mathbf{D}=\mathbf{I}$	$\mathbf{Q}=\mathbf{I}+2\tau\mathbf{L}$	$(\tau,s,\sigma_{\varepsilon}^{2})$
fem	$\gamma=0$	$\mathbf{Q}=\mathbf{D}^{-1}+2\tau\mathbf{L}$	$(\tau,s,\sigma_{\varepsilon}^{2})$
het	$\gamma=1$	$\mathbf{Q}=\mathbf{D}^{-1}+2\tau\mathbf{D}^{-1/2}\mathbf{L}\mathbf{D}^{-1/2}$	$(\tau,s,\sigma_{\varepsilon}^{2})$
lgamma	learned scalar $\gamma$	$\mathbf{Q}=\mathbf{D}^{-1}+2\tau\mathbf{D}^{-\gamma/2}\mathbf{L}\mathbf{D}^{-\gamma/2}$	$(\tau,s,\sigma_{\varepsilon}^{2},\gamma)$
lgamma_block	piecewise $\gamma_{k}$ , $K$ blocks	$\mathbf{Q}_{\gamma}$ with $\mathbf{L}\to\mathrm{blockdiag}(\mathbf{L}_{1},\dots,\mathbf{L}_{K})$	$(\tau,s,\sigma_{\varepsilon}^{2},\gamma_{1},\dots,\gamma_{K})$
lgamma_m15	Matérn $\nu=3/2$	$\mathbf{K}=s\,\mathbf{Q}_{\gamma}^{-2}$	same as lgamma

Table 1: Members of the VCGP family. Top: special-value members of lgamma . Bottom: extensions.

Dataset	method	NLL/pt	RMSE	recNLL/pt	cat
METR-LA $n_{\mathrm{test}}{=}63$	lgamma_fisher	$\mathbf{1.2018\pm 0.0287}^{\ast}$	$0.8369\pm 0.0238$	$1.1763\pm 0.0232$	VCGP
	lgamma	$1.2025\pm 0.0294$	$0.8468\pm 0.0247$	$\mathbf{1.1759\pm 0.0228}^{\wedge}$	VCGP
	het	$1.2201\pm 0.0265$	$0.8469\pm 0.0231$	$1.1975\pm 0.0234$	VCGP
	lgamma_fisher_m15	$1.2202\pm 0.0344$	$\mathbf{0.8265\pm 0.0235}^{+}$	$1.1816\pm 0.0247$	VCGP
	lgamma_m15	$1.2234\pm 0.0349$	$0.8354\pm 0.0240$	$1.1835\pm 0.0244$	VCGP
	fuglreg_lam1.0	$1.2382\pm 0.0294$	$0.8603\pm 0.0232$	$1.2064\pm 0.0224$	base

Table 2: Top-

5

methods by NLL, plus the strongest outer baseline. Entries are mean

\pm

standard error (SE); ∗ , + and ∧ mark the best NLL, RMSE and recalibrated NLL among the displayed rows. “cat” distinguishes family members from baselines. block-aug lists

(K,w)

; it and spectral-BIC abbreviate the corresponding lgamma block members. Auxiliary observations are disjoint from targets ( Section 5.1 ).

Figure 2: CA Weather seasonal heatmap of $\hat{\gamma}_{k}$ by month ( lgamma_block , $K=10$ , not the augmented leader; rows are sorted ranks, not fixed clusters). Most months contain both amplification and suppression regimes, changing substantially across seasons.

Dataset	$d_{i}$	stat	fem	het	leader	$\Delta_{1}$	$\Delta_{2}$	$\Delta_{3}$
METR-LA	CV	1.2612	1.2537	1.2201	1.2018	0.0075	0.0336	0.0183
PEMS-BAY	CV	1.3594	1.3245	1.2767	1.2720	0.0349	0.0478	0.0047
LAQN	site-type	1.6013	1.5590	1.5234	1.4519	0.0423	0.0355	0.0715
PM2.5	annual std	1.1329	1.1557	1.2403	0.9756	-0.0228	-0.0846	0.2647
CA Weather	daily-var	1.0436	1.0673	1.1661	0.8685	-0.0237	-0.0988	0.2976

Table 3: Family ablation (NLL/pt). The leader is the lowest-NLL family member in the per-dataset leaderboard. Positive

\Delta

means improvement:

