Relative-Error TV Distance Approximation for Graphical Models

Develop polynomial-time algorithms that approximate, to a prescribed relative error, the total variation distance between two probability distributions specified as graphical models.

Background

The current work resolves relative-error TV distance approximation for multivariate Gaussians via reduction and discretization techniques. Graphical models (e.g., Bayesian networks and Markov random fields) are pervasive representations of high-dimensional distributions.

The authors explicitly identify efficient relative-error TV distance approximation for graphical models as an open direction, indicating the need for algorithmic techniques that can handle structured dependencies beyond the product or Gaussian settings.

References

Several directions remain open; including TV distance estimation for general log-concave distributions, graphical models, and Gaussian-perturbed distributions; and approximations for other notions of distance such as the Wasserstein distance.

— Approximating the Total Variation Distance between Gaussians  (2503.11099 - Bhattacharyya et al., 14 Mar 2025) in Conclusion

It also leaves open the more structural question of comparing latent-tree models with different underlying trees, which is closely related to learning and testing latent-tree Ising models.

— On Deterministically Computing Total Variation Distance via Zonotope Compression  (2609.24235 - Fu, 21 Sep 2026) in Section 1, paragraph following Theorem 1.3 (Same-topology latent-tree Ising)