t3_entries_derivatives

t3toolbox.backend.sampling_derivatives.t3_entries_derivatives(index, pp, x, order)

def t3_entries_derivatives(
        index:  NDArray,                # int, shape=(d,)+W -- the grid points (one multi-index per W sample)
        pp:     typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
        x:      typ.Tuple[
            typ.Sequence[NDArray],      # tucker_cores. len=d, elm_shape=C+(nUi,Ni)
            typ.Sequence[NDArray],      # tt_cores.     len=d, elm_shape=C+(rLi,nUi,rR(i+1))
        ],                              # = TuckerTensorTrain.data
        order:  int,                    # highest derivative order
) -> NDArray:                           # entries-derivative jets, shape=(order+1,)+W+C

Symmetric derivatives of an entry of a Tucker tensor train, in one repeated direction.

The entries analog of t3_apply_derivatives() – apply-derivatives with the up-index frame jet from slicing Tucker-core fibers at index (order 0) and contracting P (order 1). Returns y^(t) = d^t/ds^t apply(X, e_{index} + s P)|_0 for t=0..order: the Taylor data of the tensor’s multilinear extension at grid corner index, in direction P. Index 0 is the ordinary entry X[index]. Stacks order + W + C. Verified vs dense_entries_derivatives().

Parameters:

• index (NDArray)

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

• x (t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Sequence[NDArray], t3toolbox.backend.common.typ.Sequence[NDArray]])

• order (int)

Return type:

NDArray