\Delta_{1}=\texttt{stat}\!\to\!\texttt{fem}

\Delta_{2}=\texttt{fem}\!\to\!\texttt{het}

\Delta_{3}=\texttt{het}\!\to\!\text{leader}

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Member	Line	Edit
lgamma_block	4	replace $\mathbf{L}$ with $\mathbf{L}_{\mathrm{bl}}=\mathrm{blockdiag}(\mathbf{L}_{1},\dots,\mathbf{L}_{K})$ ; learn $\gamma\in\mathbb{R}^{K}$
lgamma_m15	5	$\mathbf{K}_{\gamma}\leftarrow s\,(\mathbf{Q}_{\gamma}+\varepsilon\mathbf{I})^{-2}$
lgamma_fisher	6	add $\lambda(\gamma-\gamma_{\mathrm{ref}})^{2}$ to the loss ( $\lambda=0.5/\mathrm{Var}(\hat{\gamma}_{\mathrm{vanilla}})$ , $\gamma_{\mathrm{ref}}=1$ )

Appendix

Table 4: Each variant is a single-line edit of Algorithm 1 .

Figure 3: Identifiability on a

20\times 20

regular grid. Recovered

\hat{\gamma}

(mean

\pm

standard deviation over

10

seeds) versus training fraction

|\mathbf{S}|/n

, for each of four ground truths

\gamma_{\mathrm{true}}\in\{-1,0,+1,+3\}

. The dashed line marks

\hat{\gamma}=\gamma_{\mathrm{true}}

. Recovery is within

0.2

|\mathbf{S}|/n\geq 0.5

for every ground truth.

Figure 4: Two-regime SBM. Left: per-method test NLL/pt. Right: recovered

\hat{\gamma}_{k}

vs. ground truth.

Method	NLL/pt	RMSE	recNLL/pt	$p$ vs. leader	cat
lgamma_fisher	$1.2018\pm 0.0287$	$0.8369\pm 0.0238$	$1.1763\pm 0.0232$	—	family (leader)
lgamma	$1.2025\pm 0.0294$	$0.8468\pm 0.0247$	$1.1759\pm 0.0228$	$0.437$	family
het	$1.2201\pm 0.0265$	$0.8469\pm 0.0231$	$1.1975\pm 0.0234$	$0.002$	family
lgamma_fisher_m15	$1.2202\pm 0.0344$	$0.8265\pm 0.0235$	$1.1816\pm 0.0247$	$0.049$	family
lgamma_m15	$1.2234\pm 0.0349$	$0.8354\pm 0.0240$	$1.1835\pm 0.0244$	$0.039$	family
fuglreg_lam1.0	$1.2382\pm 0.0294$	$0.8603\pm 0.0232$	$1.2064\pm 0.0224$	$<\!10^{-3}$	base

Appendix

Table 5: METR-LA: full

20

-seed leaderboard with mean

\pm

SE;

p

-values from a paired one-sided

t

-test asking whether the dataset leader has significantly lower NLL/pt than the row. The

p

column is the test against the dataset leader; the last column flags whether the row is a family member or an outer baseline.

Method	NLL/pt	RMSE	recNLL/pt	$p$ vs. leader	cat
lgamma_fisher	$1.2720\pm 0.0346$	$0.9015\pm 0.0360$	$1.2282\pm 0.0371$	—	family (leader)
het	$1.2767\pm 0.0329$	$0.8906\pm 0.0329$	$1.2344\pm 0.0362$	$0.166$	family
lgamma	$1.2835\pm 0.0370$	$0.9019\pm 0.0360$	$1.2374\pm 0.0381$	$0.010$	family
lgamma_fisher_m15	$1.2911\pm 0.0359$	$0.8963\pm 0.0329$	$1.2459\pm 0.0371$	$<\!10^{-3}$	family
spectral_bic_lgamma_block	$1.2934\pm 0.0330$	$0.9249\pm 0.0359$	$1.2582\pm 0.0329$	$0.009$	family
lgamma_m15	$1.2965\pm 0.0366$	$0.8965\pm 0.0329$	$1.2504\pm 0.0371$	$<\!10^{-3}$	family

Appendix

Table 6: PEMS-BAY: full

20

-seed leaderboard with mean

\pm

SE;

p

-values from a paired one-sided

t

-test asking whether the dataset leader has significantly lower NLL/pt than the row. The

p

column is the test against the dataset leader; the last column flags whether the row is a family member or an outer baseline.

