SAIL: A Learnable Generalisation of Spatial Autocorrelation over Graph-Structured Data

Piotr Keller; Fayyaz Minhas

cs.CVOct 20, 2026

SAIL: A Learnable Generalisation of Spatial Autocorrelation over Graph-Structured Data

Authors: Piotr Keller, Fayyaz Minhas

Organizations: Predictive Systems in Biomedicine (PRISM) Lab, Department of Computer Science, University of Warwick, Coventry CV4 7AL, United Kingdom

Abstract

In vision and scientific domains involving spatial data, a key question is whether node feature values vary independently across space or form coherent spatial structures. Classical spatial autocorrelation statistics such as Moran's $I$ and Geary's $C$ quantify this phenomenon but operate on a single scalar variable with fixed neighbourhood weights, limiting their use for high-dimensional graph-like data. We introduce SAIL (Spatial Autocorrelation Index Learner), a differentiable framework that generalises spatial autocorrelation into a learnable operator by jointly learning feature projections defining candidate scalar fields and inter-node neighbourhood affinities. SAIL produces interpretable local and global spatial coherence descriptors, enabling supervised prediction and unsupervised discovery of spatial autocorrelation modes and hotspots. We show that SAIL exactly recovers classical spatial statistics under appropriate parametrisations and, for tasks driven by latent spatial autocorrelation, can approximate the Bayes-optimal predictor with an inductive bias matched to such tasks. Experiments on synthetic data and large-scale biomedical graphs demonstrate statistical correctness and discovery of spatial coherence with applications across vision, geospatial analysis, and biomedicine. The code repository is available at: https://github.com/pkeller00/SAIL.

Figures & tables

Fig. 1: Overview of the SAIL framework. (i) SAIL takes graph-structured data whose nodes carry $d$ -dimensional features. Inside the SAIL module, a learnable feature projection maps node features to $h$ scalar fields (modes), and a learnable neighbourhood affinity assigns strong or weak weights to node pairs. Each projected field is combined with these affinities to give node-wise local spatial autocorrelation (SAC) scores for every mode, which can also be aggregated into an $h$ -dimensional global SAC descriptor per graph. (ii) SAIL modes can be learned through supervised or unsupervised optimisation. In the unsupervised setting, SAIL identifies feature projections that maximise spatial autocorrelation (SAC), revealing which combinations of features exhibit the strongest spatial coherence across the dataset. Conversely, in supervised learning, global SAC descriptors are used to train a downstream predictor, enabling SAIL to identify spatial autocorrelation patterns associated with the target. In both settings, the learned feature projections provide interpretable combinations of features that define each mode. Finally, a formal statistical framework enables the detection of significant hot and cold spots on the learned modes.

Fig. 2: SAIL recovers classical SAC on synthetic graphs with known structure. Left: a graph with unstructured, independently sampled node values (colour bar shows feature value), for which SAIL correctly reports near-zero global SAC (Moran’s $I=-0.01$ , Geary’s $C=1.12$ ) and classifies most nodes as non-significant under hotspot analysis. Right: a spatially homogeneous graph with a smooth imposed gradient, for which SAIL correctly detects strong positive autocorrelation (Moran’s $I=0.99$ , Geary’s $C$ $=0.13$ ) and hotspot analysis correctly localises High–High and Low–Low clusters matching the imposed gradient.

Fig. 3: Results demonstrating distinction of SAIL from traditional message-passing architectures. In a regression benchmark where models predict Moran’s $I$ of heterogeneous test graphs, SAIL achieves significantly higher Pearson correlation between predicted and true values than classical message-passing networks with comparable or larger parameter counts ( $p<0.0001$ , box plots over 1000 train–validation–test runs).

Region/DICE Score	SAIL	Univariate	Mean	MULTISPATI	Hotspot
Nucleolus	$\mathbf{0.84}$	$0.75$	$0.50$	$0.79$	$0.71$
Endosomes/Golgi 1	$\mathbf{0.58}$	$0.38$	$0.34$	$0.38$	$0.56$
Cell periphery 1	$\mathbf{0.53}$	$0.51$	$0.51$	$0.50$	$0.50$
Cell periphery 2	$\mathbf{0.59}$	$0.49$	$0.50$	$0.49$	$0.50$

TABLE I: SAIL segmentation performance on the 4i dataset. We compare SAIL against classical Moran’s

I

computed on the best-performing individual protein among the

20

proteins with the highest global Moran’s

I

(Univariate), the multivariate extension of Moran’s

I

obtained by averaging the univariate Moran’s

I

statistic across all proteins (Mean), MULTISPATI-PCA [ 15 ] and Hotspot [ 21 ] . For each compartment, we report the Dice score between LISA-style hotspot mask and region annotation.

