Finite-dimensional Gaussian vector squared-error rate-distortion function

Write ${\delta }_i=\mathbb{E}(X_i-Y_i{)}^2$, so ${\sum }_i{\delta }_i\le D$. Repeated use of [F2] expands $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X^m;Y^m)$ as ${\sum }_i{I}_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y^m\mid X_1^{i-1})$. Put $W_i=X_1^{i-1}$; for $i=1$ this is a constant empty tuple and the conditional identities reduce to their unconditioned forms, while for $i>1$ its Euclidean value space is standard Borel by [F11]. Reorder $Y^m$ as $(Y_i,Y_{[m]\setminus \{i\}})$ by the coordinate-bijection invariance in [F2]. Its derived second-argument conditional rule gives $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y^m\mid W_i)=I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i\mid W_i)+I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_{[m]\setminus \{i\}}\mid W_i,Y_i)\ge I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i\mid W_i),$ by nonnegativity [F3]; the empty tuple has zero information.

If ${\lambda }_i>0$ and $0<{\delta }_i<{\lambda }_i$, [F4] gives $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i)\ge \frac{1}{2}\ln ({\lambda }_i\mathrm{/}{\delta }_i)$. If ${\delta }_i\ge {\lambda }_i$, nonnegativity gives $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i)\ge 0=\frac{1}{2}{\ln }^{+}({\lambda }_i\mathrm{/}{\delta }_i)$; if ${\lambda }_i=0$, the same zero lower bound holds. If ${\lambda }_i>0={\delta }_i$, then $X_i=Y_i$ almost surely. Writing $f_i$ for its bounded normal density, the product law assigns probability at most $2h\mathrm{\parallel }{f}_i{\mathrm{\parallel }}_{\mathrm{\infty }}$ to $\mathrm{\mid }{x}_i-y_i\mathrm{\mid }\le h$, for each $h>0$; taking $h=1\mathrm{/}n$ and applying [F8] to these decreasing neighborhoods gives product probability zero on the diagonal. The joint law gives the diagonal probability one, so the KL is infinite.

Independence gives $P_{X_i{W}_i}=P_{X_i}\otimes P_{W_i}$, so $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;W_i)=0$ from the KL definition. The derived second-argument rule in [F2] gives $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;(W_i,Y_i))=0+I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i\mid W_i)=I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i\mid W_i)\mathrm{.}$ A coordinate swap and the other expansion give $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;(W_i,Y_i))=I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;(Y_i,W_i))=I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i)+I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;W_i\mid Y_i)\ge I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i)\mathrm{.}$ These identities remain valid when an information value is infinite, since only the explicitly zero term is removed and all other terms are nonnegative. Thus $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X^m;Y^m)\ge {\sum }_i{I}_{\mathrm{n}\mathrm{a}\mathrm{t}}(X_i;Y_i)$.

For each coordinate put ${\bar{\delta }}_i=\min ({\delta }_i,{\lambda }_i)$ when ${\lambda }_i>0$, and ${\bar{\delta }}_i=0$ when ${\lambda }_i=0$. Then ${\sum }_i{\bar{\delta }}_i\le D$. step 1.2 and step 2.1 give the lower bound $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X^m;Y^m)\ge \frac{1}{2}{\sum }_{i:{\lambda }_i>0}\max \{0,\ln ({\lambda }_i\mathrm{/}{\bar{\delta }}_i)\}$ nats, with the sum interpreted as $+\mathrm{\infty }$ if any positive-variance coordinate has ${\bar{\delta }}_i=0$. Dividing by $\ln 2$ converts this to the corresponding coordinate-rate sum in bits.

If $0<D<{\sum }_i{\lambda }_i$, applying [F5] to $({\bar{\delta }}_i)$ minimizes the bit-valued coordinate-rate sum under ${\sum }_i{\bar{\delta }}_i\le D$ and gives the stated water-level lower bound for $I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X^m;Y^m)\mathrm{/}\ln 2$, hence for the rate in [F1]. If $D=0<{\sum }_i{\lambda }_i$, some positive-variance coordinate has zero error, so step 1.2 makes the information infinite. If $D\ge {\sum }_i{\lambda }_i$, nonnegativity is the lower bound zero; the all-zero variance case has the same lower bound for every $D$.

In the middle case let ${\delta }_i^{\ast }=\min ({\lambda }_i,\theta )$. For ${\lambda }_i\le \theta$, choose the constant reconstruction $Y_i=0$. For ${\lambda }_i>\theta$, put $a_i=\sqrt{{\lambda }_i-\theta }$, $b=\sqrt{\theta }$, and take independent standard normals $U_i,V_i$ using the finite product law in [F10]; set $Y_i=a_i{U}_i$, $X_i=a_i{U}_i+b{V}_i$. The orthogonal rotation with first row $(a_i,b)\mathrm{/}\sqrt{{\lambda }_i}$ and [F9] gives the required $N(0,{\lambda }_i)$ source; the squared error has mean $b^2=\theta$. By [F6], [F7] and the product normal density, the joint density of $(X_i,Y_i)$ is $(2\pi a_{i}b{)}^{-1}\exp [-y_i^2\mathrm{/}(2a_i^2)-(x_i-y_i{)}^2\mathrm{/}(2b^2)]$; dividing by the product of the source and reconstruction densities gives expected log ratio $\frac{1}{2}\ln ({\lambda }_i\mathrm{/}\theta )$. Choose all coordinate pairs independently. Their joint and product-marginal laws factor, and the log density ratio is the sum of the coordinate log ratios, each integrable; [F1] therefore makes the vector mutual information the sum $\frac{1}{2}{\sum }_{i:{\lambda }_i>0}{\ln }^{+}({\lambda }_i\mathrm{/}\theta )$ nats. By [F1], the rate is this value divided by $\ln 2$, so the construction attains the bit-valued lower bound.

If $D\ge {\sum }_i{\lambda }_i$, choose $Y=0$; the error is the total variance and mutual information is zero. If every ${\lambda }_i=0$, the same constant reconstruction has zero error and zero mutual information for all $D\ge 0$. These constructions attain the remaining lower bounds and complete the proof.