The information geometry of product-reference discrete diffusion: Interaction growth complexity and optimal scheduling

Abstract:We study a class of product-reference diffusion algorithms for sampling from a discrete distribution. We show that their sampling performance can be characterized using a path-based measure of data geometry that we call the interaction growth complexity (IGC). We show that a bivariate IGC kernel gives an exact representation of both the KL discretization error and a simple one-step upper bound. The simpler univariate IGC density can be used to study the effect of stepsize choices on the iteration complexity required to obtain $\epsilon$-accurate samples in KL divergence. Samplers that traverse the path with equi-spaced steps in log-squared-reliability-odds have performance that depends on the aggregate IGC mass, whereas refined choices of stepsizes have a lower complexity depending on a square-root functional. In the fine-grid limit, both of these characterizations become sharp. We also allow general product reference distributions and show that the reference law can substantially reshape the IGC profile and the resulting sampling complexity; in particular, references far from both the uniform and the data marginals can yield dimension-dependent improvements. Finally, the aggregate IGC mass admits bounds in terms of total correlation and dual total correlation, thereby connecting the pathwise geometry to classical measures of multivariate dependence.

Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Statistics Theory (math.ST)

arXiv:2608.28949 [stat.ML]

(or arXiv:2608.28949v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2608.28949 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.