utv_weighted_norm#

t3toolbox.backend.utv_operations.utv_weighted_norm(variations, weights, n_stack)#
def utv_weighted_norm(
        variations: typ.Tuple,  # UT3Variations .data: (tkv, ttv, shape, masks), stack = K + C
        weights:    typ.Tuple,  # UT3FrameWeights .data: (up, down, left, right, masks), stack = C
        n_stack:    int,        # leading K+C stack axes to keep; 0 -> a single scalar
) -> NDArray:                   # weighted coordinate norm, shape = stack_shape[:n_stack]

Weighted (Grasedyck-Kramer) coordinate norm of a uniform tangent’s variations: absorb the metric into the variation supercores, then take the corewise stack-norm. Uniform twin of fv_weighted_norm.

The frame (orthonormal) is not needed and not touched, so this is O(ranks). The inserted diagonal is squared by the norm, so weights = 1/sigma penalises by 1/sigma^2. Like corewise_norm this is the coordinate metric (= Hilbert-Schmidt only on an orthonormal, gauged frame) – and note that absorbing weights breaks the gauge, so the HS reading does not survive the weighting.

Weighting does not mask; the reduction does (utv_corewise_inner masks its own input), so garbage padding is zeroed where the sum happens. Precondition: the weight’s masks (broadcast over K) must equal the variations’ – ufv_weights_consistent; the frontend enforces it.

Lives here, not in ufv_operations beside ufv_absorb_weights, because the corewise reduction it needs is utv_corewise_inner and utv_operations already imports ufv_operations – the other placement would be a circular import. (Ragged has no such constraint: its fv_weighted_norm reaches a standalone corewise module.)

Parameters:

  • variations (t3toolbox.backend.common.typ.Tuple)

  • weights (t3toolbox.backend.common.typ.Tuple)

  • n_stack (int)

Return type:

NDArray