Lenticulum — theory vault

Lenticulum.jl learns relations instead of functions: a model of a joint space $Z=X\times Y\times U$ that decides at query time which coordinates are inputs $X$, outputs $Y$ and latents $U$. This vault is the larger half of the project: the mathematics, the papers, the design decisions, and an honest record of what does not work yet. Pick the door that matches what you already know.

Coming from machine learning

You know neural networks, backpropagation, perhaps diffusion models and deep equilibrium models. The short version: a trained denoiser defines a vector field whose stable roots are a relation; inference is root-finding with the inputs clamped, as a DEQ is evaluated; the backward pass is the implicit function theorem, one adjoint solve, nothing unrolled. The networks are small (a few thousand parameters over a handful of coordinates), and any Lux model with any AD backend works.

1. Implicit Learners — what “learning a relation” means, and the three families
2. Implicit Diffusion Learners — a diffusion model as a relation; the residual field
3. Backpropagation through Implicit Inference — the adjoint, and a parabola learned from a circle
4. DEQ as a Relation — the same idea for equilibrium models
5. backends — Zygote, Enzyme or Reactant; a 5k-parameter MLP end to end

Or start with code: the tutorials (also Jupyter notebooks).

Coming from statistics or robotics

You know Bayesian inference, Gaussian posteriors, perhaps factor graphs and GTSAM. The short version: every factor is a joint model whose conditioning direction is chosen per query; message passing on a factor graph is Bayesian inversion of each factor; the exact results are on the linear-Gaussian fragment, and the learned factors (diffusion, equilibrium, adversarial) plug into the same graph.

1. Factor Graphs and Everything is a Factor — the setting
2. Beliefs — what flows along the edges: Gaussian, Dirac, samples
3. The Linear Gaussian Chain — the case where everything is exact and checked
4. Messages are Inversions and Bethe Free Energy — inference and its objective
5. SLAM and Sensor Fusion — the application that motivates the design

Coming from category theory

You know lenses, Para, Markov categories, perhaps AutoBayes’ statistical games. The short version: a factor is a parameterized statistical game once a polarity is chosen, and a Bayesian lens after that; acausal composition is a hypergraph category; the code mirrors these definitions type for type. Notation note: this vault’s inputs $X$ are AutoBayes’ $Y$ and vice versa (Channels and Polarity §“Notation”).

1. Factors are Parameterized Statistical Games — the central correspondence
2. Channels and Polarity — open models, cups and caps, and why direction is chosen late
3. Inversions and Bayesian Lenses — what inference is, categorically
4. Acausal Composition is a Hypergraph Category — how factors compose
5. The Implicit Diffusion Factor as a Statistical Game — a learned factor, checked against the definitions

The general theory these notes build on is in the CT-ML wiki.

Everything else

• Map of Content — every note, in reading order.
• Start Here — how the vault is organised, its conventions, and how to read it in Obsidian.
• README — the project pitch.
• API documentation — the Julia packages themselves, built with Documenter.

The vault is written for Obsidian and rendered here with Quartz, including its tikz diagrams, which are compiled to SVG at build time. The graph view is Quartz’s rather than Obsidian’s.