This paper addresses graph learning in Gaussian Graphical Models (GGMs). In this context, data matrices often come with auxiliary metadata (e.g., textual descriptions associated with each node) that is usually ignored in traditional graph estimation processes. To fill this gap, we propose a graph learning approach based on Laplacian-constrained GGMs that jointly leverages the node signals and such metadata. The resulting formulation yields an optimization problem, for which we develop an efficient majorization-minimization (MM) algorithm with closed-form updates at each iteration. Experimental results on a real-world financial dataset demonstrate that the proposed method significantly improves graph clustering performance compared to state-of-the-art approaches that use either signals or metadata alone, thus illustrating the interest of fusing both sources of information.
Graphs are fundamental mathematical structures in both theoretical and applied sciences, providing a natural framework to model entities (nodes) and their relationships (edges). This representation enables the development of a wide range of methodologies for classical tasks such as signal filtering, anomaly detection, clustering, and classification. Representative examples include graph clustering methods [1], [2], graph signal processing [3]- [5], and graph neural networks [6].
Most of the aforementioned tools, however, are built upon the assumption that the underlying graph topology is known. In many practical scenarios, this assumption does not hold, and the graph structure must instead be inferred from data, giving rise to the problem of graph learning. A prominent line of research in statistical learning relates the graph topology to the conditional dependency structure among variables associated with the nodes. Within this framework, Gaussian graphical models (GGM), also referred to as Gaussian Markov random fields (GMRF), assume that the data is sampled from a multivariate Gaussian distribution and allow us to estimate the graph from the support of the precision matrix (the inverse of the covariance matrix) [7]. Furthermore, Laplacian-constrained GGMs impose the estimated precision to be a Laplacian matrix [8]. This framework bridges statistics (Gaussian models) to the field of graph signal processing [9]: the resulting learned graphs are interpreted as the ones who favor smooth signal This work was supported by the MASSILIA project (ANR21-CE23-0038-01) of the French National Research Agency (ANR). representations [10], [11]. Learned eigenvectors of the precision matrix are also linked to a graph Fourier basis [4], [12].
In many real-world applications, additional side information describing node attributes is often available. For example, textual metadata such as variable descriptions can be appended to the data matrix [13]- [16]. Classical graph learning methods usually discard such auxiliary information. This could result in degraded graph estimates and limit their effectiveness in downstream tasks. To address this limitation:
• We investigate the problem of learning a Laplacian matrix within the GMRF framework [8], [14], while explicitly incorporating side information in the objective function. • We develop an efficient optimization algorithm based on the majorization-minimization (MM) principle [17], leading to a computationally efficient algorithm with closed-form solutions at each iteration step. • We illustrate the interest of our method on the realworld financial dataset. Notably, it allows for improving the clustering performance compared to state-of-the-art approaches that use either the graph signals, or the metadata only.
We consider an undirected, weighted graph represented by the triplet G = (V, E, W), where V = {1, 2, . . . , p} denotes the set of vertices (nodes), and E ⊆ {{u, v} : u, v ∈ V, u ̸ = v} is the edge set, i.e., a subset of all possible unordered pairs of nodes. The matrix W ∈ R p×p + denotes the symmetric weighted adjacency matrix satisfying W ii = 0, W ij > 0 if {i, j} ∈ E, and W ij = 0 otherwise. The combinatorial graph Laplacian matrix L is defined as L ≜ D -W, where D ≜ diag(W1) is the degree matrix.
In this work, we restrict our attention to estimate a combinatorial Laplacian graph matrix, where the set of Laplacian matrices associated with connected graphs can be defined as
The objective of the sparse graph learning under the Laplacianconstrained GMRF is to estimate the precision of a Gaussian model x ∼ N (0, Σ) under the constraint that Σ + = L ∈ S L .
To define a parameterization that respects this constraint, we will adopt the linear operator L(•) from Definition 3.2 of [14], which maps a vector w ∈ R m (m = p(p -1)/2) to a matrix L(w) ∈ R p×p . Concretely, this operator simply maps the vectorization of the upper triangular element of adjacency matrix W to the corresponding Laplacian matrix. The Laplacian set S L , defined in (1), can thus be equivalently expressed as
where the element-wise constraint w ≥ 0 enforces the nonnegativity of all edge weights. In the following, Lw will be used in place of L(w) for notional simplicity. In conclusion, instead of directly optimizing the Laplacian matrix L, we will optimize the vector w
The data matrix is denoted by X = [x 1 , x 2 , . . . , x n ] ∈ R p×n , with x i ∈ R p being one observation of a graph signal on all the p nodes. The objective is to learn the nonnegative graph weight vector w from the signals X. In many practical scenarios, additional side-information associated with the graph nodes is also available. We denote the embedding of this side-information for node i by a vector y i ∈ R d , where d may vary depending on the embedding method.
A representative example we will use in this paper’s experiments is SP500 stock dataset. In that case node represent companies, graph signals are their stock market
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