CITATION.cff

cff-version: 1.2.0
message: "If you use this software, please cite it using the metadata from this file."
type: software
title: "Lenticulum.jl: learning relations on factor graphs"
version: "20261008-proto"
authors:
  - family-names: Boigk
    given-names: Daniel
license: MIT
repository-code: "https://github.com/MathStruct/Lenticulum.jl"
url: "https://mathstruct.github.io/Lenticulum.jl/dev/"
keywords:
  - implicit learning
  - diffusion models
  - factor graphs
  - message passing
  - Bayesian lenses
  - category theory
  - probabilistic programming
  - Julia
abstract: >-
  Lenticulum.jl is a Julia library for learning relations instead of functions.
  A neural network learns a map from inputs to outputs and runs in one direction.
  Lenticulum learns a relation on a joint space and decides at query time which
  variables are observed and which are inferred, so one trained model answers
  "y given x", "x given y" and "complete what is missing", including queries with
  several answers.


  The main learner is an implicit diffusion model. A denoising diffusion model,
  trained in the standard way, defines a field whose stable roots are the
  relation, and a query is answered by root finding. Gradients for training
  through a query come from the implicit function theorem, as in deep equilibrium
  models. Answers come with Laplace uncertainty: a Gaussian per answer, or a
  mixture over all answers. Several learned relations combine as a product of
  experts, so their intersection is found without retraining.


  For statisticians, learned relations are factors in a factor graph. They
  exchange Gaussian, categorical, mixture and particle beliefs by message
  passing, alongside linear-Gaussian and constraint factors, with a Bethe free
  energy, and they convert to RxInfer.jl's distribution types.


  For category theorists, the design follows compositional accounts of Bayesian
  inference. Each factor is a parameterised statistical game built from a
  Bayesian lens, and which channels are observed (the polarity) is part of its
  type.


  The repository includes executable tutorials, a documentation site and a
  linked notes vault on the theory, references and open problems. Lenticulum is
  a research prototype under active development, with features added and
  improved continuously. Each archived version is a snapshot of that work, and
  the concept DOI always resolves to the latest version.