Expected spectral error for real and normalized complex Gaussian range finders

Given: Countable Choice for the finite Gaussian law, and the fixed finite matrix and the real or normalized-complex Gaussian draw under the stated $k,p,\mathrm{\ell }$ constraints.

The real-field statement with the displayed constants, for $k\ge 2$, $p\ge 2$, and $\mathrm{\ell }=k+p\le \min (m,n)$, is proved in Expected spectral error of the real Gaussian range finder.

The normalized-complex range finder, with the same fixed-matrix, dimension, variance-one Gaussian, and Countable Choice hypotheses, satisfies the displayed expected-error bound (Direct expected spectral bound for the normalized complex range finder).

Standard real and normalized complex Gaussian matrix laws: real entries are standard Gaussian and complex entries are $(X+iY)\mathrm{/}\sqrt{2}$, with $\mathbb{E}\mathrm{\mid }g{\mathrm{\mid }}^2=1$; the finite-dimensional laws are product-density laws.

Borel measurability and range correctness of the finite scan proves that the finite-scan algorithm Exact Gaussian randomized range finder returns a measurable orthogonal projector onto $\mathrm{Ran}(A\mathrm{\Omega })$. If $Q$ and $Q_0$ are any two orthonormal bases of that same subspace, then $Q{Q}^{\ast }=Q_0{Q}_0^{\ast }$: for every vector $x$, both $Q{Q}^{\ast }x$ and $Q_0{Q}_0^{\ast }x$ lie in the subspace, while their residuals lie in its orthogonal complement; uniqueness of this orthogonal decomposition makes the projections equal.

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