| \section{A distribution-free conditional recovery bridge} |
| The following transfers a uniform frame inverse bound into the actual conditional-expectation formulation of VESPERS. The estimation and quadratic-completion identities are classical; this is their application to the certified operator setting, without a Gaussian posterior assumption. |
|
|
| Let $Z$ be square-integrable, $\mathcal G$ the current memory, and |
| \[ |
| y=A_JZ+\eta,\qquad |
| \mathcal H=\mathcal G\vee\sigma(J,y),\qquad |
| \|A_Jz\|\ge\rho\|z\|\quad\text{for every }z,\qquad\rho>0. |
| \] |
| Thus every observation map is injective, with a uniform full-column-rank lower bound. Assume $A_J$ is known from $\mathcal H$ and |
| $\mathbb E[\|\eta\|^2\mid\mathcal G]\le\varepsilon^2$ almost surely. The $\mathcal H$-measurable estimate $\widetilde Z=A_J^\dagger y$ obeys |
| \[ |
| \mathbb E[\|Z-\widetilde Z\|^2\mid\mathcal G] |
| \le\varepsilon^2/\rho^2. |
| \] |
| Since $m_{\mathcal H}=\mathbb E[Z\mid\mathcal H]$ is the conditional least-squares estimate and $\mathcal G\subseteq\mathcal H$, conditional orthogonality gives |
| \[ |
| \boxed{\mathbb E[\|Z-m_{\mathcal H}\|^2\mid\mathcal G] |
| \le\varepsilon^2/\rho^2.} |
| \] |
| This is pre-evidence conditional Bayes risk, averaged over the new observation given current memory. A pointwise posterior bound conditioned on $\mathcal H$ needs the stronger assumption $\mathbb E[\|\eta\|^2\mid\mathcal H]\le\varepsilon^2$, as supplied, for example, by $\|\eta\|\le\varepsilon$ almost surely. A current-memory or unconditional noise moment does not establish that stronger statement. Independence and zero-mean noise are not required. |
|
|
| For a linear task $BZ$, multiply the risk bound by $\|B\|_{\mathrm{op}}^2$. An everywhere-defined $L$-Lipschitz task similarly has risk at most $L^2\varepsilon^2/\rho^2$, by comparing with the task applied to $\widetilde Z$. |
|
|
| To connect the result to VESPERS, let $T$ be the fixed available correction map, $C\succ0$, and $\lambda>0$. Put |
| \[ |
| R=T(T^*T+\lambda C)^{-1}T^*,\qquad Q=I-R, |
| \quad m_{\mathcal G}=\mathbb E[Z\mid\mathcal G], |
| \] |
| and define |
| \begin{align*} |
| S_{\mathcal G}&=\mathbb E[(Z-m_{\mathcal G})(Z-m_{\mathcal G})^*\mid\mathcal G],\\ |
| S_{\mathcal G}^+&=\mathbb E[(Z-m_{\mathcal H})(Z-m_{\mathcal H})^*\mid\mathcal G],\\ |
| \Omega_{\mathcal G}&=\mathbb E[(m_{\mathcal H}-m_{\mathcal G})(m_{\mathcal H}-m_{\mathcal G})^*\mid\mathcal G]. |
| \end{align*} |
| Conditional orthogonality gives $S_{\mathcal G}=S_{\mathcal G}^++\Omega_{\mathcal G}$. The exact optimal $\mathcal H$-measurable correction and conditional cost are |
| \[ |
| u^*=(T^*T+\lambda C)^{-1}T^*(m_{\mathcal H}-m_{\mathcal G}), |
| \] |
| \[ |
| J_{\mathcal G} |
| =\frac12\tr[S_{\mathcal G}^++Q\Omega_{\mathcal G}] |
| =\frac12\tr[Q S_{\mathcal G}]+\frac12\tr[R S_{\mathcal G}^+], |
| \] |
| where |
| \[ |
| J_{\mathcal G}=\operatorname*{ess\,inf}_{u\;\mathcal H\text{-measurable}} |
| \frac12\mathbb E[\|Z-m_{\mathcal G}-Tu\|^2 |
| +\lambda u^*Cu\mid\mathcal G]. |
| \] |
| The frame risk certificate therefore implies |
| \[ |
| \boxed{J_{\mathcal G}\le\frac12\tr[Q S_{\mathcal G}] |
| +\frac{\|R\|_{\mathrm{op}}\varepsilon^2}{2\rho^2}.} |
| \] |
| This exact conditional control identity follows by first conditioning on $\mathcal H$, separating the orthogonal residual $Z-m_{\mathcal H}$, and completing the remaining quadratic in $u$. It requires no density or Gaussian law. The risk constant is sharp under the stipulated moment-only information: in a scalar example $A=\rho$, $\eta=-\rho Z$, and centered $Z=\pm\varepsilon/\rho$ with equal probabilities, the observation is identically zero and the bound is attained. |
|
|
