Graph Condensation via Tensor Decomposition (GCTD)

• The paper introduces GCTD, which condenses graphs by decomposing a multi-view tensor, thereby avoiding the complex bilevel and triple-loop optimization typical in prior methods.
• GCTD employs non-negative sparse RESCAL to extract latent factors and clusters them into synthetic nodes, ensuring explicit and interpretable node correspondence.
• Experiments demonstrate that GCTD achieves competitive accuracy and efficiency, with notable performance gains on citation datasets compared to existing condensation techniques.

Searching arXiv for the named method and closely related graph condensation work. Graph Condensation via Tensor Decomposition (GCTD) is a graph condensation method for node classification that synthesizes a small graph $$\mathcal{G}'=\{\mathbf{A}',\mathbf{X}',\mathbf{Y}'\}$$ from an original graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ by decomposing a tensor of augmented adjacency views rather than optimizing the synthetic graph through the bilevel GNN-matching objectives that dominate much of the literature (Santos et al., 20 Aug 2025). Introduced as "Multi-view Graph Condensation via Tensor Decomposition," GCTD uses non-negative sparse RESCAL to compress multi-view structural information, then derives synthetic nodes, features, labels, and an explicit mapping from original nodes to synthetic nodes through clustering of latent factors (Santos et al., 20 Aug 2025). The 2024 graph condensation surveys do not explicitly mention GCTD by name; instead, they describe nearby decomposition-oriented mechanisms such as low-rank adjacency parameterization and eigenbasis-based condensation, which place GCTD within the broader lineage of decomposition-based condensed graph synthesis (Gao et al., 2024, Xu et al., 2024).

1. Position within the graph condensation literature

The contemporary graph condensation literature is largely organized around two perspectives. One survey classifies methods by evaluation criteria—effectiveness, generalization, efficiency, fairness, and robustness—and separately analyzes optimization strategies such as gradient matching, trajectory matching, kernel ridge regression, and distribution matching, along with condensed graph generation strategies such as attribute initialization and structure construction (Gao et al., 2024). A second survey organizes the field by condensation objective—graph-guided, model-guided, and hybrid—and by formulation—modification versus synthetic generation (Xu et al., 2024).

GCTD is absent from both 2024 surveys because it appeared later, but the surveys provide a vocabulary for situating it. The closest explicitly named neighbors in the first survey are KIDD, which “directly optimize the low-rank decomposed matrices of the adjacency matrix,” and GCEM, which “decomposes the adjacency matrix into eigenbasis” (Gao et al., 2024). This suggests that GCTD belongs most naturally to the decomposition/parameterization branch of condensed structure generation rather than to the mainstream families centered on gradient-, trajectory-, or distribution-matching objectives.

In the terminology of the second survey, GCTD most plausibly aligns with graph-guided condensation because its core training signal is tensor reconstruction rather than direct downstream model matching (Xu et al., 2024). A plausible implication is that it also straddles the boundary between modification and synthetic formulations: the factor matrix induces a grouping of original nodes, while the final output is an explicitly synthesized compact graph.

2. Problem formulation and multi-view tensor construction

GCTD considers an original graph

$\mathcal{G} = (\mathcal{V}, \mathcal{E}),$

with adjacency matrix $\mathbf{A} \in \mathbb{R}^{N \times N}$, node features $\mathbf{X} \in \mathbb{R}^{N \times d}$, and labels $\mathbf{Y} \in \{0,\ldots,C-1\}^N$. Its goal is to construct a condensed graph

$$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$

where $\mathbf{A}' \in \mathbb{R}^{N' \times N'}$, $\mathbf{X}' \in \mathbb{R}^{N' \times d}$, $\mathbf{Y}' \in \{0,\ldots,C-1\}^{N'}$, and $\mathcal{G}=(\mathcal{V},\mathcal{E})$0, such that a GNN trained on $\mathcal{G}=(\mathcal{V},\mathcal{E})$1 attains predictive performance close to training on the full graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$2 (Santos et al., 20 Aug 2025). The condensed size is set by a target ratio $\mathcal{G}=(\mathcal{V},\mathcal{E})$3, with

$\mathcal{G}=(\mathcal{V},\mathcal{E})$4

The paper motivates GCTD by contrasting it with the standard bilevel formulation used in many prior graph condensation methods:

$\mathcal{G}=(\mathcal{V},\mathcal{E})$5

The authors emphasize that, in practice, this often becomes a “triple-loop” procedure because the synthetic graph and the downstream GNN are jointly optimized across multiple random initializations (Santos et al., 20 Aug 2025).

