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Roadmap: Versor & GBN

Versor and the Geometric Blade Network (GBN) grew from recognizing an invisible ceiling in standard Linear Algebra for Deep Learning. By moving from unconstrained projections to isometric rotations via Clifford Rotors, we preserve the geometric structure that matrices can inadvertently discard.

While the core engine is operational, the vision for a universal Geometric Computing Engine is vast. We are officially opening this roadmap to the global community.

For completed work, see the "What's Built" section in the README.

1. Core Architecture Evolution

  • Real-time Adaptive Rotors (Online SGD): Online learning framework for time-series data (e.g., Finance) where rotors adapt dynamically to streaming inputs.
  • Multi-head & Dynamic Rotors: Input-dependent rotation axes to enable a fully geometric attention mechanism.
  • Geometric Transformer (GAT): Integration of "Geometric Attention" where Q, K, V interactions are defined by Clifford blade products.

Partial progress: GeometricTransformerBlock and GeometricProductAttention already exist in layers/adapters/ and are used by the DEAP EEG task.

2. Mathematical Rigor & Stability (Open Research)

  • Convergence Proofs: Establishing UUB (Uniformly Ultimately Bounded) stability using Lyapunov Candidate Functions (\(Z\)-Energy).
  • Loss Landscape Analysis: Investigating the convexity of the Spin Group manifold to prove the absence of traditional local minima in GBN.
  • High-Dimensional Lifting & Ambiguity: Researching the "Flip Ambiguity" and gauge transformations when lifting manifolds to higher dimensions.

3. Performance & Hardware Optimization

Analysis: Why High-IPC CPUs May Outperform GPUs for GBN Inference

GBN's computational profile is fundamentally different from standard deep learning:

  • Heavy branching: The geometric product requires per-blade sign lookups (Cayley table), grade-conditional projections (popcount-based masking), and scaling-and-squaring with dynamic iteration counts. These cause GPU warp divergence.
  • Small, structured computations: A rotor sandwich product in \(Cl(3,0)\) is \(8 \times 8 = 64\) multiply-accumulates — far below the threshold where GPU parallelism pays off.
  • Low arithmetic intensity: GBN layers are memory-bound. The ratio of FLOPs to bytes transferred is low. CPUs with large caches and high IPC handle this more efficiently.
  • Sequential dependencies: exp(-B/2) via scaling-and-squaring is inherently sequential.

Planned Optimizations

  • Native CUDA Kernels: Fused kernels for \(Cl(3,0)\) and \(Cl(1,3)\) that minimize warp divergence by precomputing sign tables in shared memory.
  • JIT Compilation: Aggressive loop unrolling and metric-aware operation graph optimization — compile-time specialization for specific \((p, q)\) signatures.
  • CPU-Centric Acceleration: SIMD (AVX-512/AMX on x86, NEON on ARM) implementations of the geometric product. Exploiting the fixed sparsity pattern of the Cayley table for vectorized execution.
  • Formal Benchmarking: Systematic CPU vs. GPU latency/throughput comparison across signatures, batch sizes, and hardware targets.
  • Geometric Subspace Retrieval: Moving beyond point-to-point distance (Cosine/L2) to subspace-to-subspace relationship mapping.
  • Explainable Search Axes: Assigning geometric meaning to individual blades (e.g., temporal change vs. spatial gradient).
  • Orthogonality-based Indexing: Utilizing geometric orthogonality as a proxy for "non-similarity" for deterministic explainability.

5. Geometric Integrity & Invariance

  • Pure Invariant Design: Models inherently invariant to rotation and scaling without data augmentation.
  • Preservation of Information: Avoiding information collapse from lossy projections and investigating quantization impact on geometric purity.
  • Metric Autonomy: Enhancing Automatic Metric Search for complex datasets (Molecular, 3D Shapes, multi-modal data).