Overview
This presentation demonstrates how functional analysis provides a unified geometric language for understanding two seemingly different domains: signal processing and hyperbolic deep learning.
Part I: Signal Processing
Establishes that image compression (DCT) and analytic signal theory (Hilbert transform, AM-FM decomposition) are instances of orthogonal projection onto closed subspaces of L²(𝕋).
Part II: Hyperbolic Geometry
Develops the Riemannian geometry of the Poincaré ball, proving that hyperbolic neural networks naturally encode hierarchical structure through their metric tensor.
Presentation Slides
Full Technical Report
Complete rigorous mathematical treatment with all proofs and theorems.
Download Report →Interactive Visualizations
Signal Processing Notebook
Hyperbolic Geometry Notebook
Visual Media
3D Analytic Signal Visualization
The analytic signal traces a spiral in the complex plane, encoding both amplitude (distance from origin) and instantaneous frequency (rotation rate).
Tangent Space Geometry
Visualization of the tangent space structure in hyperbolic geometry showing how the Riesz isomorphism converts gradients to directions.
