Can Generative Models Form Groups, Rings, or Fields?
Published:
Notes to My Future Self
Why I Even Asked
While reading yet another paper on preference-aligned fine-tuning, it hit me: we keep pushing LLMs and diffusion models toward “the right answer” or “human-preferred output,” but can their outputs be organized into a real algebraic structure? If I define an operation—say, compose(text₁, text₂)—are the results closed? Could there be an associative “addition” or even an “inverse” that turns the space of generations into a group, ring, or field? It’s an intoxicating thought: import centuries of algebra straight into model interpretability.
Immediate Reality Check
The daydream popped as soon as I tried to formalize it:
- Black-box opacity – I only see inputs and outputs; the internal mapping is a 100-billion-parameter fog.
- No affine invariance – Even basic geometric invariances are fragile. Expecting full algebraic closure is orders of magnitude harder.
- Empirical hacks ≠ Theorems – Current “manifold tricks” feel like patching a leaking ship, not discovering a pristine mathematical coastline.
My Manifold-Training Detour
I spent weeks forcing a vision model to “stay on a nicer manifold.” Two stark outcomes:
- Big model, big headaches – More parameters → gradient noise → violent oscillations.
- Small model, strange attractors – Fewer parameters → it converges, but to what? The learned manifold looks like modern art; I can’t prove anything about its smoothness, let alone closure.
Half my sanity evaporated just getting that to work. The thought of elevating it to ring axioms feels absurd—at least right now.
Theory vs. Engineering
| Theoretical dream | Practical reality | |
|---|---|---|
| Goal | Stable algebraic structure | “Doesn’t explode” on training run |
| Tools | Group axioms, topology, category theory | Adam W, learning-rate decay, countless restarts |
| Guarantees | Closure, associativity, inverses | Vague empirical “it usually works on the dev set” |
I adore the dream. But when each gradient step can throw the model off a cliff, the dream feels decades away.
What I’m Taking Forward
- Keep the question alive – Even if the answer is “not yet,” asking shapes better loss functions and priors.
- Focus on invariances first – If I can’t nail affine invariance, a ring is fantasy.
- Respect the black box – Interpretability methods may one day expose hidden structure, but today they mostly expose my GPU bill.
- Document the struggle – Future-me will thank present-me for archiving both the romantic ideas and the bruises from chasing them.
Closing Line to Myself
Until manifold training feels routine, I’ll treat “LLM outputs form a field” as a philosophical postcard—nice to look at, not yet a destination in my research itinerary.