Linear Graph Convolution (LGC)

• Linear Graph Convolution (LGC) is a family of linear operators that filter graph signals via spectral, polynomial, and normalized-adjacency methods.
• LGC leverages linear propagation and diffusion processes to improve node classification while mitigating over-smoothing issues.
• LGC models extend from collaborative filtering to semi-supervised learning by employing closed-form solvers and adaptive operator designs.

Linear Graph Convolution (LGC) denotes a family of graph-convolution operators in which the central propagation or filtering step is linear in the graph signal, node features, or embeddings. Across the literature, the term covers several related constructions: spectral filters of the form $g(L)=U g(\Lambda) U^\top$, polynomial propagators such as $\sum_{k=0}^K \alpha_k L^k$ or $\sum_{k=0}^K \alpha_k \tilde A^k$, linear collaborative-filtering models such as LightGCN, and fully linear graph learners trained by closed-form solvers rather than gradient descent (Edwards et al., 2016, Pasa et al., 2021, He et al., 2020, Cai et al., 2021). This suggests that LGC is best understood not as a single canonical layer, but as a recurring operator principle: graph structure enters through a linear map, while modeling flexibility is moved to spectral coefficients, operator choice, layer aggregation, or downstream linear estimators.

1. Spectral origin and terminological scope

A foundational formulation appears in “Graph Based Convolutional Neural Network” (Edwards et al., 2016). There, the graph is a weighted undirected graph $G=\{V,W\}$ with unnormalized Laplacian $L=D-W$, eigen-decomposition $L=U\Lambda U^\top$, graph Fourier transform $\tilde f=U^\top f$, and spectral filter $k\in\mathbb{R}^N$. The graph convolution is defined spectrally as

$$y = U(\tilde f \odot k)=U\,g(\Lambda)\,U^\top f = g(L)f.$$

In the multi-channel case,

$$y_{s,o}=U\sum_{i=1}^{I}\big(U^\top f_{s,i}\odot k_{i,o}\big).$$

The paper also states that smooth spectral multipliers correspond to localized filters in the vertex domain, and that smoother multipliers provide sharper feature maps. Its architecture combines graph convolution, ReLU, and Algebraic Multigrid pooling, and reports $\sum_{k=0}^K \alpha_k L^k$0 on regular-grid MNIST and $\sum_{k=0}^K \alpha_k L^k$1 on spatially irregular subsampled MNIST (Edwards et al., 2016).

Later work uses “LGC” in broader but structurally related ways. In “Simple Graph Convolutional Networks” (Pasa et al., 2021), LGC is a single-layer polynomial Laplacian filter; in “LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation” (He et al., 2020), it is linear neighborhood aggregation on a user–item graph; in “A Unifying Generative Model for Graph Learning Algorithms” (Jia et al., 2021), it is the conditional expectation $\sum_{k=0}^K \alpha_k L^k$2; and in “Generalized Linear Graph Representation” (Zhao et al., 12 Jun 2026), it becomes propagation with a learnable linear operator family $\sum_{k=0}^K \alpha_k L^k$3.

Context	Representative operator	Representative papers
Spectral graph signal processing	$\sum_{k=0}^K \alpha_k L^k$4	(Edwards et al., 2016)
Linearized node classification	$\sum_{k=0}^K \alpha_k L^k$5	(Pasa et al., 2021)
Collaborative filtering	$\sum_{k=0}^K \alpha_k L^k$6 or $\sum_{k=0}^K \alpha_k L^k$7	(He et al., 2020, Shen et al., 2021)
Closed-form fully linear GNNs	$\sum_{k=0}^K \alpha_k L^k$8	(Cai et al., 2021)
Learnable operator families	$\sum_{k=0}^K \alpha_k L^k$9	(Zhao et al., 12 Jun 2026)

2. Linearization in node learning and statistical interpretation

In node classification, one prominent use of LGC is as a polynomial spectral filter. “Simple Graph Convolutional Networks” (Pasa et al., 2021) defines a general polynomial graph filter

$\sum_{k=0}^K \alpha_k \tilde A^k$0

and specializes Linear Graph Convolution to

$\sum_{k=0}^K \alpha_k \tilde A^k$1

This formulation generalizes SGC, which corresponds to a fixed monomial filter such as $\sum_{k=0}^K \alpha_k \tilde A^k$2, while retaining a single linear propagation stage followed by a linear classifier. The paper reports that LGC outperformed SGC and GCN on Citeseer, Cora, Pubmed, and Reddit under validation-based selection (Pasa et al., 2021).

