UT3Tangent.probe_transpose

static t3toolbox.uniform_manifold.UT3Tangent.probe_transpose(ztildes, ww, frame, sum_over_probes=False)

def probe_transpose(
        ztildes,                   # probe residuals, len=d, ith elm_shape=W+K+C+(Ni,)
        ww,                        # probe vectors,   len=d, ith elm_shape=W+(Ni,)
        frame:  ubv.UT3Frame,      # the orthogonal frame the tangent attaches at
        sum_over_probes:  bool = False,   # True: sum the probe stack W (Gauss-Newton 𝒥ᵀr); False: keep it
) -> 'UT3Tangent':  # tangent stack W+K (sum_over_probes=False) or K (True); frame stack C

Apply the transpose 𝒥ᵀ of the probe map to residuals; returns a UT3Tangent at frame. The adjoint of probe(). Uniform mirror of probe_transpose(): the residuals live in the forward probe space (W+K+C); sum_over_probes=False keeps the probe stack W as the result’s tangent stack (W+K), True sums it (K, the Gauss-Newton 𝒥ᵀr). The bare 𝒥ᵀ (no gauge projector Π).

See also

probe, apply_transpose, probe_derivatives_transpose

Examples

The defining property – the adjoint identity <r, 𝒥v> = <𝒥ᵀr, v> (summed over probes):

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.uniform_tucker_tensor_train as ut3
>>> import t3toolbox.uniform_manifold as ut3m
>>> np.random.seed(0)
>>> x = ut3.UniformTuckerTensorTrain.from_t3(t3.TuckerTensorTrain.randn((10, 11, 12), (5, 6, 4), (1, 2, 3, 1)))
>>> v = ut3m.UNIFORM_COREWISE.randn(ut3m.UNIFORM_MANIFOLD.frame(x))
>>> ww = [np.random.randn(2, N) for N in (10, 11, 12)]              # probe stack W=(2,)
>>> zz = v.probe(ww)
>>> r = [np.random.randn(*np.asarray(z).shape) for z in zz]
>>> JTr = ut3m.UT3Tangent.probe_transpose(r, ww, v.frame, sum_over_probes=True)
>>> lhs = sum(float(np.sum(ri * np.asarray(zi))) for ri, zi in zip(r, zz))
>>> print(bool(abs(lhs - float(JTr.corewise_inner(v))) < 1e-9))
True
>>> keep = ut3m.UT3Tangent.probe_transpose(r, ww, v.frame, sum_over_probes=False)
>>> print(keep.tangent_stack_shape, JTr.tangent_stack_shape)        # keep W (=W+K) vs sum W (=K)
(2,) ()

Parameters:

• frame (UT3Frame)

• sum_over_probes (bool)

Return type:

UT3Tangent