= Signal-subspace preservation for several spectral amplitudes
<Signal-subspace preservation for several spectral amplitudes> requires a compression to retain every noise-weighted mean direction relevant to those amplitudes. Let $\mathbf y$ have mean $A\boldsymbol\theta$, fixed positive-definite noise <covariance> $C$, and a full-column-rank shape matrix $A$. Compress it as $B\mathbf y$, with $B$ full row rank. The original and compressed amplitude-information matrices are
$$
J=A^TC^{-1}A,\qquad
J_B=A^TB^T(BCB^T)^{-1}BA.
$$
For Gaussian noise these are <Fisher information matrices>. Without that distributional assumption, they still describe the corresponding known-covariance linear estimation problem.
Whiten the shapes as $T=C^{-1/2}A$ and put $M=BC^{1/2}$. The <orthogonal projection> onto the row space of $M$ is $P=M^T(MM^T)^{-1}M$, giving
$$
J-J_B=T^T(I-P)T\succeq0.
$$
This difference is <positive semidefinite>. It vanishes exactly when every column of $T$ lies in that row space, equivalently
$$
\operatorname{range}(C^{-1}A)\subseteq\operatorname{range}(B^T).
$$
Thus preserving one selected shape does not establish preservation of the whole signal subspace. Linear covariance propagation supplies $BCB^T$ for the compressed observations.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
For an original example, take $C=\operatorname{diag}(1,4,9)$ and shape columns $(1,2,0)^T$ and $(0,0,3)^T$. Then $J=\operatorname{diag}(2,1)$. Choosing
$$
B=\begin{pmatrix}1&1/2&0\\0&0&1/3\end{pmatrix}
$$
gives $BA=\operatorname{diag}(2,1)$ and $BCB^T=\operatorname{diag}(2,1)$, hence $J_B=J$. Keeping only the first compressed row retains information2 for the first amplitude but destroys sensitivity to the second. An unweighted first row $(1,1,0)$ would instead reduce the first information entry to1.8.
At least $p$ independent compressed directions are needed to preserve $p$ independent shape amplitudes. One construction is $B=A^TC^{-1}$, whose compressed signal has mean $J\boldsymbol\theta$ and covariance $J$. When $J$ is invertible, $J^{-1}B\mathbf y$ has mean $\boldsymbol\theta$ and covariance $J^{-1}$. In the fixed-covariance Gaussian model, expanding the <likelihood function> shows that its parameter dependence is retained through $A^TC^{-1}\mathbf y$.
Unknown nuisance shapes, changing covariance or additional spectral parameters can alter the subspace that needs preservation. The proof concerns the specified family of mean shapes and fixed noise model, rather than guaranteeing that one compression preserves every future spectral question.
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