A distinct statistical derivation appears in “A Unifying Generative Model for Graph Learning Algorithms” (Jia et al., 2021). There, node attributes and labels are generated by a Gaussian Markov random field with precision

$\sum_{k=0}^K \alpha_k \tilde A^k$3

Under this model, Linear Graph Convolution is the conditional expectation of labels given features,

$\sum_{k=0}^K \alpha_k \tilde A^k$4

with $\sum_{k=0}^K \alpha_k \tilde A^k$5 and $\sum_{k=0}^K \alpha_k \tilde A^k$6. The same framework derives label propagation as $\sum_{k=0}^K \alpha_k \tilde A^k$7 and LGC/RP as $\sum_{k=0}^K \alpha_k \tilde A^k$8, so LGC is not merely a heuristic smoothing rule but the Bayes-optimal predictor for squared loss under the stated model (Jia et al., 2021).

A third theoretical view is provided by “Graph Convolution for Semi-Supervised Classification: Improved Linear Separability and Out-of-Distribution Generalization” (Baranwal et al., 2021). That work studies a contextual stochastic block model and analyzes a deliberately simple linear graph convolution: fixed linear smoothing of node features over the graph followed by an ordinary linear classifier. It shows that graph convolution extends the regime in which the data is linearly separable by a factor of roughly $\sum_{k=0}^K \alpha_k \tilde A^k$9, where $G=\{V,W\}$0 is the expected degree of a node, and that the resulting linear classifier generalizes to out-of-distribution data where unseen graphs have different intra- and inter-class edge probabilities (Baranwal et al., 2021). Within this line of work, LGC is valuable not because it is expressive in the deep-learning sense, but because it alters the geometry of the classification problem in a quantifiable way.

3. Diffusion, closed-form training, and fully linear graph networks

Several papers recast LGC as a diffusion process rather than as a stack of hidden layers. “Dissecting the Diffusion Process in Linear Graph Convolutional Networks” (Wang et al., 2021) interprets SGC-style propagation,

$G=\{V,W\}$1

as forward Euler discretization of the graph heat equation

$G=\{V,W\}$2

with step size $G=\{V,W\}$3 and terminal time $G=\{V,W\}$4. This analysis identifies two coupled issues: increasing the number of propagation steps increases the diffusion time, which drives over-smoothing, and it also keeps the discretization coarse. Decoupled Graph Convolution (DGC) separates terminal time $G=\{V,W\}$5 from propagation steps $G=\{V,W\}$6, using

$G=\{V,W\}$7

so that one may increase $G=\{V,W\}$8 to improve numerical accuracy without forcing $G=\{V,W\}$9 (Wang et al., 2021).

“Simple yet Effective Gradient-Free Graph Convolutional Networks” (Zhu et al., 2023) pushes linearization further by replacing gradient-based training with closed-form optimization. It treats linearized GNNs as a graph filter $L=D-W$0 followed by a linear predictor,

$L=D-W$1

where $L=D-W$2 may be SGC, SSGC, or DGC. The paper links over-smoothing to vanishing gradients and replaces cross-entropy training with a regularized mean-squared formulation in an RKHS, yielding closed-form primal or dual solvers. Its core claim is that gradient-free training significantly overcomes over-smoothing and enhances generalization while costing much less training time (Zhu et al., 2023).

“Fully Linear Graph Convolutional Networks for Semi-Supervised Learning and Clustering” (Cai et al., 2021) makes this design explicit. FLGC uses the propagation equation

$L=D-W$3

followed by a closed-form supervised or unsupervised linear model. In the semi-supervised case,

$L=D-W$4

and in the clustering case,

$L=D-W$5

The paper presents FLGC as a natural generalization of ridge regression and subspace clustering to the non-Euclidean domain, and uses an initial residual strategy to aggregate long-range neighborhoods and alleviate over-smoothing (Cai et al., 2021). In this strand of the literature, LGC is inseparable from decoupling: graph propagation is a fixed linear preconditioner, and the trainable part is a classical linear estimator.

