Proposition: Linear nuisance annihilation

Let a response bank be $r\in {\mathbb{R}}^J$, with complete permitted nuisance class $r\mapsto r+Ba$. A linear readout $w^{\mathsf{T}}r$ is nuisance invariant exactly when

An invariant readout with nonzero response exists exactly when

If component errors are bounded by ${\eta }_i$, the readout error is at most

Proof. Invariance is $w^{\mathsf{T}}Ba=0$ for every $a$. The orthogonal complement of $\mathrm{range}(B)$ is $\ker B^{\mathsf{T}}$. If $r$ lies outside the range, its orthogonal projection onto that kernel provides a nonzero invariant response. The error allowance is the triangle inequality.