Graph-Based Correlation Matrix Generation: A Convex Optimization Approach

This work addresses the generation of theoretical correlation matrices with prescribed sparsity patterns associated to graph structures. We propose a novel convex optimization framework in which an initial matrix is projected onto an elliptope under a positive semidefiniteness constraint. Several numerical schemes are implemented and compared. The problem falls within the broader class of matrix completion, where off-diagonal entries corresponding to absent edges are fixed to zero and diagonal entries are fixed to one. Beyond this structural constraint, the approach offers greater flexibility than existing methods by allowing control over the mean of the off-diagonal entry distribution, enabling the generation of correlation matrices that better reflect realistic data. This procedure is not designed to yield a uniform distribution over the feasible set; rather, it provides a principled and tunable way to construct correlation matrices suitable for benchmarking statistical methods for graphical model inference. Theoretical guarantees on the existence of solutions are established, both in the general setting and under the additional mean constraint. Simulation studies illustrate the properties of the generated matrices with respect to graph structure. The methodology is applied to two real-world datasets from neuroscience and finance, and a comparison with GAN-based correlation matrix generation is provided.

A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding

Electroencephalography (EEG) offers noninvasive, millisecond resolution recordings of neuronal activity and is widely used in neuroscience and healthcare. Many EEG decoding pipelines rely on covariance descriptors for their robustness to noise, but such representations are sensitive to channel-wise scaling. Recent studies have therefore advocated full-rank correlation matrices as a scale-invariant alternative for EEG decoding. In this paper, we study Sliced-Wasserstein (SW) discrepancies for probability distributions on the manifold of full-rank correlation matrices. We adopt the pullback-Euclidean formulation of SW, referred to as Pullback Euclidean Metric Sliced-Wasserstein (PEMSW), and instantiate it under two recently introduced correlation geometries, \textit{i.e.}, the Off-Log Metric (OLM) and Log-Scaled Metric (LSM). This yields two Correlation Sliced-Wasserstein (CorSW) discrepancies with closed-form slicing coordinates and efficient computation through one-dimensional Wasserstein distances. Building on CorSW, we further develop a domain generalization (DG) framework for EEG decoding. Experiments on three EEG datasets demonstrate improved generalization under distribution shifts, with low training overhead and no additional inference cost. The source code is available at github.com/ChenHu-ML/CorSW.