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From my morning chat with openai. Is there anything actually going on with hypercomplex neural networks?
Plato’s cave?
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This thinking is already happening, but not usually under the single label “octonionic AI.” It is scattered across several research streams. The “octonionic atlas” idea sits at the intersection of these fields:
1. Hypercomplex neural networks
This is the closest direct match.
Researchers are building neural networks that use complex numbers, quaternions, octonions, and related number systems instead of ordinary real numbers. The motivation is that these richer numbers can bind several related dimensions together, which may improve parameter efficiency and representation power. A 2025 review explicitly frames quaternion, octonion, and higher-dimensional algebra networks as a frontier in deep learning. (ScienceDirect)
The most direct example is Deep Octonion Networks, proposed in 2019, which tested octonion-valued layers against real, complex, and quaternion models and reported better convergence and classification performance in their experiments. (arXiv)
In plain English:
Some researchers are already asking whether AI should compute with richer “number packages” instead of isolated numbers.
2. Quaternion neural networks
This is a more mature cousin of octonion networks.
Quaternions are 4-dimensional numbers often used for 3D rotation, graphics, robotics, and signal processing. In AI, quaternion networks try to keep related channels tied together rather than treating each feature as separate. Recent work continues to explore quaternion models for parameter efficiency and structured representation. (arXiv)
In plain English:
Before octonions, researchers are already using quaternions as a way to make AI understand rotation, orientation, and multi-channel structure more naturally.
3. Geometric deep learning
This is the broad field your article belongs to philosophically.
Geometric deep learning asks:
What if neural networks should respect the shape of the data?
Instead of assuming all data lives on a flat spreadsheet-like grid, geometric deep learning studies data on graphs, groups, manifolds, spheres, meshes, molecules, social networks, and curved spaces. A major survey describes geometric deep learning through group equivariant and gauge equivariant neural networks, including models on arbitrary manifolds. (Springer)
In plain English:
This field says AI should not force every problem into a flat box. The model should match the geometry of the thing it is learning.
4. Equivariant neural networks
This is one of the most important practical versions of the idea.
An equivariant neural network is designed so that when the input changes in a predictable way, the output changes in the corresponding predictable way. For example, if you rotate an image, the model’s internal understanding rotates consistently too.
Maurice Weiler, a researcher in geometric deep learning, explains that equivariant models can generalize learned patterns across transformed inputs, reduce parameter counts, improve data efficiency, and reduce overfitting. (Maurice Weiler)
In plain English:
Instead of making AI relearn the same object at every angle, we teach the AI that rotation is part of the rules of the world.
That is very close to the spirit of your article: build the structure of reality into the model instead of forcing the model to rediscover it by brute force.
5. Gauge-equivariant neural networks
This is a deeper, more physics-like version.
Gauge-equivariant networks borrow ideas from physics, especially gauge theory, where local coordinate systems can change but the underlying physical reality remains consistent. The survey on geometric deep learning discusses gauge-equivariant convolutional networks on manifolds using principal bundles and vector bundles. (Springer)
In plain English:
This is AI that understands that different local “maps” can describe the same underlying object.
That connects strongly to the “atlas” idea. An atlas is not one map. It is many local maps stitched together.
6. Lie group and symmetry-based AI
Another major stream is using Lie groups and Lie algebras in neural networks. These are mathematical tools for describing continuous transformations: rotation, scaling, translation, Lorentz transformations, affine transformations, and other symmetries.
Recent papers explore equivariant neural networks on Lie groups, reductive Lie groups, and general linear symmetries. These are being applied to particle physics, molecular modeling, shape recognition, drone state estimation, and other domains where symmetry matters. (arXiv)
In plain English:
Researchers are teaching AI the legal moves of the world: rotate, shift, stretch, transform — but preserve the underlying meaning.
7. Manifold learning and representation geometry
This is another nearby area.
The idea is that high-dimensional data may actually lie on lower-dimensional curved surfaces called manifolds. Faces, language meanings, biological forms, and physical states may not be random clouds of points; they may lie on shaped surfaces.
Geometric deep learning and gauge-equivariant models directly build on this manifold idea. (Springer)
In plain English:
AI is trying to learn the shape of meaning, not just memorize examples.In plain English:
AI is trying to learn the shape of meaning, not just memorize examples.
8. Physics-inspired AI
This is where your article’s curvature, torsion, symmetry, and algebra language most strongly resonates.
Physics-inspired AI tries to build models that respect conservation laws, symmetries, geometry, and field-like structure. Examples include models for molecules, fluids, particle physics, robotics, quantum systems, and lattice gauge theories. Lattice gauge equivariant convolutional networks, for instance, are designed so models can operate on non-Abelian lattice gauge theories without violating gauge symmetry. (OpenReview)
In plain English:
Instead of letting AI invent bad physics, researchers build some physics into the AI from the beginning.
Where this is happening institutionally
This thinking is especially active around:
Geometric deep learning communities connected to researchers such as Michael Bronstein, Joan Bruna, Taco Cohen, Max Welling, Petar Veličković, Maurice Weiler, and others. A course on geometric deep learning has been taught by Bronstein, Bruna, Cohen, and Veličković, showing how central this topic has become in modern AI theory. (YouTube)
Scientific machine learning labs, especially those working on molecules, materials, physics simulations, protein geometry, robotics, and particle physics.
Hypercomplex neural network researchers, who are directly exploring quaternion, octonion, Clifford, and other algebra-valued networks. (ScienceDirect)
The key distinction
The octonionic part is still more speculative and less mainstream.
The geometric part is very real and very active.
So I would place the article like this:
Mainstream today: geometric deep learning, equivariant networks, manifold learning, symmetry-aware AI.
Emerging today: hypercomplex neural networks, especially quaternion and Clifford-based models.
More speculative but real: octonion-valued neural networks and octonionic geometry as a deep AI framework.
The simplest answer
This thinking is already taking place wherever researchers are asking:
“Can we make AI smarter by giving it the right geometry, instead of just giving it more parameters?”
That includes geometric deep learning, equivariant neural networks, gauge-equivariant networks, Lie-group neural networks, manifold learning, scientific machine learning, and hypercomplex neural networks.
Your “octonionic atlas” idea is best understood as a bold extension of that movement:
from flat AI, to geometric AI, to algebraically structured AI.
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-notary sojac –
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