Fig. 4: Unsupervised SAIL Modes recover known subcellular compartments in the 4i spatial proteomics dataset. (a) Two data-driven SAIL Modes are visualised over the field of view. SAIL Mode 1 (showing the highest Dice score with Cell periphery 2) is defined by positive weights on p-MEK, TSG101 and Climp63 and a negative weight on ISG15. SAIL Mode 8 (showing the highest Dice score with the Nucleolus) is defined by positive weights on GSK3A/B, CAV1 and SARA and negative weights on PCNA, FBL and LC3B. (b) Visualisations of LISA High–High and Low–Low scoring locations for both SAIL Modes, which were used for defining spatial regions. We also show the expression of the most important proteins for each SAIL Mode and the reference annotation mask.

Model	AUC (95% CI)	Mann–Whitney $U$ ( $p$ -value)
SAIL (Mode 18)	0.86 (0.64–1.00)	0.005
Favourable Niches	0.78 (0.55–0.95)	0.021
PD-L1+ (untreated slide)	0.78 (0.49–1.00)	0.072
PD-L1+ (all slides)	0.69 (0.42–0.92)	0.19

TABLE II: Prediction of response to PD-1/PD-L1 blockade in the cSCC cohort. We report discriminative performance as the area under the ROC curve (AUC) with 95% confidence interval, together with the two-sided Mann–Whitney U p-value for separating responders from non-responders. SAIL is compared against the Favourable Niches classifier and pre-treatment PD-L1 status [ 22 ] .

Fig. 5: Association of an unsupervised SAIL Mode with response to PD-1/PD-L1 blockade. (a) Per-patient global Geary-style dissimilarity SAIL score for SAIL Mode 18, compared between responders and non-responders. Each dot represents a patient; responders show a higher score ( $p=0.005$ ), suggesting greater local cell heterogeneity. (b) Gene weights defining SAIL Mode 18. (c) Visualisation of responder and non-responder patients. We show the local $f_{\mathrm{map}}$ and Geary-style dissimilarity score per cell.

Metric	SAIL	Univariate	Mean	MULTISPATI	Hotspot
C-Index	$\mathbf{0.74}$	$0.69$	$0.68$	$0.70$	$0.60$
$\Delta$ RMST (days)	$\mathbf{782}$	$385$	$362$	$617$	$777$
HR (95% CI)	$\mathbf{5.23}$ (1.45–18.89)	$0.44$ (0.15–1.31)	$3.15$ (0.98–10.08)	$3.30$ (1.03–10.56)	$4.89$ (1.36–17.59)
Log-rank $p$ -value	$\mathbf{0.005}$	$0.13$	$0.04$	$0.03$	$0.007$

TABLE III: Prognostic performance of SAIL on the ORION cohort. We compare SAIL against classical Geary’s

C

computed on individual protein markers (Univariate) and the multivariate extension of Geary’s

C

obtained by averaging the univariate Geary’s

C

(Mean), MULTISPATI-PCA [ 15 ] , and Hotspot [ 21 ] . Patients are stratified into groups via a median split on each score; C-Index, hazard ratio (HR) with 95% CI, restricted mean survival time difference (

\Delta

RMST), and log-rank

p

-value are reported.

Fig. 6: Unsupervised SAIL Modes link spatial coherence to outcome. (a) A SAIL Mode discovered in ORION-CRC (mIF). Bar chart shows $f_{\mathrm{map}}$ weights. The corresponding projected scalar field $f_{\mathrm{map}}(x)$ is overlaid on a tissue section to visualise its spatial pattern, and the associated global SAIL scores stratify patients into distinct survival groups. (b) SAIL learns a scalar field $f_{\mathrm{map}}(x)$ on TCGA-BRCA WSI deep-feature patch graphs (Mode 1); slides are visualised by $f_{\mathrm{map}}(x)$ alongside hotspot clusters. The corresponding global SAIL score (median split) defines risk groups with significant Kaplan–Meier separation in disease-specific survival. (c) Similarly to (b), we analyse the TCGA-CRC cohort.

Model	AUC $\pm$ $\sigma$	AP $\pm$ $\sigma$
All features ( $d=617$ )
SAIL	$\mathbf{0.902\pm 0.035}$	$\mathbf{0.849\pm 0.055}$
MesoGraph	$0.863\pm 0.065$	$0.832\pm 0.051$
PINS	$0.844\pm 0.017$	$0.821\pm 0.036$
Max-MIL	$0.782\pm 0.050$	$0.698\pm 0.055$
Interpretable features only ( $d=105$ )

TABLE IV: Performance of supervised SAIL for mesothelioma subtype prediction [ 37 ] , using all

617

cell features (top) or only the

105

interpretable features (bottom). We report AUC and average precision (AP) with their standard deviation (

\sigma

) across cross-validation folds. Comparisons include graph-based state of the art (MesoGraph) and standard multiple-instance learning pipelines (PINS, Max-MIL).

