ICE: Task-Aligned Clifford Latent Fields for Multimodal Graph Foundation Models

Xunkai Li; Xu Wang; Yinlin Zhu; Xiong Yongfu; Yi Liu; Rong-Hua Li; Guoren Wang

cs.LGSep 24, 2026

ICE: Task-Aligned Clifford Latent Fields for Multimodal Graph Foundation Models

Authors: Xunkai Li, Xu Wang, Yinlin Zhu, Xiong Yongfu, Yi Liu, Rong-Hua Li, Guoren Wang

Organizations: Department of Computer Science, Beijing Institute of Technology, Beijing, China · School of Airspace Science and Engineering, Shandong University, WeiHai, China · School of Computer Science and Engineering, Sun Yat-sen University, GuangZhou, China

Abstract

Multimodal attributed graphs connect entities, visual content, language, and observed relations. Learning one foundation across such graphs requires more than compressing each node into a fused Euclidean vector. The representation must preserve entity semantics, construct interaction state from graph neighborhoods, and expose that state to prediction units with different geometry. Our empirical study shows why these requirements are inseparable. Higher-grade channels recover pair relations across the foundation graphs, specialized queries reveal information hidden by a generic readout, and rigid blade isolation removes cross-grade capacity. We therefore introduce ICE (Interaction-aware Clifford Encoder), a multimodal graph foundation model built on a node-indexed Clifford latent field. Topology, text, and images enter explicit Cl(3) addresses. Edge-aware geometric products transform these directions into scalar, bivector, and trivector relations over observed neighborhoods. A protected Grade-1 route preserves entity semantics, while the full grade and depth bank remains available to fresh node and link heads. We establish exact cross-grade reachability, node-permutation equivariance, and a bound on the task residual around the semantic score. Experiments span one shared foundation over eleven graphs, six node-classification datasets, three link-prediction datasets, and matched few-shot tasks. ICE ranks first in all 30 reported supervised and few-shot comparisons. Core removals reduce every task summary, and mechanism controls connect the gains to higher-order transport, retained multidepth structure, semantic protection, and direct field access.

Figures & tables

Figure 1: Empirical study. (a) Pair-relation AUC after grade expansion and a second transport layer. (b) Accuracy and Macro-F1 gains (percentage points) from direct grade–depth access over a generic readout. (c) Rigid blade isolation reports the retained active share (“Act.”), deepest

e_{12}

energy (“

e_{12}

”), and matched validation wins (“Win.”).

Figure 2: ICE framework. One dataset-agnostic Clifford field connects multimodal input, edge-aware graph transport, foundation learning, and fresh node and link heads.

Traditional multimodal
Method	RedditS		Movies		Grocery		Toys		Ele-F.		Books-NC
	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1
MMGCN	90.27 $\pm$ 0.34	84.22 $\pm$ 0.73	53.41 $\pm$ 1.03	41.66 $\pm$ 2.03	82.56 $\pm$ 0.49	73.83 $\pm$ 0.93	80.02 $\pm$ 0.64	76.36 $\pm$ 1.23	86.59 $\pm$ 0.08	68.85 $\pm$ 0.35	83.22 $\pm$ 0.10	71.28 $\pm$ 0.22
MGAT	92.78 $\pm$ 0.50	87.27 $\pm$ 0.53	53.87 $\pm$ 0.50	44.09 $\pm$ 1.60	83.74 $\pm$ 0.62	74.77 $\pm$ 1.11	79.61 $\pm$ 0.74	77.09 $\pm$ 0.87	84.84 $\pm$ 0.08	69.62 $\pm$ 0.21	82.91 $\pm$ 0.04	71.45 $\pm$ 0.11
Self-supervised graph models
GRACE	93.01 $\pm$ 0.53	88.39 $\pm$ 1.12	48.09 $\pm$ 0.97	37.18 $\pm$ 1.33	70.83 $\pm$ 0.81	60.69 $\pm$ 1.05	72.82 $\pm$ 0.66	69.09 $\pm$ 0.63	83.58 $\pm$ 0.11	70.09 $\pm$ 0.47	74.96 $\pm$ 0.06	70.09 $\pm$ 0.11

Table 1: Node classification. Accuracy and Macro-F1 for each dataset over eight seeds. Best means are bold and second-best means are underlined.

