Generating Correlation Matrices with Graph Structures Using Convex Optimization

TL;DR

This work addresses generating theory-based correlation matrices that conform to prescribed graph sparsity while controlling the mean of off-diagonal entries. It introduces a convex quadratic program that minimizes $\frac{1}{2}\|\mathbf{C} - \bar{\mathbf{C}}\|_F^2$ subject to $\mathbf{C} \succeq 0$, $\mathrm{diag}(\mathbf{C})=1$, sparsity constraints $c_{ij}=0$ for non-edges, and a mean constraint $\frac{1}{2|\mathcal{E}|}\sum_{i\neq j} c_{ij} \ge b$, with $\bar{\mathbf{C}}$ from data. The method supports non-chordal graphs, implements a practical feasibility remedy via a diagonal shift, and demonstrates competitive performance against existing approaches across several graph models, including neuroscience-inspired data. This yields more realistic synthetic correlation matrices for benchmarking graph-inference methods, at the cost of higher computational effort. Overall, the approach provides a flexible, PSD-guaranteed framework for graph-structured correlation matrix generation with tunable mean behavior, suitable for simulations in neuroscience and related fields.

Abstract

This work deals with the generation of theoretical correlation matrices with specific sparsity patterns, associated to graph structures. We present a novel approach based on convex optimization, offering greater flexibility compared to existing techniques, notably by controlling the mean of the entry distribution in the generated correlation matrices. This allows for the generation of correlation matrices that better represent realistic data and can be used to benchmark statistical methods for graph inference.

Figures (4)

• Figure 1: Density of non-diagonal, non-zero entries in generated correlation matrices using different methods (diagonal dominance, partial orthogonalization, and our method) compared to the correlation matrix obtained from rat fMRI data.
• Figure 2: Proportion of successfully finding a correlation matrix $\mathbf{C} \in \mathbb{R}^{51 \times 51}$ per bin over different threshold values of $b$, using an Erdős-Rényi pattern. White areas indicate cases where no solution was found.
• Figure 3: Density of non-zero, non-diagonal elements in the correlation matrix generated by our algorithm for different random graph models and the Chordal graph, given a graph edge density of $d=0.5$ for $\mathbf{C} \in \mathbb{R}^{51 \times 51}$. Results are obtained over 50 runs, with a threshold constraint of $b = 0.2$.
• Figure 4: Execution time (in seconds) of our method for computing the correlation matrix $\mathbf{C} \in \mathbb{R}^{51 \times 51}$, averaged over 50 runs for each box plot (representing a different graph type) as a function of the density of non-zero, non-diagonal elements in the correlation matrix. The parameter $b$ is set to $-1$.