Mostly the Norm: Decomposing Frequency-Domain Forecast Losses into Basis, Norm, and Residual Weighting

Frequency-domain forecast losses such as FreDF change three things at once: the basis in which the residual is measured, the norm applied to it and the weight of each coordinate. We separate the three. (i) Under a squared norm any unitary basis is inert (Parseval), and training confirms it. (ii) Under ${\ell }_1$, on iTransformer and at an equal mix of the two terms, the same ${\ell }_1$ term with no transform recovers most of FreDF's gain and is behind FreDF on none of nineteen held-out dataset–horizon cells. At FreDF's released weights it is ahead on ten of twenty cells and behind on four, mostly at the short horizons of the hourly datasets. (iii) To measure explicit weighting we use \method as an instrument, a squared loss on the residual whitened by detached, shrunk, Kronecker-factored second-moment estimates. Linear theory gives the invariance condition under which a fixed quadratic metric can move the fitted predictor. Empirically, the estimated weighting is ahead of the time-domain ${\ell }_1$ term on no iTransformer cell. Its largest gain, $5.3\%$ on PatchTST at ETTh1 and horizon $720$ over three seeds, does not survive a Holm correction, depends on the metric's eigenvectors rather than its spectrum, and disappears once the test period moves three months later. A time-domain ${\ell }_1$ term at matched strength is therefore the baseline against which frequency-domain and whitened losses should be judged.