Traditional multimodal
Method	Link prediction			Few-shot link classification
	Sports	Cloth	Books-LP	Sports 3	Sports 5	Sports 10	Cloth 3	Cloth 5	Cloth 10
MMGCN	23.44 $\pm$ 0.43	17.74 $\pm$ 0.38	20.73 $\pm$ 0.48	55.86 $\pm$ 2.96	53.70 $\pm$ 1.20	56.66 $\pm$ 1.60	64.36 $\pm$ 4.07	68.61 $\pm$ 5.38	67.27 $\pm$ 4.20
MGAT	21.74 $\pm$ 0.96	15.47 $\pm$ 0.32	21.82 $\pm$ 0.53	54.92 $\pm$ 2.40	56.55 $\pm$ 2.69	57.92 $\pm$ 1.80	67.05 $\pm$ 1.45	70.34 $\pm$ 3.36	68.66 $\pm$ 2.94
Self-supervised graph models
GRACE	25.31 $\pm$ 0.16	18.27 $\pm$ 0.15	19.30 $\pm$ 0.27	56.42 $\pm$ 1.36	57.94 $\pm$ 2.58	58.50 $\pm$ 1.43	62.36 $\pm$ 2.75	64.94 $\pm$ 1.09	64.96 $\pm$ 1.86

Table 2: Link prediction and few-shot link classification. MRR for supervised link prediction and Accuracy for balanced 2-way classification over eight seeds.

Method	Grocery 5-way			Ele-Fashion 5-way			Books-NC 5-way
	3	5	10	3	5	10	3	5	10
GFT	61.45 $\pm$ 2.22	64.60 $\pm$ 3.54	63.12 $\pm$ 4.07	59.08 $\pm$ 3.95	61.38 $\pm$ 3.41	62.28 $\pm$ 2.62	48.70 $\pm$ 4.96	50.95 $\pm$ 4.35	51.62 $\pm$ 4.73
UniGraph2	60.05 $\pm$ 4.13	61.25 $\pm$ 3.09	66.27 $\pm$ 3.11	53.98 $\pm$ 4.07	58.73 $\pm$ 2.95	60.38 $\pm$ 2.08	59.67 $\pm$ 4.23	61.55 $\pm$ 3.97	63.80 $\pm$ 4.02
PLANET	77.85 $\pm$ 4.06	79.88 $\pm$ 3.27	81.93 $\pm$ 3.48	70.50 $\pm$ 4.37	72.97 $\pm$ 3.55	74.85 $\pm$ 3.73	63.62 $\pm$ 4.11	67.59 $\pm$ 4.68	69.18 $\pm$ 3.90
ICE	78.96 $\pm$ 3.78	81.02 $\pm$ 3.08	83.10 $\pm$ 3.21	71.61 $\pm$ 4.01	74.07 $\pm$ 3.33	76.02 $\pm$ 3.42	64.83 $\pm$ 3.86	68.78 $\pm$ 4.32	70.42 $\pm$ 3.61

Table 3: Few-shot node classification. Focused 5-way comparison across 3, 5, and 10 shots; the complete matrix appears in Appendix Table 8 .

Figure 3: Mechanism behavior. (a–b) NC and LP access mass by grade and depth. (c–d) Accuracy (Acc.), Macro-F1 (F1), and MRR drops (percentage points) after edge rolling and protected Grade-1 route removal, respectively.

Configuration	NC Acc.	NC F1	LP MRR
ICE	83.21 $\pm$ 0.31	74.90 $\pm$ 0.38	26.24 $\pm$ 0.18
w/o Clifford product	82.00 $\pm$ 0.46	73.54 $\pm$ 0.52	24.73 $\pm$ 0.29
w/o layer 2	82.71 $\pm$ 0.37	74.34 $\pm$ 0.44	25.58 $\pm$ 0.21
w/o grade-depth bank	82.52 $\pm$ 0.39	74.12 $\pm$ 0.45	25.26 $\pm$ 0.24
w/o Grade-1 route	82.08 $\pm$ 0.43	73.50 $\pm$ 0.49	25.69 $\pm$ 0.25
w/o task access	82.29 $\pm$ 0.41	73.86 $\pm$ 0.47	25.16 $\pm$ 0.26

Table 4: Core ablation (eight-seed mean).

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Graph	Nodes	Edges	Role	Classes	Exposure
Movies	16,672	218,390	NC	20	5.0
Toys	20,695	126,886	NC	18	5.0
Grocery	17,074	171,340	NC	20	5.0
Sports	50,250	356,202	LP	–	1.0
Cloth	125,839	951,271	LP	–	1.0
Ele-Fashion	97,766	199,602	NC	12	1.0

Appendix

Table 5: Foundation graph inventory and pretraining exposure.

Stage	Configuration	Value
Foundation	Input width per modality	3,584
Foundation	Hidden width and blades	256 and 8
Foundation	Transport layers	2
Foundation	RWPE steps	4
Foundation	Root batch and fanout	128 and $(5,10)$
Foundation	Epochs and optimizer	5 and AdamW

Appendix

Table 6: Foundation and downstream configuration.

