Gaussian CBO on Bures–Wasserstein Space

Gaussian CBO on Bures–Wasserstein space runs consensus-based optimisation (CBO) for Gaussian variational inference directly on the Bures–Wasserstein manifold, with no reference measure, no linearisation and no regularisation of the objective.

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🎯 The problem

Borghi & Carrillo (ICML 2026) brought CBO to Gaussian variational inference: a swarm of Gaussian particles, each with a mean $m$ and covariance $\Sigma$, searches for the best Gaussian approximation of a target. To keep the objective finite near singular covariances, they work in a linearisation of the Bures–Wasserstein manifold and introduce eigenvalue clipping, a regularised entropy and a penalty value for singular particles. Their outlook asks whether the dynamics can be run in the full Bures–Wasserstein space instead.

💡 The idea

The Bures–Wasserstein boundary is only the singularity of $a \mapsto a^2$ at the origin. Writing each particle through a square root $\Sigma = AA^\top$ removes it: the flat metric on square roots induces the Bures–Wasserstein metric exactly, and singular Gaussians become an ordinary subvariety rather than a boundary.

The tangent space of the square roots splits into a horizontal part, which moves the Gaussian, and a vertical part, which does not. Ambient noise wastes about half its budget on the vertical part. The horizontal space is exactly the set of matrices $SA$ with $S$ symmetric, so noise generated as $SA$ is horizontal by construction, with no projection and no eigenvalue denominators. Weighting its coordinates recovers Borghi & Carrillo’s anisotropic noise in full.

As a result:

• each Euler step is a congruence, so the rank of $\Sigma$ is preserved and the boundary is repelled;
• the coefficients stay finite at rank-deficient covariances;
• no regularisation length needs to be chosen by hand.

📈 Results

On Borghi & Carrillo’s released $d = 10$ benchmark (fitting a Gaussian to a 5-component mixture, 30 instances, 100 particles):

The repository contains the full proposal and the convergence report.