The defining step in GCTD is the construction of a multi-view tensor from augmented graph structures. Here, “multi-view” does not denote multimodal data; it refers to multiple perturbed adjacency matrices over the same node set (Santos et al., 20 Aug 2025). Starting from $\mathcal{G}=(\mathcal{V},\mathcal{E})$6, the method generates $\mathcal{G}=(\mathcal{V},\mathcal{E})$7 perturbed views

$\mathcal{G}=(\mathcal{V},\mathcal{E})$8

by randomly dropping existing edges with probability $\mathcal{G}=(\mathcal{V},\mathcal{E})$9 and adding absent edges with probability $\mathcal{G} = (\mathcal{V}, \mathcal{E}),$0. If the original graph has $\mathcal{G} = (\mathcal{V}, \mathcal{E}),$1 edges, then after dropping,

$\mathcal{G} = (\mathcal{V}, \mathcal{E}),$2

and after adding,

$\mathcal{G} = (\mathcal{V}, \mathcal{E}),$3

The original adjacency matrix is used as the first frontal slice, and the perturbed matrices are stacked as the remaining slices of a third-order tensor (Santos et al., 20 Aug 2025).

3. RESCAL decomposition and condensation mechanism

GCTD reframes graph condensation as decomposition of the multi-view graph tensor rather than synthetic-graph optimization with a relay GNN. The paper reviews Tucker decomposition for a third-order tensor $\mathcal{G} = (\mathcal{V}, \mathcal{E}),$4,

$\mathcal{G} = (\mathcal{V}, \mathcal{E}),$5

and specializes to RESCAL, a Tucker variant for relational multi-slice data:

$\mathcal{G} = (\mathcal{V}, \mathcal{E}),$6

where $\mathcal{G} = (\mathcal{V}, \mathcal{E}),$7 is the shared factor matrix and $\mathcal{G} = (\mathcal{V}, \mathcal{E}),$8 is the $\mathcal{G} = (\mathcal{V}, \mathcal{E}),$9-th frontal slice of the core tensor $\mathbf{A} \in \mathbb{R}^{N \times N}$0 (Santos et al., 20 Aug 2025).

The single-view precursor is the symmetric matrix tri-factorization

$\mathbf{A} \in \mathbb{R}^{N \times N}$1

GCTD extends this to the multi-view case by reconstructing each slice through

$\mathbf{A} \in \mathbb{R}^{N \times N}$2

or, under sparse observed-entry fitting,

$\mathbf{A} \in \mathbb{R}^{N \times N}$3

To avoid overfitting only to positive entries, negative entries are sampled in a 1:1 ratio with observed nonzero values (Santos et al., 20 Aug 2025).

The optimization objective is a normalized reconstruction loss. For a single adjacency matrix, the loss is

$\mathbf{A} \in \mathbb{R}^{N \times N}$4

and for the tensor case,

$\mathbf{A} \in \mathbb{R}^{N \times N}$5

This reconstruction objective is the main training criterion in GCTD; the method does not introduce an additional explicit soft regularization term, relying instead on hard nonnegativity constraints (Santos et al., 20 Aug 2025).

Nonnegativity is enforced by initialization and projection:

$\mathbf{A} \in \mathbb{R}^{N \times N}$6

followed after each gradient step by

$\mathbf{A} \in \mathbb{R}^{N \times N}$7

The paper explicitly describes these as hard constraints and notes that ReLU also induces sparsity because negative values are zeroed (Santos et al., 20 Aug 2025).

A common misconception is that GCTD is a variant of gradient matching or trajectory matching. It is not framed that way. Its condensation stage is a one-stage decomposition-and-extraction procedure centered on tensor reconstruction, and the downstream GNN is trained only after the condensed graph has been constructed (Santos et al., 20 Aug 2025).

4. Construction of the condensed graph and interpretability

Once the factorization is learned, GCTD constructs the condensed adjacency by averaging the core tensor across views:

$\mathbf{A} \in \mathbb{R}^{N \times N}$8

If averaging yields a non-symmetric matrix, the paper states that the result is symmetrized by adding corresponding off-diagonal values (Santos et al., 20 Aug 2025).

The factor matrix $\mathbf{A} \in \mathbb{R}^{N \times N}$9 is then used to induce explicit node correspondence. GCTD clusters the rows of $\mathbf{X} \in \mathbb{R}^{N \times d}$0 with K-Means, using

$\mathbf{X} \in \mathbb{R}^{N \times d}$1

subject to

$\mathbf{X} \in \mathbb{R}^{N \times d}$2

Here $\mathbf{X} \in \mathbb{R}^{N \times d}$3, so the number of clusters equals the number of synthetic nodes (Santos et al., 20 Aug 2025).

This clustering step gives GCTD one of its distinctive properties: each original node is assigned to a synthetic node, and each synthetic node represents a cluster of original nodes sharing similar structural patterns across graph views (Santos et al., 20 Aug 2025). The assignment is hard rather than soft. The paper explicitly identifies soft membership as future work, which indicates that current GCTD does not model fractional node membership (Santos et al., 20 Aug 2025).