Fig. 7: Supervised SAIL discriminates mesothelioma subtype from the spatial coherence of interpretable cell features (model trained on the $105$ interpretable features only). (a) $f_{\mathrm{map}}$ loadings of the two SAIL Modes with the largest weights in the prediction head (Mode 11, weight $+0.57$ ; Mode 12, weight $+0.46$ ; both towards non-epithelioid), shown for the $12$ features with the largest absolute loading. (b) Local predicted value $\hat{y}$ and local Geary-style SAIL scores for these modes, shown for a representative epithelioid and non-epithelioid sample.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

Compartment	Mode	LISA cluster	Dice
Nucleolus	$8$	Low–Low	$0.839$
Cell periphery 2	$1$	Low–Low	$0.587$
Endosomes/Golgi 1	$3$	High–High	$0.577$
Cell periphery 1	$4$	Low–Low	$0.533$
Nucleus	$5$	High–High	$0.529$
ER/mitochondria 1	$14$	High–High	$0.488$

Appendix

TABLE V: Best-matching unsupervised SAIL Mode per annotated subcellular compartment on the 4i dataset. For each compartment, we report the mode whose LISA hotspot mask (High–High or Low–Low) attains the highest Dice overlap with the expert annotation.

Compartment	SAIL	Univ.	Mean	MULTISPATI	Hotspot
Nucleolus	$\mathbf{0.84}$	$0.75$ (DAPI)	$0.50$	$0.79$	$0.71$
Cell periphery 2	$\mathbf{0.59}$	$0.49$ (CRT)	$0.50$	$0.49$	$0.50$
Endosomes/Golgi 1	$\mathbf{0.58}$	$0.38$ (SARA)	$0.34$	$0.38$	$0.56$
Cell periphery 1	$\mathbf{0.53}$	$0.51$ (CRT)	$0.51$	$0.50$	$0.50$
Nucleus	$\mathbf{0.53}$	$0.47$ (PCNA)	$0.41$	$0.40$	$0.46$
ER/mitochondria 1	$0.49$	$0.45$ (ABCD3)	$0.24$	$\mathbf{0.50}$	$0.47$

Appendix

TABLE VI: Full per-compartment Dice comparison on the 4i dataset, extending the four-compartment summary of the main text to all

10

annotated compartments. SAIL is the best-matching unsupervised mode (Supplementary Table V ); Univariate (Univ.) is classical Moran’s

I

with LISA computed on the single best-performing individual protein (named in parentheses) among the

20

proteins with the highest global Moran’s

I

; Mean is the traditional multivariate extension of Moran’s

I

, i.e. the mean of the univariate Moran’s

I

statistics across all

43

proteins; MULTISPATI is MULTISPATI-PCA [ 1 ] with local Moran’s

I

LISA computed on the axes and Hotspot is Hotspot [ 2 ] with local Moran’s

I

LISA computed on the scores of its protein modules, all using the same

k_{\mathrm{NN}}=32

log-distance kNN graph. For each compartment we report the Dice overlap between the derived High–High/Low–Low hotspot mask and the expert annotation.

Fig. 8: All $20$ unsupervised SAIL Modes learned on the 4i dataset. For each mode we show the learned $f_{\mathrm{map}}$ projection weights over the $43$ protein markers (bar chart, as in Figure 4 ), alongside the corresponding High–High and Low–Low LISA cluster maps overlaid on the tissue. Together these define, for every mode, both the molecular program it emphasises and the spatial regions where that program is coherently high (High–High) or low (Low–Low). The best-matching mode per annotated compartment is quantified in Supplementary Table V .

Fig. 9: All $43$ proteins in the 4i dataset. For each protein we show its localised expression, to help understand its expected spatial distribution and how it contributes to SAIL Modes that overlap with known subcellular compartments. We also show the ground-truth annotation masks for all compartments.

Mode	Med. $R$	Med. $NR$	AUC	$U$	$p$
18	0.692	0.556	0.864	95	0.005
11	0.589	0.554	0.809	89	0.018
4	0.588	0.513	0.791	87	0.027
8	0.634	0.541	0.782	86	0.032
14	0.638	0.549	0.773	85	0.038
7	0.611	0.519	0.764	84	0.045

Appendix

TABLE VII: SAIL niche discrimination of PD-1/PD-L1 blockade response (see Supplementary Section H ). Each row is one SAIL Mode, scored at the subject level across 11 responder and 10 non-responder patients and ranked by discriminative AUC. Med.

