apply_derivatives_model#

t3toolbox.fitting.apply_derivatives_model(geometry, x, ww, pp, order, residual, weight=None, regularizer=None)#
def apply_derivatives_model(
        geometry,                            # MANIFOLD / COREWISE
        x:          t3.TuckerTensorTrain,    # the current point
        ww:         typ.Sequence[NDArray],   # probe vectors X,        len=d, elm_shape=W+(Ni,)
        pp:         typ.Sequence[NDArray],   # perturbation vectors P, len=d, elm_shape=W+(Ni,)
        order:      int,                     # highest derivative order
        residual:   NDArray,                 # RAW r = apply_derivatives(x) − y, shape (order+1)+W+C
        weight:     typ.Optional[typ.Any] = None,  # ORDER-only residual weight ω, (order+1,); None = 1
        regularizer: typ.Any = None,         # optional regularizer, e.g. optimizers.IdentityRegularizer(λ)
) -> typ.Union[GaussNewtonModel, UniformGaussNewtonModel]:

The Gauss-Newton model of an apply-derivatives least-squares objective at x: the symmetric directional derivatives (orders 0..order) of the all-modes apply, in direction P.

The all-modes apply contracts every mode into a scalar, so weight is order-only (a per-mode weight is a structural error – mode weighting is defined only for probe; see probe_derivatives_model()).

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.fitting as fitting
>>> np.random.seed(0)
>>> x = t3.TuckerTensorTrain.randn((6, 7, 8), (2, 3, 2), (1, 2, 2, 1))
>>> ww = [np.random.randn(15, N) for N in (6, 7, 8)]    # 15 samples
>>> pp = [np.random.randn(15, N) for N in (6, 7, 8)]    # one direction P per sample
>>> r = np.random.randn(4, 15)                          # RAW residual jet, (order+1, W) for order 3

A per-order weight ω balances the wildly-different-magnitude orders – it weights the objective ½‖ω⊙r‖² inside the kind; the gradient stays a gauged Riemannian tangent, and the Gauss-Newton quadratic form pᵀHp = ‖J p‖² agrees with the Hessian action:

>>> model = fitting.apply_derivatives_model(t3m.MANIFOLD, x, ww, pp, 3, r, weight=[1.0, 0.5, 0.3, 0.2])
>>> print(model.gradient.is_gauged())
True
>>> p = t3m.MANIFOLD.randn(model.frame)
>>> bool(np.allclose(float(model.gn_quadratic(p)), float(p.corewise_inner(model.gn_hessian(p)))))
True
Parameters:

  • x (TuckerTensorTrain)

  • ww (t3toolbox.backend.common.typ.Sequence[NDArray])

  • pp (t3toolbox.backend.common.typ.Sequence[NDArray])

  • order (int)

  • residual (NDArray)

  • weight (t3toolbox.backend.common.typ.Optional[t3toolbox.backend.common.typ.Any])

  • regularizer (t3toolbox.backend.common.typ.Any)

Return type:

t3toolbox.backend.common.typ.Union[GaussNewtonModel, UniformGaussNewtonModel]