IdentityRegularizer
===================

.. toctree::
   :hidden:

   /autoapi/t3toolbox/backend/regularization/IdentityRegularizer.value
   /autoapi/t3toolbox/backend/regularization/IdentityRegularizer.gradient
   /autoapi/t3toolbox/backend/regularization/IdentityRegularizer.hessian
   /autoapi/t3toolbox/backend/regularization/IdentityRegularizer.quadratic

.. py:class:: t3toolbox.backend.regularization.IdentityRegularizer

   Bases: :py:obj:`Regularizer`


   Identity (Tikhonov) regularization ``ρ(X) = ½·strength·‖X‖²`` in the geometry's own tangent metric.

   On the **manifold** this is the Hilbert-Schmidt ridge ``½λ‖X‖²_HS`` -- ``H_R = λ·Π`` (``= λp`` on a
   gauged tangent), which is the **exact** Riemannian Hessian (``∇ρ = X`` is fully tangent, so the
   curvature term vanishes). On **corewise** it is core weight-decay ``½λ Σ‖core_i‖²``, which also makes
   the gauge-singular corewise Gauss-Newton Hessian strictly positive-definite -- a better-conditioned
   Newton system (though CG already converges on the singular ``H`` since the gradient ``g = 𝒥ᵀr`` lies in
   ``range(H)``). ``X_ref = 0``. The ``point_tangent`` = ``v_X`` construction (the attachment point as a
   single gauged tangent term) is in ``dev/regularization_design.md`` §4.


   .. py:attribute:: strength
      :type:  float

Methods
-------

.. autoapisummary::

   t3toolbox.backend.regularization.IdentityRegularizer.value
   t3toolbox.backend.regularization.IdentityRegularizer.gradient
   t3toolbox.backend.regularization.IdentityRegularizer.hessian
   t3toolbox.backend.regularization.IdentityRegularizer.quadratic