where W_A is the output and W_B is the input. A detailed justification for using this measure is given in ARENA. The justification is based on the SVD. If you do an SVD for each term, the numerator ends up containing a cosine similarity between the right singular output vectors and the left singular input vectors, so the norm is maximized when the output and input are aligned. Here are the subspace scores between the embedding and positional encodings against each layer 0 head’s QK circuit:
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