Linear Is Not Invariant

TL;DR for operators

If two parameterizations compute exactly the same model function, an interpretability conclusion presented as a property of that model needs a reason to remain valid when the internal coordinates change. The Linear Representation Hypothesis Needs a Group Action by Louie Hong Yao, Yuhao Li, and Shengchao Liu¹ makes that requirement explicit.

The paper’s symmetry audit shows why this matters. Exact rank, spans, incidence relations, and CCA can survive affine transformations. Cosines, angles, principal angles, and several normalized spectral quantities generally require a narrower similarity geometry. Absolute norms, distances, reconstruction losses, and intervention magnitudes require fixed Euclidean scale. These are not interchangeable measurements of one coordinate-free object.

For operators, the main use is methodological governance. Before promoting a probe, direction, similarity score, PCA result, or intervention into evidence about the model itself, ask what transformations the claim survives, whether the architecture permits those transformations without changing the model function, and whether later preprocessing or calibration narrows the symmetry further.

The boundary is equally concrete: the paper establishes formal invariance and non-invariance results. It does not show that the audited interpretability methods necessarily produce worse predictions, explanations, or interventions in practice.

The metric already contains a hypothesis

Suppose two analysts inspect the same activation space.

One reports that two directions have high cosine similarity. Another reports that two subspaces have the same span. Both calculations may be correct. But they are not necessarily describing the same underlying property.

The paper’s Table 1 makes the distinction systematic. Some quantities depend only on linear dependence or incidence structure and therefore survive a broad class of affine coordinate changes. Angular quantities require an inner product preserved up to scale. Euclidean distances and magnitudes require still more structure because absolute scale must remain fixed.

That turns metric selection into more than a tooling decision. Choosing cosine similarity asserts that angle is meaningful. Choosing Euclidean distance asserts that length and scale are meaningful. Choosing coordinatewise sparsity assumes particular latent axes matter.

The authors’ core reframing follows from this observation: there is no single Linear Representation Hypothesis. There is a family of claims whose meaning depends on the object being studied and the transformations regarded as equivalent descriptions.

A representation claim needs four pieces

The formal framework starts by specifying an equivalence group $G$: the transformations that count as alternative descriptions of the same representation.

The representation object then lives in a space on which those transformations act. A procedure $F$ extracts that object from the model representation, and a predicate $P$ expresses the final property being asserted.

For a coordinate-independent claim, the construction must transform consistently when the representation coordinates change, while the final truth claim must remain unchanged:

The first condition is equivariance. The second is invariance.

This matters because representation objects that look identical in storage can obey different transformation laws. The array shape alone does not determine what mathematical object an analysis has produced.

A probe coefficient is not automatically a steering vector

This distinction becomes concrete in the paper’s treatment of linear probes and activation steering.

A linear probe coefficient naturally acts as a linear functional: it maps an activation to a scalar score. Mathematically, it belongs to the dual space. An additive steering vector is instead a displacement applied directly to the activation space. It belongs to the primal space.

Both may be stored as length-$d$ arrays.

That storage-level similarity does not supply a coordinate-independent identification between them. The paper proves that, for dimension at least two, there is no nonzero identification from the dual space to the primal space that remains equivariant under arbitrary general-linear changes of coordinates. An inner product can provide such a map, but then the analysis has introduced additional geometric structure and reduced the symmetry it claims to respect.

For steering systems, this changes the review question. A pipeline that learns a probe and then reuses the probe coefficients as an intervention direction needs to document the geometry or mapping that connects diagnosis to intervention. Tensor compatibility is not enough.

The architecture sets a floor on acceptable symmetry

The equivalence group is not entirely a matter of analyst preference.

The paper defines function-preserving parameter transformations: changes to the model parameters that leave the computed input-output function unchanged. When such a transformation changes an internal representation, a claim presented as a property of the model function has to accommodate that induced representation change.

This creates what the authors describe as an architectural floor.

The consequence is reading-point specific. A function-preserving transformation may leave one internal location unchanged while inducing a nontrivial transformation elsewhere. In attention, the paper derives query/key reparameterization symmetries, including residual freedom under RoPE, under which Euclidean norms, cosines, PCA quantities, and some spectral summaries in query or key coordinates can change even though the model computation does not.

That result does not make attention-head geometry unusable. It identifies what the geometry belongs to. A quantity that varies across functionally equivalent parameterizations is a property of the chosen parameterization or metric unless further structure justifies a stronger interpretation.

The paper also shows a constructive alternative: key interchangeability can be measured using a query-distribution-weighted distance tied to the expected squared change in attention logits. That quantity is designed around the computation that remains invariant under the relevant gauge transformation.

Invariance has to survive the whole pipeline

Even an estimator with attractive symmetry properties does not guarantee that the resulting analysis preserves them.

A representation workflow may estimate a direction, normalize it, reduce dimension, calibrate a score, compare it with another representation, and finally convert the result into an intervention. Any later stage can impose additional geometric assumptions.

The paper therefore treats symmetry compositionally. The final claim inherits the restrictions introduced by the complete procedure, not merely those of its first component.

For model-evaluation teams, Cognaptus infers a practical audit pattern:

This is a governance use of the paper rather than an empirical result reported by it. The framework supplies the checks; organizations still need to decide which invariance standard their assurance process requires.

The framework narrows claims; it does not invalidate geometry

There are two important boundaries.

First, the paper does not argue that Euclidean geometry inside a particular trained network is meaningless. A norm, cosine, PCA direction, or reconstruction loss can be perfectly well defined for a fixed parameterization and chosen metric. The constraint appears when that quantity is generalized into a claim about the model representation independently of those choices.

Second, formal admissibility can be made weak by declaring a very small equivalence group. If almost no transformations count as equivalent, many coordinate-dependent predicates become invariant by construction. The authors therefore associate a stronger representation claim with survival under the largest justified transformation group.

Nor does the analysis establish that coordinate-dependent methods fail operationally. Whether a probe predicts well, an intervention changes behavior reliably, or a diagnostic transfers across training runs remains an empirical question. The paper establishes what those results can legitimately be interpreted as without additional evidence.

Interpretability claims need a symmetry specification

The paper changes the unit of scrutiny.

Instead of asking only whether a model contains a linear direction, whether two representations have high similarity, or whether a steering vector changes behavior, the analyst also has to specify what representation object was constructed, which coordinate transformations count as equivalent, how that object transforms, and which part of the final conclusion is invariant.

For businesses using interpretability in safety reviews, steering systems, or model diagnostics, that distinction can prevent a subtle category error: treating geometry of one coordinate realization as evidence about the model function itself.

The immediate deliverable is not a new interpretability score. It is a specification discipline for deciding what an existing score is allowed to mean.

Cognaptus: Automate the Present, Incubate the Future.