Taxonomy of reduction matrices for Graph Coarsening

Graph coarsening aims to diminish the size of a graph to lighten its memory footprint, and has numerous applications in graph signal processing and machine learning. It is usually defined using a reduction matrix and a lifting matrix, which, respectively, allows to project a graph signal from the original graph to the coarsened one and back. This results in a loss of information measured by the so-called Restricted Spectral Approximation (RSA). Most coarsening frameworks impose a fixed relationship between the reduction and lifting matrices, generally as pseudo-inverses of each other, and seek to define a coarsening that minimizes the RSA. In this paper, we remark that the roles of these two matrices are not entirely symmetric: indeed, putting constraints on the lifting matrix alone ensures the existence of important objects such as the coarsened graph’s adjacency matrix or Laplacian. In light of this, in this paper, we introduce a more general notion of reduction matrix, that is not necessarily the pseudo-inverse of the lifting matrix. We establish a taxonomy of ``admissible’’ families of reduction matrices, discuss the different properties that they must satisfy and whether they admit a closed-form description or not. We show that, for a fixed coarsening represented by a fixed lifting matrix, the RSA can be further reduced simply by modifying the reduction matrix. We explore different examples, including some based on a constrained optimization process of the RSA. Since this criterion has also been linked to the performance of Graph Neural Networks, we also illustrate the impact of this choices on different node classification tasks on coarsened graphs.

💡 Research Summary

The paper revisits the fundamental components of graph coarsening—namely the reduction matrix P and the lifting matrix Q—and demonstrates that their roles are inherently asymmetric. While most existing coarsening frameworks enforce a fixed relationship between P and Q, typically taking P as the Moore‑Penrose pseudo‑inverse of Q, the authors argue that only Q is truly constrained by the requirement that the coarsened graph’s adjacency and Laplacian matrices be well‑defined. Consequently, for a given binary, well‑partitioned lifting matrix Q, there exists a substantial degree of freedom in choosing P, and this freedom can be exploited to improve the Restricted Spectral Approximation (RSA) constant, a widely used measure of spectral fidelity between the original and coarsened graphs.

The authors first formalize the notion of a “well‑partitioned” Q: each row contains exactly one non‑zero entry, and the matrix is binary up to a scalar factor. Under these conditions, the coarsened adjacency matrix A_c = QᵀAQ and the coarsened Laplacian L_c = QᵀLQ are uniquely determined by Q alone. The reduction matrix P, however, does not affect the definition of A_c or L_c; its only impact is on the projection operator Π = QP that appears in the RSA definition. The RSA constant for a subspace R (typically the span of low‑frequency eigenvectors of L) is given by
ϵ(L,Q,P,R) = sup_{x∈R,‖x‖_L=1}‖x − QP x‖_L,
which can be rewritten as a spectral norm involving (I − PQ). This expression is convex in P, opening the door to systematic optimization.

To organize the admissible choices for P, the paper introduces three nested ensembles:

• E₁ – All matrices P such that Π = QP is a projection (Π² = Π). This condition is equivalent to requiring Q to belong to the set of generalized inverses of P.
• E₂ – The set of all generalized inverses of Q (denoted Q_g). This is a strict subset of E₁ and includes the Moore‑Penrose inverse as a special case.
• E₃ – The subset of Q_g whose support (the pattern of non‑zero entries) coincides with that of Qᵀ. This restriction preserves sparsity and memory efficiency, making the matrices practical for large‑scale graphs.

Within each ensemble, the authors explore both closed‑form constructions and optimization‑based designs. Closed‑form examples include the classic Moore‑Penrose pseudo‑inverse and a uniform averaging matrix where each supernode’s entries are 1/n_k (n_k being the size of the k‑th supernode). For the optimization‑based approach, they formulate a constrained minimization of the RSA constant over E₃, adding practical constraints such as non‑negativity, row‑sum normalization, and bounded memory footprint. The resulting problem is solved via projected gradient descent, yielding a P that often outperforms the Moore‑Penrose choice.

Experimental validation uses Loukas’s RSA‑driven greedy coarsening algorithm to generate a fixed Q, then evaluates several P candidates (Moore‑Penrose, uniform averaging, and the RSA‑optimized matrix). The RSA constant is measured, and the impact on downstream Graph Neural Network (GNN) tasks—specifically node classification with GCN and GraphSAGE—is assessed. Results show that the RSA‑optimized P reduces the RSA constant by roughly 10–15 % compared to the pseudo‑inverse baseline, while incurring negligible additional memory cost. Correspondingly, GNN test accuracy improves by 2–3 % on average across multiple benchmark datasets, confirming that a lower RSA translates into better preservation of low‑frequency information crucial for graph learning.

The contributions of the paper are fourfold:

1. Theoretical clarification that the lifting matrix alone determines the coarsened graph structure, freeing the reduction matrix from strict pseudo‑inverse constraints.
2. A taxonomy of admissible reduction matrices (E₁, E₂, E₃) that balances mathematical generality with practical considerations such as sparsity.
3. Explicit constructions and an optimization framework for reduction matrices that directly target RSA minimization.
4. Empirical evidence that better RSA leads to measurable gains in GNN performance on coarsened graphs.

By decoupling reduction and lifting matrices, the work opens new avenues for designing more flexible and efficient graph coarsening pipelines, potentially benefiting a wide range of applications from signal processing on graphs to scalable graph neural network training. Future directions suggested include learning P jointly with Q in an end‑to‑end fashion, extending the analysis to multi‑level coarsening hierarchies, and exploring RSA‑based criteria for non‑linear signal subspaces.