R

and Med.

NR

are the median per-subject global Geary-style SAIL scores among responder (

R

) and non-responder (

NR

) patients, respectively. AUC is the area under the ROC curve for separating the two groups by SAIL score;

U

and

p

are the Mann–Whitney–Wilcoxon rank-sum statistic and its two-sided

p

-value.

Fig. 10: Gene weights defining all $20$ unsupervised SAIL Modes learned on the cSCC PD-1/PD-L1 CosMx cohort. For each mode, bars show the top genes by absolute loading in the learned $f_{\mathrm{map}}$ projection, with positively and negatively weighted genes coloured separately; these weights specify the gene combination whose spatial Geary-style dissimilarity each mode minimises.

Mode	C-Index	$p$	$\Delta$ RMST
1	0.68	$0.403$	213
2	0.66	$0.403$	213
3	0.52	$0.300$	288
4	0.72	$0.040$	502
5	0.68	$0.048$	482
6	0.51	$0.502$	-153

Appendix

TABLE VIII: Survival association (overall survival) of all

20

unsupervised SAIL Modes in ORION-CRC. Patients are split at the median of each mode’s global SAIL score. C-Index, log-rank

p

-value and

\Delta

RMST (restricted mean survival time of the low- minus the high-score group, in days) is reported.

	TCGA-BRCA				TCGA-CRC
Mode	C-Index	$p$	HR (95% CI)	$\Delta$ RMST	C-Index	$p$	HR (95% CI)	$\Delta$ RMST
1	0.64	$0.005$	2.39 (1.28–4.46)	386	0.59	$0.117$	1.45 (0.91–2.30)	102
2	0.55	$0.450$	1.26 (0.69–2.28)	195	0.60	$0.057$	1.56 (0.98–2.48)	653
3	0.58	$0.288$	1.37 (0.76–2.47)	159	0.54	$0.576$	1.14 (0.72–1.80)	107
4	0.57	$0.053$	1.80 (0.98–3.28)	302	0.56	$0.059$	1.55 (0.98–2.47)	365
5	0.54	$0.690$	1.13 (0.62–2.07)	104	0.55	$0.020$	1.74 (1.09–2.79)	689

Appendix

TABLE IX: Survival association of all

20

unsupervised SAIL Modes in TCGA-BRCA (disease-specific survival) and TCGA-CRC (overall survival). Patients are split at the median of each mode’s global SAIL score. C-Index is reported as

\max(c,1-c)

;

p

is the log-rank

p

-value; HR (95% CI) compares the high- with the low-score group;

\Delta

RMST is the restricted mean survival time of the low- minus the high-score group, in days.

Covariate	C-Index	Log-rank $p$ -value	HR (high vs low)	$\Delta$ RMST (low–high, days)
Mean aggregate	0.68	0.04	3.15	362
CD68	0.61	0.05	3.00	643
CD4	0.69	0.13	2.28	385
Grade	0.61	0.13	2.20	393
E-cadherin	0.59	0.14	2.22	440
CD3e	0.63	0.15	2.18	460

Appendix

TABLE X: Baseline survival associations in the ORION cohort using inverse-distance spatial weighting to compute univariate global spatial autocorrelation using Geary’s

C

. For the multivariate case, we report the mean of the univariate spatial autocorrelations across proteins (Mean aggregate). Results are also compared with immune groups (IFM1–4) defined in the original ORION study.

Fig. 11: Unsupervised SAIL Modes link spatial coherence to clinical outcome. A SAIL Mode was identified in the BRAF-CRC cohort by minimising Geary’s $C$ over one-hot encoded cell types (cancer epithelial cells, normal epithelial cells, fibroblasts, lymphocytes, neutrophils, macrophages, endothelial cells, and myocytes). The bar chart shows the learned $f_{\mathrm{map}}$ weights assigned to each cell type. Global SAIL scores derived from this mode stratify patients into distinct survival groups.

Covariate	C-Index	Log-rank $p$ -value	HR (high vs low)	$\Delta$ RMST (low–high, months)
Mean aggregate	0.59	0.01	1.58	10.4
Endothelial cell	0.58	0.02	1.57	11.8
Lymphocyte	0.59	0.06	1.42	7.2
Cancer epithelial cell	0.57	0.06	1.42	9.2
Fibroblast	0.57	0.07	1.41	7.5
Normal epithelial cell	0.55	0.26	1.23	0.9

Appendix

TABLE XI: Baseline survival associations in the BRAF-CRC cohort using inverse-distance spatial weighting to compute univariate global spatial autocorrelation using Geary’s

C

on each one-hot cell-type channel. The multivariate covariate (Mean aggregate) corresponds to the mean of the univariate cell-type spatial autocorrelations.

