Tabular data is widely applied in critical domains, where tree-based models remain dominant, yet their performance is often constrained by manual heuristic splitting criteria and limited ensemble diversity. While hyperbolic geometry naturally fits the hierarchical topology of tabular data, existing hyperbolic algorithms are hindered by complex Riemannian optimization and the difficulty of defining subtree priors in non-Euclidean spaces. To address these challenges, we propose HOT, a hyperbolic random forest framework based on data decoupling. First, we leverage intrinsic geometric structures and LLM reasoning to guide subtree construction, replacing traditional manual heuristics with geometric-semantic priors. Second, we introduce an orthogonal hyperbolic subspace projection mechanism via tangent space isometry, effectively decoupling feature dependencies and maximizing ensemble diversity. Finally, we establish an end-to-end collaborative training paradigm where gradient residuals from a graph network directly guide the growth of new hyperbolic trees, bypassing expensive iterative optimization. Experiments on 13 benchmark datasets demonstrate that HOT significantly outperforms state-of-the-art baselines in both computational efficiency and predictive performance.

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

Multimodal learning beyond two modalities commonly leverages a specific modality (e.g., text) to bind other modalities. However, how to establish a more balanced representation space that approximates shared semantics while respecting the holistic geometry of $n$-modal data remains challenging. In this work, we present BaryBind, which aims to transport the specific modality towards the Wasserstein barycenter (WB) optimized across all modalities and introduces a volumetric alignment objective to establish a unified semantic space around the WB embedding. Specifically, we project specific modalities to the WB, which minimizes the average Wasserstein distances to multimodal distributions and serves as the anchor for subsequent alignment. We then construct a barycenter simplex, whose volume is taken as a similarity metric for global alignment centered at the WB. Extensive experiments show that BaryBind achieves competitive performance in text-video-audio retrieval, classification, videoQA, and cross-modal generation tasks, along with robustness under modality absence and scalability to more than three modalities.

We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class $\mathcal{F}$ and every $\gamma>0$, uniform convergence at scale $\gamma$, agnostic learnability at scale $\gamma/2$, and finiteness of the fat-shattering dimension at every scale $\gamma'>\gamma$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press ’99) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS ’98), which incur such a loss. The key technical ingredient is a direct bound on empirical $\ell_\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $\gamma$: an $O(\log^2 n)$ bound holds already at scale $\gamma/2$, while an $O(\log n)$ bound holds at scale $2\gamma$. We further show that the $O(\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM ’97) and Rudelson and Vershynin (Ann. of Math. ’06). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML '26). We conclude with several open questions on quantitative sample complexity and evaluability.