= Rank deficiency from deterministic spectral upsampling
{title2=$\operatorname{rank}(ACA^T)\leq\operatorname{rank}(A)$}
\b[Rank deficiency from deterministic spectral upsampling] occurs because a deterministic map cannot manufacture independent measurements. For $\mathbf y=A\mathbf n$ and input <covariance matrix> $C$,
$$
C_y=ACA^T,\qquad \operatorname{rank}(C_y)\leq\operatorname{rank}(A).
$$
When the output has more channels than the input, its <covariance matrix> is necessarily singular. <Covariance of count-conserving rebinned Raman channels> gives the conserved-count example and its full propagated uncertainty.<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}
Using that parent's two-to-three channel map, the output always satisfies $y_1-y_2+y_3=0$. If the input readings are $20$ and $30$, the rebinned vector is $(10,25,15)^T$. Recovering the original readings as $n_1=2y_1$ and $n_2=2y_3$ shows that the third displayed number is not a third independent observation.
For an original fitting example, suppose both input means equal an unknown $\theta$ and their independent measurement-error <variances> are known to be $25$. Fitting in the original channels gives $\widehat\theta=(20+30)/2=25$, with <variance> $25/2=12.5$. A correct <generalized least squares> fit after rebinning gives the same result and uncertainty. It must use the singular <covariance> on its supported range, for example through the <Moore-Penrose inverse>, together with the exact linear constraints.
The <range restriction in singular-covariance spectral fitting> matters if a proposed output model fails those constraints. A residual along $(1,-1,1)^T$ cannot arise from this rebinning model at all. Merely inserting a pseudoinverse into a quadratic objective would assign zero penalty to that impossible direction. Either enforce the range condition or supply a physically justified additional noise model; adding a small diagonal term solely to make the matrix invertible changes the statistical assumptions.
The motivating Raman apparatus measures detector-array spectra before any subsequent numerical grid change.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Instrumentation, array detection and spectral calibration; Data Collection, equation 4]{ref} Working on the original grid often avoids this singular fitting problem while retaining the actual measurement information.
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