4. Recommendation and collaborative filtering

In recommender systems, LGC became closely associated with the removal of feature transformation matrices and nonlinear activations from GCN-based collaborative filtering. “LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation” (He et al., 2020) defines layerwise propagation on the user–item interaction graph as

$L=D-W$6

with final embedding aggregation

$L=D-W$7

There are no feature transformation matrices, no activation functions, and no trainable parameters in the convolution layers; the only trainable parameters are the initial user and item ID embeddings. Under exactly the same experimental setting, LightGCN reports about $L=D-W$8 relative improvement on average over NGCF (He et al., 2020). In this context, LGC denotes linear neighborhood aggregation on a bipartite graph.

“How Powerful is Graph Convolution for Recommendation?” (Shen et al., 2021) generalizes this perspective by treating collaborative filtering through graph signal processing. On the item–item graph, it defines a unified framework

$L=D-W$9

and identifies Linear Graph Convolution with polynomial low-pass filters

$L=U\Lambda U^\top$0

equivalently

$L=U\Lambda U^\top$1

Within this framework, neighborhood-based CF, low-rank matrix factorization, linear auto-encoders, LightGCN, LGCN-IDE, and GF-CF all become instances of linear low-pass graph filtering. The paper further proposes GF-CF, a fully linear closed-form baseline, and reports a $L=U\Lambda U^\top$2 performance gain over LightGCN on the Amazon-book dataset (Shen et al., 2021).

A specialized extension appears in “Criteria Tell You More than Ratings: Criteria Preference-Aware Light Graph Convolution for Effective Multi-Criteria Recommendation” (Park et al., 2023). CPA-LGC constructs an MC expansion graph, runs two parallel light graph convolution branches, and keeps propagation linear: $L=U\Lambda U^\top$3 where $L=U\Lambda U^\top$4 is PairNorm rather than a learned nonlinearity. The model incorporates user-specific criteria-preference embeddings and item-specific criterion embeddings, uses BPR loss, and reports superiority over benchmark multi-criteria recommendation methods and benchmark recommendation methods using GNNs with tremendous gains, together with computational efficiency (Park et al., 2023). Here, “light” and “linear” are closely aligned: graph structure is exploited through normalized propagation, while expressive power is moved to graph construction and embedding design.

5. Generalized operator spaces, multi-graph formulations, and local filters

More recent work broadens LGC from fixed adjacency- or Laplacian-based propagation to learnable operator families. “Generalized Linear Graph Representation: A Compact Operator Space for Graph Signal Processing and Graph Neural Networks” (Zhao et al., 12 Jun 2026) introduces

$L=U\Lambda U^\top$5

This compact two-parameter family recovers adjacency, degree, combinatorial Laplacian, signless Laplacian up to scaling, Nikiforov’s $L=U\Lambda U^\top$6 family, Wang’s $L=U\Lambda U^\top$7 family, and transition/PageRank-like operators without asymptotic parameters. Adaptive GLGR Convolution then uses

$L=U\Lambda U^\top$8

with learnable $L=U\Lambda U^\top$9 and $\tilde f=U^\top f$0. The paper reports that GLGR improves both fixed-operator representation search and adaptive graph learning across multiple backbones (Zhao et al., 12 Jun 2026). In this formulation, LGC is no longer tied to a single canonical graph matrix; the operator itself becomes part of the hypothesis space.

“What Can We Learn From MIMO Graph Convolutions?” (Roth et al., 16 May 2025) argues that standard graph convolutions are typically approximated in the single-input single-output case even though GNNs operate in the MIMO regime. It derives true MIMO graph convolution and identifies multiple computational graphs, or equivalently distinct feature transformations for each pair of nodes, as the key MIMO-specific property. Its localized MIMO graph convolution has the form

$\tilde f=U^\top f$1

and generalizes many linear message-passing neural networks. The paper proves that, for almost every choice of edge weights, LMGCs with a single computational graph are injective on multisets, and that the resulting representations are linearly independent when more than one computational graph is used (Roth et al., 16 May 2025). This suggests that the expressive bottleneck of many linear models lies less in linearity itself than in the restriction to a single propagation operator.

