fitting
=======


.. py:module:: t3toolbox.backend.fitting

.. autoapi-nested-parse::

   Sampling-kind primitives for the geometry-generic Gauss-Newton fitting model.

   The Gauss-Newton model ``m(p) = c + gᵀp + ½ pᵀ(JᵀJ)p`` of a least-squares objective factors over two
   *independent* choices, and the fitting layer is generic in both:

     * the sampling **kind** (apply / entries / probe) -- the bare single-sample Jacobian ``𝒥`` / its
       transpose ``𝒥ᵀ`` (from :py:mod:`t3toolbox.backend.probing`), plus the kind-specific ``‖·‖²``
       reduction over the sample stack; and
     * the **geometry** (manifold / corewise) -- the gauge projection ``Π`` (``geometry.project``), with
       the Riemannian forward ``J = 𝒥∘Π`` and gradient ``Jᵀr = Π∘𝒥ᵀr``.

   This module supplies the **kind** half: a :py:class:`SamplingKind` descriptor bundling the bare probing
   primitives + the reduction per kind (:py:data:`APPLY` / :py:data:`ENTRIES` / :py:data:`PROBE`). The
   **geometry** half lives in the geometry (:py:data:`t3toolbox.manifold.MANIFOLD` /
   :py:data:`~t3toolbox.manifold.COREWISE`); the frontend :py:class:`t3toolbox.fitting.GaussNewtonModel`
   composes the two. The §6.3 corewise substitution ``(U,O,P,Q) -> (U,G,G,G)`` is now nothing but
   ``CorewiseGeometry.frame`` -- there is no separate corewise backend.

   ``probe`` is vector-valued (one free mode per probe), so its residual / forward output is a sequence of
   ``d`` arrays and its reduction sums the free mode too; apply / entries are the scalar all-modes special
   case. The objective constant ``c = ½‖r‖²`` is just ``½ · sumsq(r)`` (the same reduction as the model's
   quadratic term ``½‖𝒥Πp‖²``), so the kind needs no separate objective function.



Attributes
----------

.. autoapisummary::

   t3toolbox.backend.fitting.APPLY
   t3toolbox.backend.fitting.ENTRIES
   t3toolbox.backend.fitting.PROBE


Classes
-------

.. toctree::
   :hidden:

   /autoapi/t3toolbox/backend/fitting/SamplingKind

.. autoapisummary::

   t3toolbox.backend.fitting.SamplingKind


Functions
---------

.. toctree::
   :hidden:

   /autoapi/t3toolbox/backend/fitting/sumsq_over_samples
   /autoapi/t3toolbox/backend/fitting/sumsq_over_probes
   /autoapi/t3toolbox/backend/fitting/block_sumsq_over_samples
   /autoapi/t3toolbox/backend/fitting/block_sumsq_over_probes
   /autoapi/t3toolbox/backend/fitting/probe_kind
   /autoapi/t3toolbox/backend/fitting/apply_derivatives_kind
   /autoapi/t3toolbox/backend/fitting/entries_derivatives_kind
   /autoapi/t3toolbox/backend/fitting/probe_derivatives_kind

.. autoapisummary::

   t3toolbox.backend.fitting.sumsq_over_samples
   t3toolbox.backend.fitting.sumsq_over_probes
   t3toolbox.backend.fitting.block_sumsq_over_samples
   t3toolbox.backend.fitting.block_sumsq_over_probes
   t3toolbox.backend.fitting.probe_kind
   t3toolbox.backend.fitting.apply_derivatives_kind
   t3toolbox.backend.fitting.entries_derivatives_kind
   t3toolbox.backend.fitting.probe_derivatives_kind


Module Contents
---------------

.. py:data:: APPLY

.. py:data:: ENTRIES

.. py:data:: PROBE