Synthetic features, labels, and splits are derived from assigned original nodes rather than directly optimized. The feature vector of a synthetic node is obtained by averaging embeddings or features of its assigned nodes. The split is selected as the most frequent split among assigned original nodes, with preference to underrepresented splits. The class label is chosen from the assigned nodes in the selected split, using frequencies while prioritizing underrepresented classes to preserve the target class distribution (Santos et al., 20 Aug 2025). Because $\mathbf{X} \in \mathbb{R}^{N \times d}$4, $\mathbf{X} \in \mathbb{R}^{N \times d}$5, and split metadata are computed from assigned original nodes, the provenance of each synthetic node is traceable.

The paper presents this traceability as a central interpretability advantage over black-box synthetic-node methods. In its own account, $\mathbf{X} \in \mathbb{R}^{N \times d}$6 exposes latent co-clusters of original nodes, while $\mathbf{X} \in \mathbb{R}^{N \times d}$7 captures interactions among those latent groups across views (Santos et al., 20 Aug 2025).

5. Optimization profile, empirical evaluation, and ablations

Algorithmically, GCTD avoids bilevel optimization entirely. The paper describes a single reconstruction-fitting stage followed by graph extraction: initialize $\mathbf{X} \in \mathbb{R}^{N \times d}$8 and $\mathbf{X} \in \mathbb{R}^{N \times d}$9, enforce nonnegativity, repeatedly reconstruct sparse observed entries, minimize $\mathbf{Y} \in \{0,\ldots,C-1\}^N$0 by gradient descent, apply ReLU projection, compute $\mathbf{Y} \in \{0,\ldots,C-1\}^N$1, cluster rows of $\mathbf{Y} \in \{0,\ldots,C-1\}^N$2 with K-Means, derive $\mathbf{Y} \in \{0,\ldots,C-1\}^N$3, and assign synthetic-node splits and classes (Santos et al., 20 Aug 2025). Adam is used for optimization. Learning rates are tuned over $\mathbf{Y} \in \{0,\ldots,C-1\}^N$4, weight decay over $\mathbf{Y} \in \{0,\ldots,C-1\}^N$5, augmentation probabilities $\mathbf{Y} \in \{0,\ldots,C-1\}^N$6 over $\mathbf{Y} \in \{0,\ldots,C-1\}^N$7, and number of views over $\mathbf{Y} \in \{0,\ldots,C-1\}^N$8. Hyperparameter search is performed with Weights & Biases Bayesian optimization. Convergence is declared after 200 epochs or when the absolute difference in reconstruction loss between consecutive epochs falls below $\mathbf{Y} \in \{0,\ldots,C-1\}^N$9 (Santos et al., 20 Aug 2025).

The paper decomposes time complexity into tensor decomposition, clustering, and adjacency generation:

$$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$0

where $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$1 is approximately the number of edges per view, $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$2 is the number of views, and $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$3 in experiments. Space complexity includes $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$4 for $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$5, $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$6 for $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$7, and graph storage terms for the observed and multi-view data (Santos et al., 20 Aug 2025).

The empirical study covers six node-classification datasets—Cora, Citeseer, Pubmed, Ogbn-arxiv, Flickr, and Reddit—under both transductive and inductive settings. Evaluation follows prior work: a 2-layer GCN with 256 hidden units and no dropout is trained for 600 epochs on the condensed graph, the checkpoint with lowest validation loss is selected, and test performance on the original graph is reported as mean and standard deviation over 10 runs (Santos et al., 20 Aug 2025).

Dataset	Ratios reported	Headline outcome
Citeseer	0.9%, 1.8%, 3.6%	76.8 ± 0.4, 76.5 ± 2.5, 76.7 ± 0.2
Cora	1.3%, 2.6%, 5.2%	81.4 ± 1.6, 84.0 ± 0.4, 79.9 ± 0.3
Pubmed	0.08%, 0.15%, 0.3%	79.9 ± 0.2, 79.4 ± 2.8, 80.0 ± 1.0
Flickr	0.1%, 0.5%, 1%	consistently second-best
Reddit	0.05%, 0.1%, 0.2%	competitive, second-best at 0.05%
Ogbn-arxiv	0.05%, 0.25%, 0.5%	principal weak point

On Citeseer, Cora, and Pubmed, GCTD outperforms the listed baselines, with the paper highlighting gains of up to 4.0% over prior condensation methods on Citeseer (Santos et al., 20 Aug 2025). The reported full-data GCN accuracy on Citeseer is 71.7 ± 0.4, while GCTD reaches 76.8 ± 0.4 at 0.9%; on Pubmed the full-data score is 77.1 ± 0.3, while GCTD reaches 79.9 ± 0.2 at 0.08%; and on Cora GCTD ties the full-data 81.4 ± 0.6 at 1.3% and exceeds all baselines at 2.6% with 84.0 ± 0.4 (Santos et al., 20 Aug 2025). On Flickr, GCTD is consistently second-best yet remains above the full-data score. On Reddit it is competitive but trails GDEM at all three tested ratios. Ogbn-arxiv is the main weak point, with GCTD substantially below the strongest baselines and the full-data score (Santos et al., 20 Aug 2025).