Panel of Figure 23 — Fig. 12: Zoomed-in regions of SAIL outputs. (a) Patch graphs for TCGA-BRCA and TCGA-CRC. We show two zoomed-in regions per patient, including the $I^{\mathrm{local}}$ score and LISA clusters for each graph node. (b) Cell graphs for BRAF-CRC and ORION-CRC. We show two zoomed-in regions per patient, including the $I^{\mathrm{local}}$ score and LISA clusters for each graph node.

Fig. 13: 16 learned SAC descriptors (trained using interpretable features) on the mesothelioma dataset used in Section J . For each descriptor we report the three features with the most positive and negative loadings. Here “nbhd” denotes a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ neighbourhood average.

$f_{\mathrm{map}}$	$f_{\mathrm{affinity}}$	Train step (ms)	Inference (ms)	Peak mem. (GB)	Max. $n$
Linear	none	$6.1\pm 0.0$	$4.0\pm 0.1$	$0.91$	$3\times 10^{6}$
Orthogonal	none	$37.7\pm 0.6$	$3.9\pm 0.0$	$0.92$	$3\times 10^{6}$
Linear	shared-map	$63.2\pm 0.5$	$59.2\pm 0.8$	$12.98$	$3\times 10^{5}$
Orthogonal	shared-map	$129.6\pm 3.8$	$59.2\pm 0.3$	$13.00$	$3\times 10^{5}$
Linear	query–key	$62.7\pm 0.3$	$51.0\pm 1.6$	$13.02$	$3\times 10^{5}$
Orthogonal	query–key	$94.8\pm 1.0$	$50.6\pm 1.5$	$13.03$	$3\times 10^{5}$

Appendix

TABLE XII: Computational cost of each SAIL variant at the reference configuration (

n=10^{5}

d=1{,}000

h=25

k_{\mathrm{NN}}=16

) on one NVIDIA A100-80GB GPU; mean

\pm

standard deviation over

5

repeats. Max.

n

is the largest graph size in the sweep that fitted in GPU memory.

Fig. 14: Computational scaling of SAIL in its default configuration (orthogonal $f_{\mathrm{map}}$ , no learned feature affinity, Moran-style objective with decorrelation and sparsity regularisation) on synthetic spatial data (Supplementary Section K-B ). Each panel varies one factor with the others held at $n=10^{5}$ nodes, $d=1{,}000$ features, $h=25$ modes and $k_{\mathrm{NN}}=16$ neighbours. Train step : one full-graph forward pass, backward pass and Adam update (one epoch); Inference : one forward pass of the trained model. (a) Runtime versus number of nodes $n$ : a fixed, $n$ -independent cost dominates up to $n\approx 10^{5}$ . (b) Runtime versus feature dimension $d$ : training time grows approximately as $d^{3}$ , owing to the $d\times d$ matrix exponential of the Stiefel parametrisation, while inference is essentially independent of $d$ . (c) Runtime versus number of modes $h$ and (d) versus neighbourhood size $k_{\mathrm{NN}}$ : both have only a minor effect. (e) Peak GPU memory versus $n$ grows linearly. The dashed line marks GPU capacity (NVIDIA A100, $80$ GB); $\times$ marks the first size that did not fit, joined by a dotted line to the largest size that did. Points are means over $5$ repeats with $\pm 1$ standard deviation bands; legend exponents are log–log least-squares slopes. All axes are logarithmic.

Fig. 15: Computational cost of SAIL’s optional components. Top row (a–c): linear $f_{\mathrm{map}}$ ; bottom row (d–f): orthogonal (Stiefel) $f_{\mathrm{map}}$ , the default. Colours denote the feature affinity $f_{\mathrm{affinity}}$ : none (blue; attention from spatial distance only), shared-map (orange) and query–key (green) (Supplementary Section K-B ). Other factors are held at $n=10^{5}$ , $d=1{,}000$ , $h=25$ , $k_{\mathrm{NN}}=16$ . (a, d) Time per training step versus feature dimension $d$ : nearly flat with the linear map, approximately cubic with the orthogonal map. (b, e) Peak GPU memory versus number of nodes $n$ : linear for all variants, but both learned affinities require approximately $14\times$ more memory, because they gather the $d$ -dimensional features of every neighbour ( $\mathcal{O}(nk_{\mathrm{NN}}d)$ ), and therefore exceed GPU memory at $n=10^{6}$ . (c, f) Peak GPU memory versus neighbourhood size $k_{\mathrm{NN}}$ : linear in $k_{\mathrm{NN}}$ for the learned affinities. The shared-map and query–key affinities have near-identical memory (and near-identical runtime with the linear map), since both are dominated by the same neighbour-feature gather; the shared-map curve is drawn wider underneath so that both remain visible. $\times$ marks the first configuration that did not fit in GPU memory. The dashed line marks GPU capacity. Means over $5$ repeats with $\pm 1$ standard deviation bands; all axes are logarithmic.

