Testing direction shifts under orthogonally invariant observations

We study high-dimensional shift testing when the statistician observes only an orthogonally invariant transformation of the shifted sample $Y_v=X+v$. Examples include norms or Gram matrices. Such observations discard the left angular orientation of $Y_v$, so when the noise $X$ is itself $O(d)$-invariant, they cannot distinguish a shift $v$ from any shift $Uv$. We ask when this geometric loss of information yields a dimension-dependent contraction of distinguishability. We answer this question through Fisher information. For the class of $O(d)$-invariant noise distributions, we prove that every invariant observation satisfies a Fisher-information contraction bound whose leading factor scales as $1/d$, up to constants depending only on the noise law. Then, we instantiate the result for Gaussian and Frobenius-radial Laplace distributions, and show how the Gaussian case improves existing analyses in *privacy amplification by synthetic data release*.