P-GATr: Projective Geometric Algebra Transformer

• Projective Geometric Algebra Transformer (P-GATr) is a neural architecture that unifies projective geometric algebra with transformers to process and reason about complex geometric data.
• It employs multivector representations and equivariant attention mechanisms that preserve Euclidean invariances, enabling accurate spatial reasoning.
• P-GATr demonstrates strong efficiency and generalization in applications like robotics, physics simulations, and 3D vision by leveraging built-in geometric inductive biases.

The Projective Geometric Algebra Transformer (P-GATr) is a neural architecture that combines transformer-based modeling with projective geometric algebra (PGA), aiming to encode, process, and reason about geometric data in a manner that is mathematically faithful to the structure of Euclidean (and, more broadly, Cayley-Klein) geometry. P-GATr enables spatially informed deep learning, especially in tasks requiring strong geometric inductive biases—such as robotics, physics, and 3D computer vision—by making use of a multivector representation for geometric primitives (points, lines, planes, directions, rotations) and directly integrating the algebraic operations inherent to PGA into neural network computation (Brehmer et al., 2023, Haan et al., 2023, Sun et al., 8 Jul 2025).

1. Foundations of Projective Geometric Algebra

Projective geometric algebra (PGA), typically instantiated as the Clifford algebra $\mathcal{G}_{(n,0,1)}$ for $n$-dimensional Euclidean space, augments the conventional vector space with an extra null vector $e_0$ (satisfying $e_0^2 = 0$). This degenerate direction encodes “points at infinity” and enables a homogeneous model of geometry in which translations, rotations, and all Euclidean isometries can be represented by linear, composable operations. Every geometric entity—point, direction, line, plane—can be represented as a blade (a multivector of specific grade) (Gunn, 2019, 1411.6502, Bamberg et al., 24 Aug 2024).

The algebra is constructed so that the geometric product combines both the inner (metric) and outer (wedge) products:

$ab = a \cdot b + a \wedge b$

with projective duality implemented via non-metric maps adapted for degenerate algebras (Gunn, 2022, Navrat et al., 2020).

This split extension structure is conceptually significant: every algebra element can be decomposed into a “Euclidean” (finite) part and an “ideal” (infinite, or direction-based) part, mirroring the affine geometric distinction between position and direction, and reflecting the underlying decomposition of the Clifford algebra as $$\mathrm{Cl}(V) \cong \mathrm{Cl}(V/e_0)\ltimes_\alpha \mathrm{Cl}(V/e_0)$$, where $e_0$ generates the square-zero ideal and $\alpha$ is the grade involution (Bamberg et al., 24 Aug 2024). This separation is leveraged in designing networks that can reason simultaneously about absolute positions and spatial relationships at infinity.

2. Algebraic Operations and Geometric Representations

P-GATr operationalizes all aspects of PGA in data representation and transformation. Inputs, outputs, and hidden states are represented as multivectors in the PGA; for instance, in 3D, each multivector is 16-dimensional (Brehmer et al., 2023, Sun et al., 8 Jul 2025). In this scheme:

• Points are encoded as trivectors (e.g., $x = e_0 + x_1 e_1 + x_2 e_2 + x_3 e_3$), with $e_1, e_2, e_3$ denoting the spatial basis and $e_0$ encoding homogeneity.
• Directions, rotations, and planes are represented through corresponding grades and duality.
• Complex transformations—such as isometries (rotations, translations), reflections, and screws—are efficiently handled via the geometric (sandwich) product and exponentials of bivectors:

$$X' = R X \widetilde{R}, \quad R = \exp\left(\frac{\theta}{2}B\right)$$

where $R$ is a rotor (unit even multivector), $\widetilde{R}$ its reverse, $\theta$ an angle/pitch, and $B$ the generating bivector.

The join ($\vee$) and meet ($\wedge$) operations furnish parallel-safe, coordinate-free means to compute unions and intersections of geometric entities. Duality can be implemented in software either via coordinate-free maps such as $J$ or hodge-dual maps $H$, with a bit flag tracking algebra “flavor” to maintain computational consistency (Gunn, 2022).

3. Equivariant Transformer Architecture

P-GATr replaces conventional scalar or vector tokens in transformer networks with multivectors, and its layers are crafted to be equivariant under the group of Euclidean motions $E(3)$ (rotations, translations, reflections) (Brehmer et al., 2023, Haan et al., 2023, Sun et al., 8 Jul 2025). Key architectural components include:

• Equivariant Linear Maps: Layer operations (e.g., MLPs) are constructed using only geometric algebra operations—grade projection, multiplication, contraction, and join—and act separately on each grade. These maps are rigorously characterized so that they commute with $E(3)$ actions:

$$\phi(x) = \sum_{k=0}^{d+1} w_k \langle x \rangle_k + \sum_{k=0}^{d} v_k e_0 \langle x \rangle_k$$

where $\langle x \rangle_k$ denotes projection onto grade-$k$ (Brehmer et al., 2023, Haan et al., 2023).

