Expected spectral error of the real Gaussian range finder

Given: The inherited Countable Choice assumption for the Gaussian law, and the fixed matrix, oversampling conditions, a standard real Gaussian draw, and the exact range-finder output.

Orthogonal SVD coordinates split $V^{\ast }\mathrm{\Omega }$ into independent standard Gaussian blocks (Orthogonal and unitary invariance of Gaussian matrix blocks).

The $k\times \mathrm{\ell }$ block ${\mathrm{\Omega }}_1$ has full row rank almost surely (A wide Gaussian matrix has full row rank almost surely).

$\mathbb{E}\mathrm{\parallel }{\mathrm{\Omega }}_1^{†}{\mathrm{\parallel }}_F^2=k\mathrm{/}(p-1)$ (Exact Frobenius moments of wide Gaussian pseudoinverses).

$\mathbb{E}\mathrm{\parallel }{\mathrm{\Omega }}_1^{†}{\mathrm{\parallel }}_2\le e\sqrt{k+p}\mathrm{/}p$ (Spectral moments and integrable tails for wide Gaussian pseudoinverses).

For real Gaussian $G$, $\mathbb{E}\mathrm{\parallel }SGT{\mathrm{\parallel }}_2\le \mathrm{\parallel }S{\mathrm{\parallel }}_2\mathrm{\parallel }T{\mathrm{\parallel }}_F+\mathrm{\parallel }S{\mathrm{\parallel }}_F\mathrm{\parallel }T{\mathrm{\parallel }}_2$ (Real scaled-Gaussian spectral and Frobenius product bounds).

The deterministic SVD partition gives $\mathrm{\parallel }(I-Q{Q}^{\ast })A{\mathrm{\parallel }}_2\le {\sigma }_{k+1}+\mathrm{\parallel }{\mathrm{\Sigma }}_2{\mathrm{\Omega }}_2{\mathrm{\Omega }}_1^{†}{\mathrm{\parallel }}_2$ when ${\mathrm{\Omega }}_1$ has full row rank (Deterministic SVD-partition error bound for the range finder).

Jensen's inequality gives $(\mathbb{E}X{)}^2\le \mathbb{E}{X}^2$ for nonnegative $X$ with finite second moment (Jensen's inequality for expectation).

The finite-dimensional expectation may be evaluated by iterated nonnegative integrals (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

The best rank-at-most-$k$ spectral approximation error is ${\sigma }_{k+1}$ (The best rank-at-most-k approximation in operator norm is the rank-k truncation of a singular value decomposition).

Countable Choice is used for the finite-dimensional Lebesgue Gaussian law (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The finite scan returns a Borel orthogonal projector onto the sample range (Borel measurability and range correctness of the finite scan). Every orthonormal basis of that range gives the same projector: each projection has residual orthogonal to the range, so uniqueness of the orthogonal decomposition identifies them.