Panel of Figure 28 — Fig. 16: Additional synthetic tests of SAIL. (a) Toy graph examples, coloured by both LISA clusters and the projected scalar field $f_{\mathrm{map}}(x)$ . Top-left: a smooth, organically varying spatial field generated with Perlin noise, designed to produce complex but strongly positive local spatial autocorrelation. Top-right: a checkerboard graph with a sharply alternating high-contrast pattern that induces strong negative spatial autocorrelation through repeated adjacent sign flipping. Bottom-left: a plateau graph with a two-level step surface with an abrupt spatial boundary, producing clustered positive autocorrelation within each region and a sharp discontinuity at the divide. Bottom-right: concentric rings with an oscillating radial wave pattern that alternates between positive and negative spatial dependence as distance increases from the centre. (b) A synthetic 3D point set, coloured by both LISA clusters and the projected scalar field $f_{\mathrm{map}}(x)$ . It is generated by sampling a grid and assigning heights using 2D Perlin noise, giving a smooth synthetic terrain; the resulting $40{,}000$ points are used to build a $k$ -nearest-neighbour graph for SAIL analysis.

Category	Setting	Mean Dice
$\lambda_{\mathrm{corr}}$	0	0.42
	1 †	0.44
	5	0.42
	10	0.36
	100	0.32
$\lambda_{\mathrm{sparse}}$	0	0.44

Appendix

TABLE XIII: Ablation study of SAIL components on the 4i dataset (Supplementary Section M ). For each setting we report the best Dice per subcellular region (maximum over the 20 modes and High–High/Low–Low LISA clusters), averaged over the 10 regions. Each block varies one component while holding all others at their default setting (indicated via † ).

Category	Setting	AUC (95% CI)	$p$ -value
$\lambda_{\mathrm{corr}}$	0	0.84 (0.62–1.00)	0.010
	1 †	0.86 (0.64–1.00)	0.005
	5	0.82 (0.60–0.98)	0.015
	10	0.75 (0.49–0.95)	0.062
	100	0.78 (0.55–0.96)	0.032
$\lambda_{\mathrm{sparse}}$	0	0.84 (0.63–0.99)	0.010

Appendix

TABLE XIV: Ablation study of SAIL components on the cSCC PD-1/PD-L1 blockade cohort. For each setting, we report the mode that best separates responders from non-responders (highest AUC over the

20

modes): its AUC with a

95\%

bootstrap confidence interval (patients resampled with replacement,

10{,}000

resamples) and the Mann–Whitney

p

-value. Each block varies one component while holding all others at their default setting (indicated via † ).

Category	Setting	C-Index	$p$ -value	$\Delta$ RMST
$\lambda_{\mathrm{corr}}$	0	0.67	0.001	365
	1 †	0.64	0.005	386
	5	0.60	0.046	243
	10	0.60	0.464	96
	100	0.60	0.123	188
$\lambda_{\mathrm{sparse}}$	0	0.62	0.020	347

Appendix

TABLE XV: Ablation study of SAIL components on the TCGA-BRCA cohort (

498

patients; Supplementary Section M ). For each setting we report the SAIL Mode with the highest C-Index; patients are split into high/low groups via a median split on that mode’s score, and we report the log-rank

p

-value and

\Delta

RMST (days, 10-year horizon) between groups. Each block varies one component while holding all others at their default setting (indicated via † ).

Category	Setting	C-Index	$p$ -value	$\Delta$ RMST
$\lambda_{\mathrm{corr}}$	0	0.61	0.015	474
	1 †	0.60	0.057	653
	5	0.60	0.091	178
	10	0.57	0.320	112
	100	0.57	0.021	373
$\lambda_{\mathrm{sparse}}$	0	0.60	0.002	807

Appendix

TABLE XVI: Ablation study of SAIL components on the TCGA-CRC cohort (

558

p

-value and

\Delta

RMST (days, 10-year horizon) between groups. Each block varies one component while holding all others at their default setting (indicated via † ).

Fig. 17: All $8$ SAIL Modes learned on the Global Macro Database. For each mode, bars show the top macroeconomic indicators by absolute loading in the learned $f_{\mathrm{map}}$ projection, with positively and negatively weighted indicators coloured separately; these weights specify the indicator combination whose spatial Geary-style dissimilarity each mode minimises.

