Poincaré inequalities on hyperbolic-type spaces

What the math gives to ML

The paper constructs a hyperbolic half-space metric and a dyadic spiderweb graph whose vertices have vertical parent-child edges and bounded horizontal edges. Its key transferable asset is the standard-geodesic decomposition: communication between two locations can be organized as ascent to a common scale, a short horizontal transfer at that scale, and descent. This suggests a hierarchical attention block that replaces dense all-to-all token attention with multiscale pooling, local cross-cell attention, and broadcast, while using the hyperbolic metric to select the appropriate communication scale. The construction is especially suited to long-context sequences, images, and hierarchical graphs where distant interactions should occur through coarse representatives.