Method	NLL/pt	RMSE	recNLL/pt	$p$ vs. leader	cat
lgamma_m15	$1.4519\pm 0.0374$	$1.0455\pm 0.0383$	$1.4068\pm 0.0332$	—	family (leader)
lgamma	$1.4688\pm 0.0400$	$1.0434\pm 0.0393$	$1.4171\pm 0.0346$	$0.001$	family
lgamma_fisher_m15	$1.4741\pm 0.0410$	$1.0468\pm 0.0386$	$1.4227\pm 0.0355$	$<\!10^{-3}$	family
lgamma_fisher	$1.4742\pm 0.0403$	$1.0460\pm 0.0392$	$1.4223\pm 0.0347$	$<\!10^{-3}$	family
spectral_bic_lgamma_block	$1.4779\pm 0.0528$	$1.0259\pm 0.0396$	$1.4020\pm 0.0372$	$0.142$	family
het	$1.5234\pm 0.0419$	$1.0717\pm 0.0356$	$1.4676\pm 0.0343$	$<\!10^{-3}$	family

Appendix

Table 7: LAQN: full

20

-seed leaderboard with mean

\pm

SE;

p

-values from a paired one-sided

t

-test asking whether the dataset leader has significantly lower NLL/pt than the row. The

p

column is the test against the dataset leader; the last column flags whether the row is a family member or an outer baseline.

Figure 5: LAQN spectral partition. Sensors are coloured by the two-cluster spectral partition of

\mathbf{L}

chosen by BIC; the per-cluster

\hat{\gamma}_{k}

values are shown. The partition is geographic rather than by site type: the two clusters carry almost the same mix of site classes (mean site-type ordinal

1.13

against

1.06

) and separate east from west.

Method	NLL/pt	RMSE	recNLL/pt	$p$	cat
lgamma_block_aug ( $K{=}20,w{=}1$ )	$0.9756\pm 0.0106$	$0.6541\pm 0.0055$	$0.9703\pm 0.0095$	—	family (ldr)
lgamma_block_aug ( $K{=}10,w{=}1$ )	$1.0078\pm 0.0105$	$0.6822\pm 0.0075$	$1.0032\pm 0.0099$	$<\!10^{-3}$	family
lgamma_block_aug ( $K{=}10,w{=}0.5$ )	$1.0303\pm 0.0097$	$0.7011\pm 0.0069$	$1.0269\pm 0.0090$	$<\!10^{-3}$	family
lgamma_block_aug ( $K{=}20,w{=}0.5$ )	$1.0410\pm 0.0121$	$0.6975\pm 0.0072$	$1.0338\pm 0.0102$	$<\!10^{-3}$	family
lgamma_block_aug ( $K{=}5,w{=}1$ )	$1.0597\pm 0.0099$	$0.7171\pm 0.0069$	$1.0553\pm 0.0095$	$<\!10^{-3}$	family
lgamma_block_aug ( $K{=}5,w{=}0.5$ )	$1.0698\pm 0.0082$	$0.7323\pm 0.0062$	$1.0672\pm 0.0080$	$<\!10^{-3}$	family

Appendix

Table 8: PM2.5: full

20

-seed leaderboard with mean

\pm

SE;

p

-values from a paired one-sided

t

-test asking whether the dataset leader has significantly lower NLL/pt than the row. The

p

column is the test against the dataset leader; the last column flags whether the row is a family member or an outer baseline.

Figure 6: PM2.5 spatial leader. Stations are coloured by the augmented

K

-means partition (

K=20

w=1

); the per-cluster

\hat{\gamma}_{k}

values are shown. The fitted exponents span both regimes—propagation on some clusters, isolation on others—which is what a single global

\gamma

cannot represent.

Method	NLL/pt	RMSE	recNLL/pt	$p$	cat
lgamma_block_aug $K{=}10,w{=}1$	$0.8685\pm 0.0079$	$0.5738\pm 0.0049$	$0.8572\pm 0.0076$	—	family (ldr)
ps	$0.9360\pm 0.0088$	$0.6261\pm 0.0053$	$0.9247\pm 0.0078$	$<\!10^{-3}$	base
se_kriging	$0.9511\pm 0.0061$	$0.6245\pm 0.0043$	$0.9376\pm 0.0065$	$<\!10^{-3}$	base
spectral_bic_lgamma_block	$0.9562\pm 0.0229$	$0.6151\pm 0.0094$	$0.9364\pm 0.0163$	$<\!10^{-3}$	family
fuglreg_lam1.0	$1.0004\pm 0.0087$	$0.6451\pm 0.0055$	$0.9900\pm 0.0076$	$<\!10^{-3}$	base
lgamma	$1.0031\pm 0.0086$	$0.6562\pm 0.0069$	$0.9910\pm 0.0089$	$<\!10^{-3}$	family

Appendix

Table 9: CA Weather:

20

-seed leaderboard including the Fisher, Matérn and spectral-BIC members; only the aggregate-best augmented configuration is shown (its

K

-grid is in Table 13 ). Each entry is the mean over the

12

monthly snapshots, with mean

\pm

SE across seeds;

p

-values from a paired one-sided

t

-test asking whether the dataset leader has significantly lower NLL/pt than the row. The

p

column is the test against the dataset leader; the last column flags whether the row is a family member or an outer baseline.