Fig. 18: Temporal trends of all $8$ SAIL Modes learned on the Global Macro Database. For each mode, the line shows the global Geary-style SAIL score of the mode’s projected field on each yearly country graph from 2000 to 2023; lower values indicate that neighbouring countries are more similar along that mode.

Fig. 19: Unsupervised SAIL for COVID-19 trend analysis. Here we show a line graph that reports the global Moran’s $I$ of new COVID-19 cases (y-axis) against a given week (x-axis).

Fig. 20: A choropleth map for the world coloured by LISA clustering. Interestingly, we notice that across time, COVID-19 hotspots tend to form in Western and Central European countries, reflecting a consistent pattern of spatially concentrated High–High clusters and regional synchronisation of outbreaks.

Fig. 21: SAIL tracks the mixing of two gases. (i) Loadings of the single SAIL Mode learned on a training run, over species identity, particle speed and four nuisance noise features. (ii) SAIL score over time (seconds) for the three scenarios (partition closed, small hole, partition removed); lines show the mean and shaded bands $\pm 1$ standard deviation over $5$ seeds. (iii) Gas states (top) and the corresponding LISA clusters of the learned mode (bottom) at selected time points.

Model	#Params	Pearson	MSE
SAIL (dense)	$82$	$\mathbf{0.819\pm 0.015}$	$\mathbf{0.0114\pm 0.0016}$
GCN (X, FC)	$81$	$0.730\pm 0.022$	$0.0163\pm 0.0025$
GCN (X+coords, FC)	$193$	$0.724\pm 0.022$	$0.0169\pm 0.0028$
GAT (X, FC)	$353$	$0.627\pm 0.096$	$0.0212\pm 0.0046$
GAT (X+coords, FC)	$785$	$0.459\pm 0.203$	$0.0271\pm 0.0065$
MLP (X)	$321$	$0.016\pm 0.044$	$0.0335\pm 0.0031$

Appendix

TABLE XVII: Per-model results on the synthetic Moran’s

I

regression benchmark. Each model is evaluated at its Stage-1–selected configuration over

1000

seeds; we report the mean

\pm

standard deviation of test Pearson and MSE. Rows are ordered by descending Pearson correlation.

$\gamma$	SAIL	GCN	GAT	MLP
$0.00$	$\mathbf{0.831}_{\pm 0.009}$	$0.744_{\pm 0.017}$	$0.450_{\pm 0.156}$	$0.010_{\pm 0.049}$
$0.15$	$\mathbf{0.814}_{\pm 0.010}$	$0.746_{\pm 0.018}$	$0.367_{\pm 0.160}$	$0.118_{\pm 0.075}$
$0.30$	$\mathbf{0.761}_{\pm 0.018}$	$0.755_{\pm 0.012}$	$0.505_{\pm 0.125}$	$0.356_{\pm 0.068}$
$0.50$	$0.584_{\pm 0.032}$	$\mathbf{0.789}_{\pm 0.015}$	$0.736_{\pm 0.088}$	$0.682_{\pm 0.038}$
$0.70$	$0.338_{\pm 0.045}$	$0.815_{\pm 0.014}$	$\mathbf{0.910}_{\pm 0.018}$	$0.905_{\pm 0.013}$
$0.85$	$0.177_{\pm 0.053}$	$0.822_{\pm 0.016}$	$0.975_{\pm 0.004}$	$\mathbf{0.978}_{\pm 0.003}$

Appendix

TABLE XVIII: Spatial-versus-content blend ablation. Mean test Pearson correlation (subscript:

\pm

standard deviation over

10

seeds) as the target shifts from pure spatial autocorrelation (

\gamma{=}0

) to pure feature content (

\gamma{=}1

). GCN, GAT and MLP use node coordinates (X+coords), and the GNNs use the fully connected graph (FC). SAIL is strongest at

\gamma{=}0

and degrades monotonically as the target loses its spatial component, while the content-recoverable baselines improve. The best model per row is in bold, with a clear crossover near

\gamma\approx 0.3

–

0.5

Fig. 22: Spatial-versus-content blend ablation. Mean Pearson correlation (shaded bands $\pm 1$ standard deviation over $10$ seeds) as a function of the blend weight $\gamma$ , interpolating the regression target between pure spatial autocorrelation ( $\gamma=0$ ) and pure feature content ( $\gamma=1$ ). SAIL is strongest at $\gamma=0$ and degrades monotonically as the spatial component is removed, while GCN, GAT, and MLP improve, crossing SAIL near $\gamma\approx 0.3$ – $0.5$ .

