The Learning Geometry of Deterministic Views

Adding deterministic views of an observation can change a model's predictions without adding information. This makes a standard view ablation difficult to interpret: the input expansion may also change the learning rule. We characterize this intervention for full-rank fixed linear views, each embedded by a trainable linear branch and summed before a shared receiver. Their frame operator $S=A^{\top }A$ preconditions the effective SGD update and determines the norm induced by branch regularization. Two view systems produce the same update for every loss exactly when their frame operators agree. With matched initialization, equal-$S$ systems follow identical trajectories under SGD and scalar-linear optimizer state, including momentum. The classical Parseval transform $A{S}^{-1/2}$ therefore supplies a neutral control relative to the parent-only model. Nonidentical equal-$S$ frames match exactly in our experiments, whereas equal-trace frames can separate. One multiscale frame changes subject-mean NLL by $+0.0338$ on Sleep-EDF and $-0.1215$ on ISRUC; Parsevalization removes both differences. Independent schedule selection retains a Sleep-EDF gap of $+0.0999$ (95% CI $[0.0673,0.1285]$). AdamW and branchwise nonlinearities break the tested equivalences. The frame operator thus specifies what to control when attributing a deterministic view's effect to the representation.