Transfer experiments across GCN, APPNP, ChebyNet, SGC, and GraphSAGE show strong average accuracy for GCTD-condensed graphs: 80.5 on Cora (2.6%), 75.3 on Citeseer (1.8%), 79.2 on Pubmed (0.08%), and 46.1 on Flickr (0.5%), all above the reported averages for GCond, SFGC, and GCDM on those settings (Santos et al., 20 Aug 2025). The multi-view ablation reports that, except for Cora, increasing the number of views improves over the single-view baseline; on Ogbn-arxiv, using three views instead of one yields a 31.5% performance improvement (Santos et al., 20 Aug 2025). A second ablation finds that K-Means assignment of rows of $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$8 consistently outperforms a simple row-wise argmax assignment (Santos et al., 20 Aug 2025).

Runtime is reported in seconds on a single NVIDIA RTX A6000. Representative condensation times are 32.1 on Citeseer (0.9%), 42.8 on Cora (1.3%), 48.1 on Pubmed (0.08%), 965.6 on Ogbn-arxiv (0.05%), 222.9 on Flickr (0.1%), and 2515.3 on Reddit (0.05%) (Santos et al., 20 Aug 2025). The paper interprets these results as showing that GCTD is usually much faster than SGDD, often faster than GCond and GCDM on smaller datasets and Flickr, but not always the fastest overall; SFGC and SNTK can be faster in some settings, and GCTD is not dominant in speed on the largest graphs (Santos et al., 20 Aug 2025). Storage reductions are also substantial in the reported examples: Citeseer is reduced from 47.1 MB to 0.51 MB, Cora from 14.9 MB to 0.23 MB, and Pubmed from 40 MB to 0.05 MB (Santos et al., 20 Aug 2025).

6. Comparative context, limitations, and open directions

GCTD occupies a distinctive position among recent condensation methods. Unlike CTGC, which is a self-supervised dual-branch method based on contrastive clustering, spectral positional embeddings, alternating optimization, and model inversion, GCTD contains no tensor-free centroid-inversion pipeline and no self-supervised contrastive surrogate (Gao et al., 2024). Unlike SimGC, which learns an explicit synthetic graph by aligning classwise multi-hop representation statistics and logits under a fixed pre-trained SGC, GCTD is not a teacher-guided alignment method and does not optimize condensed features and adjacency through layerwise statistic matching (Xiao et al., 2024). Its central operation is tensor factorization of multi-view structure.

The method’s principal strengths, as presented by the paper, are threefold. First, it avoids bilevel optimization and the associated triple-loop burden common in many prior methods (Santos et al., 20 Aug 2025). Second, it offers explicit node correspondence through clustering of $$\mathcal{G}' = \{\mathbf{A}', \mathbf{X}', \mathbf{Y}'\},$$9, which gives synthetic nodes interpretable provenance in terms of original-node groups (Santos et al., 20 Aug 2025). Third, it shows that multi-view structural augmentation can improve condensation quality, especially outside the smallest citation-graph regime (Santos et al., 20 Aug 2025).

Its limitations are equally explicit. Performance is not uniformly best: GCTD underperforms strong baselines on Ogbn-arxiv and trails the best methods on Reddit and Flickr (Santos et al., 20 Aug 2025). The method depends on the suitability of low-rank RESCAL structure for the graph at hand, and the paper notes that Tucker/RESCAL lacks the uniqueness guarantees often associated with CP decomposition (Santos et al., 20 Aug 2025). The current assignment mechanism is hard K-Means rather than soft membership, and the multi-view construction augments only adjacency structure rather than features (Santos et al., 20 Aug 2025). A plausible implication is that graphs whose predictive signal depends strongly on feature geometry or on relational patterns not well captured by RESCAL may be less favorable settings.

Within the broader field, the open problems emphasized by the surveys remain relevant. The surveys identify interpretability, evaluation beyond downstream performance, and support for more complex graph types such as heterogeneous and dynamic graphs as major unresolved questions in graph condensation (Gao et al., 2024, Xu et al., 2024). GCTD directly addresses the interpretability issue through node correspondence, but the paper itself points to soft membership as future work rather than claiming the problem is solved (Santos et al., 20 Aug 2025). In that sense, GCTD is best understood not as a generic replacement for matching-based condensation, but as a decomposition-based alternative whose central contribution is to show that tensor factorization can function as a viable and interpretable condensation mechanism (Santos et al., 20 Aug 2025).