Spatial Context Modeling

See all 24 papers

Oct 25, 2026

Spatially-Aware Self-Organizing Graph Autoencoders via Localized Hebbian Learning

Traditional autoencoders rely on static network architectures to learn latent representations. Recently we suggested a dynamic architecture inspired by Hebbian plasticity, enabling topological self-organization among convolutional feature maps (R. Koç et al., Proceedings of the SIU, 2026). Early implementations of this paradigm infer relationships among feature maps using global average pooling and model their interactions through scalar-valued connections. This approach falls short to fully exploit the underlying spatial information, embedded in input images. Furthermore, abrupt transitions between different loss functions and training dynamics can introduce representation shocks, leading to unstable feature learning. In this paper, we introduce a Spatially-Aware Self-Organizing Graph Autoencoder (S-LooM) to address these structural bottlenecks. By using spatial cross-correlation, and spatial kernels, S-LooM establishes dynamic connections among feature maps. We replace static scalar message passing of autoencoders with learnable spatial kernels and eliminate high-frequency deconvolutional artifacts observed in decoders using nearest-neighbor upsampling. Via a soft-blending training curriculum, the latent topology aligns smoothly during training. Our experiments show that S-LooM yields a more stable topology, compared to the state of the art dynamic architectures. Meaning, the graph structure that emerge has more resistance to feature entanglement and noise. It forms functional communities responsible for visual attributes, such as illumination, shape and texture even under high connection densities.

Robin Koç, Hazal Moğultay Özcan, Fatoş T. Yarman Vural

Computer Engineering Department, Middle East Technical University, Ankara, Türkiye

Visual Representation Learning Convolutional Autoencoder Graph Structure Learning Graph Neural Networks Hebbian Learning

Ask PDF

Jul 16, 2026

Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance

We propose a novel measure of the discrepancy between two probability distributions

f

and

g

on a graph - which we call the diffusion distance - that measures the rate of convergence of

f

g

under a graph-constrained Markov chain with stationary distribution

g

. As a default choice for this Markov chain, we use the Metropolis-Hastings transition matrix targeting

g

with proposals given by a random walk on the graph. Our primary case of interest is when the second distribution

g

is uniform, in which case the diffusion distance becomes a measure of spatial clustering in

f

. Used in this way, (Metropolis-Hastings) diffusion distance to uniformity extends Moran's

I

-type measures of spatial autocorrelation by incorporating global graph geometry rather than just local patterns. Indeed, Moran's

I

, the most well-known measure of spatial autocorrelation, can be viewed as a one-step heuristic for diffusion distance, so long as specific spatial weights are used. We establish theoretical bounds and a stability result for our measure, connecting it to graph spectra and optimal transport. We then turn our attention to outlining a statistical test for spatial clustering using diffusion distance. Under permutation null models, we derive high-probability bounds on diffusion distance underpinned by exact spectral formulas for convergence of distributions, enabling an efficient statistical test for spatial clustering on large datasets. We empirically compare diffusion distance to Moran's

I

both as a numerical measure and as a statistical test. We show that diffusion distance exhibits higher power on synthetic data using a stochastic block model. Empirical analysis of Black population distributions for 100 U.S. cities shows that diffusion distance detects subtle differences in urban segregation patterns that Moran's

I

does not.

Thomas Weighill, Chidinma Williams

Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27402

Clustering Graph Diffusion Statistical Hypothesis Testing

Ask PDF

Oct 6, 2026

How Many Independent Samples Does a Satellite Image Contain? Generalization Bounds for Spatially Dependent Data

Machine learning classifiers for remote sensing imagery are typically evaluated as though every pixel were an independent sample. Spatial autocorrelation violates this assumption, since neighboring pixels carry redundant information which inflates sample sizes. How many independent samples does a satellite image actually contain? For an

n \times n

image whose spatial correlation persists over a range of

r

pixels, the effective sample size is

Θ(n^2/r^2)

, not

n^2

. We prove this as a finite-sample upper bound for classifiers on spatially correlated data, and show via a matching lower bound that the rate is tight, and no algorithm can do better. We extend the results to images with directional correlation and spatially varying correlation structure. Our result justifies spatial cross-validation since block holdout with separation proportional to the correlation range achieves optimal generalization guarantees, while random holdout can underestimate confidence interval widths by a factor proportional to

r

. We validate the theory on synthetic data and satellite image tiles from three sensors (Landsat 8, Sentinel-2, and Sentinel-1).

Robin Young

Department of Computer Science and Technology University of Cambridge, UK

Spatial Generalization Remote Sensing Statistical Learning Theory Generalization Bounds

Ask PDF

Abstract

Spatial Context Modeling

Explore similar work

Spatially-Aware Self-Organizing Graph Autoencoders via Localized Hebbian Learning

Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance

How Many Independent Samples Does a Satellite Image Contain? Generalization Bounds for Spatially Dependent Data