Setting	$\mathrm{NLL}_{\mathrm{DE}}$	$c^{\star}_{\mathrm{DE}}$	$\mathrm{NLL}_{\mathrm{lg}}$	$c^{\star}_{\mathrm{lg}}$	$\mathrm{rec}_{\mathrm{DE}}$	$\mathrm{rec}_{\mathrm{lg}}$	$\Delta$	$p$ (DE $>$ lg)
METR-LA	$1.5820\pm 0.1465$	$2.73$	$1.2025\pm 0.0294$	$1.14$	$1.1417$	$1.1759$	$+0.3796$	$\phantom{<}0.009^{*}$
PEMS-BAY	$1.6402\pm 0.0914$	$2.49$	$1.2835\pm 0.0370$	$1.09$	$1.3196$	$1.2374$	$+0.3567$	$<0.001^{*}$
LAQN	$5.4225\pm 0.9784$	$9.97$	$1.4688\pm 0.0400$	$1.20$	$1.9130$	$1.4171$	$+3.9537$	$<0.001^{*}$
PM2.5	$1.1474\pm 0.0129$	$1.27$	$1.1261\pm 0.0086$	$1.00$	$1.1276$	$1.1230$	$+0.0213$	$\phantom{<}0.020^{*}$
CAW Jan	$0.9832\pm 0.0241$	$1.29$	$0.9718\pm 0.0169$	$0.92$	$0.9574$	$0.9588$	$+0.0114$	$\phantom{<}0.270$
CAW Feb	$0.8080\pm 0.0164$	$1.25$	$0.7692\pm 0.0129$	$0.91$	$0.7894$	$0.7620$	$+0.0388$	$\phantom{<}0.011^{*}$

Appendix

Table 10: Deep Ensemble GNN (

M=5

) versus lgamma across all

16

settings,

20

seeds each. NLL is per-test-point.

c^{\star}>1

means the predictive variance is too small (overconfident).

Month	Comparator	NLL/pt aug.	NLL/pt comp.	$\Delta$ NLL/pt	RMSE aug.	RMSE comp.
Jan	ps	$0.8561$	$0.8764$	$-0.020$	$0.5731$	$0.5800$
Feb	ps	$0.6390$	$0.6682$	$-0.029$	$0.4657$	$0.4732$
Mar	ps	$0.8470$	$0.8683$	$-0.021$	$0.5661$	$0.5867$
Apr	se_kriging	$0.9889$	$1.0292$	$-0.040$	$0.6353$	$0.6773$
May	se_kriging	$0.8846$	$0.9726$	$-0.088$	$0.5802$	$0.6368$
Jun	spectral_bic_lgamma_block	$0.8656$	$0.9308$	$-0.065$	$0.5726$	$0.5958$

Appendix

Table 11: CA Weather: the aggregate-selected augmented member

(K,w)=(10,1)

versus the lowest-NLL comparator each month (

20

-seed means). Comparators are the non-augmented members in Table 9 , including spectral-BIC and Fisher members. Other augmented configurations are excluded; the fixed member need not be the monthly optimum over that grid. RMSE uses the same comparator selected by NLL. Negative gap favours the fixed augmented member.

Dataset	$n$	mean deg.	dense (ms)	sparse (ms)	speed-up
LAQN	52	4.9	0.1	0.1	1.2 $\times$
METR-LA	207	12.7	71.2	1.0	72.9 $\times$
PEMS-BAY	325	12.8	7.0	1.6	4.5 $\times$
CA Weather	649	12.2	26.5	2.7	9.7 $\times$
PM2.5	988	10.1	47.2	4.5	10.4 $\times$

Appendix

Table 12: Factorisation of

Q_{\gamma}

for the five real graphs at

\gamma=1

\tau=1

. Minimum of five runs on a single CPU; “dense” is dense inverse, “sparse” is sparse LU.

