Tree-Sliced Orlicz Integral Probability Metric

Tree-sliced distances have emerged as scalable alternatives to Sliced Wasserstein distances by projecting probability measures onto tree metric spaces and exploiting closed-form transport on trees. Existing tree-sliced constructions, however, are largely built around first-order Wasserstein geometry. While this choice yields efficient computation, it fixes the discrepancy to an $L^1$-type aggregation of subtree mass imbalances and provides limited control over the geometry used in downstream optimization. We propose Tree-Sliced Orlicz IPM (TS-Orlicz), a tree-sliced framework that replaces the tree-level $W_1$ discrepancy with an Orlicz integral probability metric. Building on the tractable formulation of Orlicz IPMs on trees, TS-Orlicz aggregates Orlicz-induced discrepancies over random tree systems while preserving the scalability of tree-sliced computation. By varying the underlying Orlicz function, the framework recovers $L^p$-type behavior and also supports non-polynomial, tail-sensitive geometries that emphasize large subtree discrepancies. We show that TS-Orlicz preserves key properties of tree-sliced constructions and admits efficient computation via closed-form special cases and one-dimensional scalar optimization. We further extend the framework to probability measures on hyperspheres. Experiments across Euclidean and spherical settings, including gradient flows, self-supervised learning, and diffusion-based generative modeling, show that Orlicz-induced tree-sliced discrepancies achieve competitive or improved performance over sliced and tree-sliced baselines while maintaining low computational cost.