“Graph Convolution with Low-rank Learnable Local Filters” (Cheng et al., 2020) pushes the same point from a different angle. Its L3Net layer starts from a general linear graph operator

$\tilde f=U^\top f$2

and factorizes

$\tilde f=U^\top f$3

where $\tilde f=U^\top f$4 are learnable local filters supported on $\tilde f=U^\top f$5-hop neighborhoods. The paper shows that ChebNet, GCN, GAT, EdgeNet, and low-rank Euclidean CNNs are special cases; introduces local graph-Laplacian regularization

$\tilde f=U^\top f$6

and proves representation stability against graph-data perturbation (Cheng et al., 2020). Its expressiveness result is especially relevant to LGC: there exists a local filter representable by L3Net with $\tilde f=U^\top f$7 that cannot be expressed by any spectral graph convolution $\tilde f=U^\top f$8 (Cheng et al., 2020).

6. Geometric, invertible, and conceptual extensions

LGC has also been adapted to settings where the central concern is not only smoothing but geometry or invertibility. “The Convolution Exponential and Generalized Sylvester Flows” (Hoogeboom et al., 2020) defines a linear graph convolutional layer

$\tilde f=U^\top f$9

which vectorizes to

$k\in\mathbb{R}^N$0

Exponentiating this operator yields the graph convolution exponential

$k\in\mathbb{R}^N$1

an invertible linear flow with

$k\in\mathbb{R}^N$2

The construction preserves permutation equivariance because if $k\in\mathbb{R}^N$3 commutes with a symmetry $k\in\mathbb{R}^N$4, then $k\in\mathbb{R}^N$5 does as well (Hoogeboom et al., 2020). In this setting, LGC is the generator of an invertible graph mixing layer.

A geometric reformulation appears in “L$k\in\mathbb{R}^N$6GC: Lorentzian Linear Graph Convolutional Networks for Node Classification” (Liang et al., 2024). That model keeps propagation linear and simple, maps node features into hyperbolic space, and performs Lorentzian linear feature transformation in the tangent space of the hyperboloid. It uses personalized PageRank-style propagation

$k\in\mathbb{R}^N$7

then applies Lorentzian linear transformation through exponential and logarithmic maps. On standard citation datasets, it reports $k\in\mathbb{R}^N$8 on Citeseer and $k\in\mathbb{R}^N$9 on PubMed, and observes that it can be trained up to two orders of magnitude faster than other nonlinear GCN models on PubMed (Liang et al., 2024). Related hyperbolic graph-convolution work rigorously defines Lorentzian matrix-vector multiplication and centroid-based neighborhood aggregation on the hyperboloid model (Zhang et al., 2021).

Several misconceptions recur in the LGC literature. First, linearity does not denote a single operator family. Depending on context, it may mean spectral diagonal filtering, polynomial propagation, normalized-adjacency diffusion, PageRank-like propagation, closed-form linear regression on graph embeddings, or Lorentzian tangent-space transformation. Second, linearization is not uniformly advantageous. In collaborative filtering on user–item graphs with ID features, removing feature transformations and activations often improves performance and efficiency (He et al., 2020, Shen et al., 2021); in semi-supervised node classification, linearized models are competitive but still require mechanisms against over-smoothing, including decoupled diffusion, initial residuals, PairNorm, or closed-form training (Wang et al., 2021, Zhu et al., 2023, Cai et al., 2021, Park et al., 2023). Third, recent operator-centric work indicates that the main design question has shifted from whether to use nonlinearities to which linear operator family to use: compact learnable operators, multiple computational graphs, low-rank local filters, and geometry-aware transforms all change what “linear graph convolution” can express (Zhao et al., 12 Jun 2026, Roth et al., 16 May 2025, Cheng et al., 2020, Liang et al., 2024).

In aggregate, the LGC literature presents a coherent but heterogeneous picture. The invariant element is linear propagation on a graph-structured domain. What varies is the operator class—spectral multiplier, Laplacian polynomial, normalized adjacency, compact two-parameter family, local filter bank, or hyperbolic tangent-space map—and the training regime—gradient descent, propagation plus regression, residual diffusion, or exact closed-form solvers. This suggests that the enduring significance of LGC lies less in simplification alone than in making the graph operator itself mathematically transparent, analyzable, and, increasingly, learnable.