Dataset	Partition	$K{=}2$	$K{=}3$	$K{=}4$	$K{=}5$	$K{=}7$	$K{=}10$	$K{=}20$
LAQN	spectral	1.4855	1.5011	1.5559	2.1434	1.5386	—	—
PM2.5 ( $d{=}$ std)	augmented ( $w{=}1$ )	1.0803	—	—	1.0597	—	1.0078	0.9756
CA Weather	augmented ( $w{=}1$ )	0.9040	—	—	0.8866	—	0.8685	0.9081

Appendix

Table 13: Fixed-

K

NLL/pt for piecewise members. Bold entries are the aggregate NLL-selected configurations for PM2.5 and CA Weather. LAQN reports every fitted fixed

K

; its BIC-selected aggregate is stated below and is not the NLL leader. Dashes denote unrun settings. SEs and tests for the selected members are in the per-dataset leaderboards.

Dataset	honest NLL/pt	RMSE	oracle	gap	versus baselines
PM2.5 (20 seeds)	$\mathbf{0.9813\pm 0.014}$	$0.6690$	$0.9446$	$+0.037$	beats all ( fuglreg $1.0748$ , m15 $1.1194$ )
CA Weather (12 months)	$\mathbf{0.9300\pm 0.048}$	$0.6069$	$0.8637$	$+0.066$	ties ps $0.9360$ ; beats the rest
LAQN (20 seeds)	$1.6984\pm 0.150$	$1.0047$	$1.4666$	$+0.232$	below the fixed member lgamma
PEMS-BAY (20 seeds)	$1.4865\pm 0.067$	$0.9763$	$1.2445$	$+0.242$	below the fixed member het
METR-LA (20 seeds)	$1.3249\pm 0.112$	$0.8628$	$1.0463$	$+0.279$	below the fixed member het

Appendix

Table 14: Validation-fold selection. The price of honest selection is small exactly where the method works and large exactly where it does not.

Auxiliary $d_{i}$	best validation NLL/pt	selected
std (external)	$\mathbf{1.0026\pm 0.0140}$	$19/20$
spatial_k8 (signal-derived)	$1.0872\pm 0.0178$	$1/20$
spatial_k12 (signal-derived)	$1.0631\pm 0.0156$	$0/20$

Appendix

Table 15: PM2.5 auxiliary ablation. Best validation NLL/pt attained by any family member under each auxiliary field, mean

\pm

SE over

20

seeds, and how often validation selects that auxiliary. The external field wins on all but one seed.

Proxy	NLL/pt	RMSE
Oracle proxy	$0.6835\pm 0.0095$	$0.4809\pm 0.0048$
Noisy proxy ( $\sigma=0.1$ )	$0.6840\pm 0.0096$	$0.4809\pm 0.0048$
Noisy proxy ( $\sigma=0.3$ )	$0.6858\pm 0.0098$	$0.4811\pm 0.0048$
Noisy proxy ( $\sigma=0.5$ )	$0.6878\pm 0.0100$	$0.4812\pm 0.0048$
Noisy proxy ( $\sigma=1.0$ )	$0.6916\pm 0.0102$	$0.4817\pm 0.0048$
Rank-only proxy	$0.6838\pm 0.0096$	$0.4810\pm 0.0048$

Appendix

Table 16: Auxiliary-proxy control on a grid with known hyperparameters. Every non-stationary row uses the same fixed exponent; no

\gamma

is fitted. Entries are mean

\pm

SE across simulated samples. Here

\sigma

is the proxy noise standard deviation before clipping.

Dataset	NLL/pt	RMSE	recNLL/pt
METR-LA	$2.7214\pm 0.2608$	$1.1237\pm 0.0296$	$1.7718\pm 0.0441$
PEMS-BAY	$3.1423\pm 0.3538$	$1.2189\pm 0.0431$	$1.9880\pm 0.0520$
LAQN	$4.3435\pm 0.2683$	$1.2220\pm 0.0448$	$1.6085\pm 0.0419$
PM2.5	$2.1634\pm 0.0805$	$0.9692\pm 0.0131$	$1.6526\pm 0.0211$
CA Weather	$1.9763\pm 0.0689$	$0.9714\pm 0.0089$	$1.6279\pm 0.0281$

Appendix

Table 17: MLP deep-kernel GP results ( dk ), mean

\pm

SE across the locked seeds. CA Weather averages the monthly snapshots within each seed. The feature map does not propagate through graph edges.

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