• Geometric Attention Mechanisms: Multihead attention is adapted to the multivector setting, with query-key-value tensors paired via the invariant inner product or more expressive operations (e.g., join, mapping to conformal algebra) to allow distance-aware, translation-equivariant interactions (Haan et al., 2023). For example:

$$\text{Attention}(q, k, v) = \sum_{i} \text{Softmax}_i\left(\frac{\langle q_{ic}, k_{jc} \rangle}{\sqrt{8 n_c}}\right) \cdot v_{jc}$$

Scalability is achieved by leveraging optimized dot-product attention over multivector features.

• Bilinear/Multilinear Geometric Operations: Bilinear maps such as the geometric product and the join are essential to achieve expressivity—especially in projective algebra, where the inner product is degenerate (does not “see” $e_0$). Mapping PGA 3-vectors to CGA 1-vectors or including join-based attention remedies this and enables attention to capture spatial proximity (Haan et al., 2023).
• LayerNorm and Nonlinearities: Normalization and nonlinear activation functions are implemented so as to preserve equivariance, typically by acting uniformly across multivector grades or through grade-specific scaling (Brehmer et al., 2023, Haan et al., 2023).

4. Sample Efficiency, Expressivity, and Practical Impact

Comparison of P-GATr and its variants reveals that a basic PGA implementation—using only inner product-based attention and the geometric product—can lack sufficient expressivity due to the degeneracy of the inner product. Improved iP-GATr variants, which add the join operation and/or map to conformal models, overcome this constraint and match or exceed the performance of both pure Euclidean and conformal-algebra-based transformers across physical simulation and geometric learning tasks (Haan et al., 2023).

Empirical results demonstrate that iP-GATr achieves strong data efficiency and generalization in settings where both rotational and translational invariance are crucial, including $n$-body simulation, hemodynamics, and block stacking (Brehmer et al., 2023, Haan et al., 2023). The built-in geometric bias reduces the need to relearn spatial invariances and enables more interpretable latent representations.

5. Hybrid Architectures and Applications

Recent work has shown that incorporating P-GATr as both an encoder and decoder in hybrid architectures (e.g., diffusion policies for robot manipulation) yields advantages in training efficiency and task success rates (Sun et al., 8 Jul 2025). In these settings:

• Raw robot observations (states, poses) are embedded as multivectors, encoded by a P-GATr stack into a spatially structured latent.
• Denoising of action sequences is performed by a conventional U-Net or Transformer in this latent space, balancing the geometric inductive bias of P-GATr with the well-understood convergence properties of classic denoisers.
• The action decoder, again a P-GATr, translates latent vectors back to geometric control commands (positions, orientations).

This approach is found to dramatically accelerate convergence and improve task performance compared to end-to-end P-GATr denoisers or classical baselines, both in simulation and on physical robots.

6. Software and Implementation Considerations

Efficient symbolic and numerical implementations of projective and conformal geometric algebra—such as SUGAR for Matlab—facilitate the development, testing, and application of P-GATr-style models (Velasco et al., 25 Mar 2024). Key considerations for practical deployment include:

• Use of efficient multivector storage and operations, with support for both numeric and symbolic computation.
• Representation of duality and regressive products via coordinate-neutral mechanisms, ensuring that all geometric primitives can be manipulated and transformed across their multiple representations (Gunn, 2022).
• Handling the degenerate (square-zero) ideal is critical: the algebra must be organized to reflect the split structure, with geometric operations carefully designed to exploit and preserve the Euclidean–ideal decomposition (Bamberg et al., 24 Aug 2024).
• Scalability on modern hardware, notably GPU architectures, is supported by casting geometric attention and meet/join operations in terms of cross-products and determinants amenable to parallel linear algebra (Skala, 2022).

7. Significance, Limitations, and Outlook

P-GATr stands as a principled realization of geometric deep learning, integrating projective geometric algebra’s uniform, coordinate-free language for spatial entities with the scalability and flexibility of transformer models. Its advantages include built-in $E(3)$-equivariance, rich and interpretable geometric representations, and robust handling of translations, rotations, and reflections.

However, care must be taken in handling the degenerate nature of PGA (requiring special duality maps and careful numerical treatment); the basic inner product’s insensitivity to translation necessitates explicit inclusion of join or conformal-algebraic primitives for expressivity (Haan et al., 2023). Computational complexity can grow for high-dimensional extensions, and seamless integration into large-scale architectures remains an active area of development.

Applications extend from robotics (manipulation, motion planning) (Sun et al., 8 Jul 2025) and computer vision to physical simulation and engineering analysis (Brehmer et al., 2023, Haan et al., 2023, Velasco et al., 25 Mar 2024). Continued integration of geometric algebra frameworks into neural architectures promises improved inductive bias, better generalization, and enhanced interpretability for complex tasks involving geometric data.