dense_apply_derivatives

t3toolbox.backend.sampling_derivatives.dense_apply_derivatives(ww, pp, T, order)

def dense_apply_derivatives(
        ww:     typ.Sequence[NDArray],  # probe vectors X,        len=d, elm_shape=(Ni,)
        pp:     typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=(Ni,)
        T:      NDArray,                # dense tensor, shape=(N0,...,N(d-1))
        order:  int,                    # highest derivative order
) -> NDArray:                           # apply-derivative jets, shape=(order+1,)

Exact dense symmetric apply derivatives, by the all-modes multilinear subset expansion (oracle).

apply(X+sP) is a polynomial in s of degree d, so the t-th derivative at s=0 is

y^(t) = t! * sum_{|S|=t, S subset of {0..d-1}} T contracted with {p_j: j in S, x_j: else},

here contracting every mode (no free mode). Enumerates the size-t subsets S – testing only, unstacked.

Parameters:

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

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

• T (NDArray)

• order (int)

Return type:

NDArray