Noise-level noisy-network-coding inner bound for real Gaussian multicast networks

Initially choose $0<P_i^{\mathrm{\prime }}<P_i$ when $P_i>0$, and set $X_i=0$ when $P_i=0$. On positive-power nodes take independent $X_i\sim N(0,P_i^{\mathrm{\prime }})$. Use the stated reconstruction kernels. Remove constant coordinates when writing densities. The remaining joint coordinates $(X,Y,{\widehat{Y}}_{\mathcal{Q}})$ are an invertible linear image of the independent nonconstant inputs, channel noises and quantization noises: recover inputs first, channel noises from $Y-GX$, and quantization noises from ${\widehat{Y}}_{\mathcal{Q}}-Y_{\mathcal{Q}}$. They therefore have a strictly positive nonsingular Gaussian density. Its needed marginal and conditional densities are likewise Gaussian with positive conditional covariance, so all local and reference likelihood ratios in [F2] exist measurably. Every log ratio is a constant plus a quadratic polynomial in Gaussian variables.

These log ratios are square-integrable: a quadratic polynomial squared is a finite sum of monomials of degree at most four, and Gaussian fourth moments are finite. Explicitly for a scalar $N(0,v)$ density, integrating by parts twice gives $\mathbb{E}{Z}^4=3v^2$; Gaussian coordinates have those scalar marginal laws, and $\mathbb{E}(Z_i^2{Z}_j^2)\le (\mathbb{E}{Z}_i^4\mathbb{E}{Z}_j^4{)}^{1\mathrm{/}2}$. The input cost $c_i(x)=x^2$ also has finite second moment $3(P_i^{\mathrm{\prime }}{)}^2$ and mean $P_i^{\mathrm{\prime }}<P_i$. All the Gaussian cut mutual informations are finite, as differences of finite Gaussian conditional log-density expectations. Thus every Euclidean hypothesis of [F2] holds at the positive-power nodes with slack square costs; zero-power inputs are fixed zero coordinates satisfying their constraints pointwise and need no shell constraint.

Apply [F2] with constant time-sharing variable, converting $r_i=R_i\ln 2$. It gives precisely the first displayed strict cut inequalities at variances $P_i^{\mathrm{\prime }}$, since constant reconstructions may be deleted from both mutual informations. Its codes are deterministic and measurable, and every positive-power input word has empirical square cost at most a threshold strictly below $P_i$. Hence the expected average-power constraints in [F1] hold for the chosen deterministic code, independently of messages and selected compression indices. Zero-power nodes always send zero.

Increase $P_i^{\mathrm{\prime }}$ to $P_i$ simultaneously at positive-power nodes. Each mutual information in these finitely many cuts is a continuous function of the variances: its Gaussian entropy formula is a log determinant of a positive-definite conditional noise covariance plus a positive-semidefinite signal covariance. One may derive the formula directly from the Gaussian density: for positive-definite $K$, integration of its log density gives $h(N(0,K))=\frac{1}{2}\ln ((2\pi e{)}^m\det K)$ because the expected quadratic form is $m$. Conditional noise covariance is positive definite even at the limiting input variances. Therefore any tuple satisfying all strict inequalities at $P_i$ satisfies them at $P_i^{\mathrm{\prime }}$ sufficiently close to $P_i$. Use the codes from step 3.1, already satisfying the original power constraints. This proves the full chosen-$\mathcal{Q}$ achievability assertion, including arbitrary zero-power nodes.

For the target Gaussian law, conditioned on all inputs the noises at nodes in $A$ are independent of $(Y_d,{\widehat{Y}}_{A^c\cap \mathcal{Q}})$, since $d\mathrm{\notin }A$ and channel and quantization noises are independent across nodes. Each active source-side test channel adds independent unit quantization noise to unit channel noise. Using the density entropy formula from step 4.1 gives $I(Y_{A\cap \mathcal{Q}};{\widehat{Y}}_{A\cap \mathcal{Q}}\mathrm{\mid }{X}_{[N]},Y_d,{\widehat{Y}}_{A^c\cap \mathcal{Q}})=\frac{\mathrm{\mid }A\cap \mathcal{Q}\mathrm{\mid }}{2}{\log }_{2}2=\frac{\mathrm{\mid }A\cap \mathcal{Q}\mathrm{\mid }}{2}\mathrm{.}$ Constants contribute zero. This calculation holds for every destination outside $A$ and all choices of powers.

When $\mathcal{Q}=[N]$ and all powers equal $P$, condition on $X_{A^c}$. The quantized vector ${\widehat{Y}}_{A^c}$ is a known translation of $G(A)X_A$ plus an independent Gaussian vector of covariance $2I$. Its entropy minus the entropy of that noise, divided by $\ln 2$, gives $I(X_A;{\widehat{Y}}_{A^c}\mathrm{\mid }{X}_{A^c})=\frac{1}{2}{\log }_2\det (I+\frac{P}{2}G(A)G(A{)}^T)\mathrm{.}$ By [F3], adding $Y_d$ can only increase this positive mutual-information term. Subtracting the penalty $\mathrm{\mid }A\mathrm{\mid }\mathrm{/}2$ from step 5.1 proves the sufficient determinant inequality for every destination outside every admissible cut. Step 4.1 makes all tuples satisfying this finite family of strict sufficient inequalities achievable.

The closure of the strict regions follows from [F2], or by selecting successively longer reliable codes for rate tuples tending to the claimed boundary points. For the all-zero tuple give every source a singleton message set, transmit zero at all nodes, and return the unique messages at every destination; error and power are both zero. This argument applies even if a sufficient cut expression is negative.