Measure	Value
Foundation state
Foundation parameters	79.99 M
Foundation checkpoint	305.2 MiB
Fresh NC adaptation
Head parameters	22.7 to 41.3 K
Peak GPU memory	0.52 to 0.95 GiB

Appendix

Table 7: Model and adaptation scale. Architecture size and completed NC adaptation cost.

	Method	Grocery 5-way			Ele-Fashion 5-way			Books-NC 5-way
	Method	3	5	10	3	5	10	3	5	10
	MMGCN	48.13 $\pm$ 2.65	50.30 $\pm$ 3.13	53.73 $\pm$ 3.51	54.37 $\pm$ 2.99	57.05 $\pm$ 2.42	60.87 $\pm$ 3.08	52.93 $\pm$ 2.94	54.10 $\pm$ 2.90	56.42 $\pm$ 2.84
	MGAT	50.83 $\pm$ 2.88	54.07 $\pm$ 3.18	55.53 $\pm$ 2.98	60.38 $\pm$ 2.57	61.10 $\pm$ 2.20	62.82 $\pm$ 2.18	53.15 $\pm$ 3.09	57.05 $\pm$ 2.35	58.22 $\pm$ 3.40
	GRACE	57.28 $\pm$ 4.91	59.22 $\pm$ 3.50	62.22 $\pm$ 2.89	55.49 $\pm$ 4.10	59.90 $\pm$ 2.73	64.72 $\pm$ 3.04	56.84 $\pm$ 4.65	61.53 $\pm$ 3.14	59.20 $\pm$ 2.00
	GraphMAE2	51.90 $\pm$ 5.76	54.60 $\pm$ 3.83	58.00 $\pm$ 4.26	57.35 $\pm$ 3.05	60.00 $\pm$ 3.53	63.65 $\pm$ 3.48	43.20 $\pm$ 2.70	46.30 $\pm$ 2.95	49.83 $\pm$ 2.60
GFM	RiemannGFM	63.63 $\pm$ 2.48	66.23 $\pm$ 2.73	67.14 $\pm$ 2.02	59.13 $\pm$ 3.71	60.24 $\pm$ 3.55	61.73 $\pm$ 2.90	54.29 $\pm$ 4.10	57.17 $\pm$ 3.83	60.54 $\pm$ 3.92

Appendix

Table 8: Few-shot node classification. Accuracy for 5-way 3-shot, 5-shot, and 10-shot tasks.

Figure 4: Component necessity. Absolute performance changes for the matched removals.

Representation	AUC
Hidden roll	0.5496
L1 Vector	0.7714
L1 V+B	0.7833
L1+L2 V+B	0.8090

Appendix

Table 9: Relation probe. Pair-relation AUC across the transport sequence.

Dataset	Accuracy	Macro-F1
Movies	10.888	5.539
Toys	3.213	1.648
Grocery	4.130	3.880
RedditS	0.472	0.302
Ele-Fash.	2.460	5.062
Books-NC	2.995	6.752

Appendix

Table 10: Task-access probe. Improvement over the generic query in percentage points.

Quantity	Value
Active parameters under same-blade maps	26.53%
Layer-2 $e_{12}$ energy retained	2.658%
Cases improved by blade isolation	0 of 6

Appendix

Table 11: Cross-grade capacity probe. Changes induced by permanent blade isolation.

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Multimodal graph foundation models aim to learn reusable knowledge from graphs enriched with text, images, attributes, and relational topology, thereby supporting diverse graph-centric and modality-centric tasks. In practice, however, such multimodal graphs are often distributed across decentralized clients, where raw contents and local structures cannot be centrally shared due to privacy constraints. This motivates federated multimodal graph foundation learning, which requires not only transferable representation learning but also intrinsic semantic traceability under strict data isolation. Existing methods usually exchange or store knowledge through parameters, prototypes, embeddings, or compact codebooks, which support optimization and transfer but do not explicitly expose how modality evidence, node semantics, and topology context jointly support predictions. To bridge this gap, we propose FedLAB, a traceable semantic codebook framework that organizes multimodal graph knowledge into typed hierarchical codebooks for modality evidence, node semantics, and topology context. FedLAB further refines these trace units through federated semantic barycenter pre-training while keeping raw multimodal contents and graph structures local. Extensive experiments on 10 benchmarks and 6 downstream tasks show that FedLAB improves over state-of-the-art baselines by up to 7.53%, while preserving a native semantic trace interface.

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