skoro v0.0.0

Mathematics

The mathematics skoro rests on, one idea to a note, linked by what each builds on. The archive’s writeups lean on these notes rather than repeating them.

What skoro is: ContextWhat skoro is: ContextThe core: The problemThe core: The problemAutomatic differentiation over framesAutomatic differentiation over framesDerivatives in tangent chartsDerivatives in tangent chartsModels built forward, for the measure theoristModels built forward, for the measure…Product manifoldsProduct manifoldsRetractions and tangent chartsRetractions and tangent charts

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Notes

  1. Automatic differentiation over frames How a straight-line program is differentiated by composing each step's derivative maps over its computation graph, forward with frames of tangents, backwards over its backpropagation graph with frames of cotangents, and forward-over-reverse for Hessian products.
  2. Derivatives in tangent charts The derivative of a map between manifolds, its gradient and its Hessian product, all read in the tangent charts a retraction provides, and why automatic differentiation computes them unchanged.
  3. Models built forward, for the measure theorist Why an abstract probability space serves modelling as an abstract vector space serves linear algebra, why the way back to coordinates is harder in probability, and how building a model forward from densities restores it, so that every random variable factors through the quantities and every expectation is an explicit integral.
  4. Product manifolds A product of manifolds is a manifold whose tangent space stacks its factors', and whose retraction acts factor by factor; derivatives in its charts are block matrices of the factors'.
  5. Retractions and tangent charts A retraction moves a point of a manifold along a tangent vector; it makes each tangent space a chart around its point, and gives the operations ⊕ and ⊖.