A Decision-Theoretic Framework for Representation Learning: Operationalizing Le Cam Theory from Sufficiency to Deficiency

Modern machine learning relies on learned representations as substitutes for raw data, yet the classical theory of statistical experiments---which provides operational notions of information preservation---remains underutilized in this context. Motivated by representation transfer under precision shifts (e.g., across multi-site clinical trials and other heteroskedastic data-collection settings), we develop a deficiency-based framework for representation learning by treating each dataset (domain) as a statistical experiment and evaluating representations via Le Cam deficiency, thereby connecting classical experiment comparison to modern deep learning practice. Our main contributions are: \begin{enumerate} \item[(1)] A \textbf{Risk-Bounded Representation Learning Theorem} showing that reconstruction loss $\varepsilon^2$ implies deficiency at most $\varepsilon/\sqrt{2}$, which bounds risk degradation for any bounded downstream task. \emph{(This combines Pinsker's inequality with Le Cam's randomization criterion; the contribution is the explicit operationalization for encoder-decoder architectures.)} \item[(2)] A \textbf{No-Free-Transfer Theorem} proving that strict marginal invariance between heteroskedastic experiments forces information loss. For linear representations, this yields exact collapse ($A=0$); for nonlinear representations, Fisher information is bounded by the intersection of the source and target information cones. \emph{(Part (a) is algebraic; Part (b) extends to differentiable nonlinear maps via the data processing inequality.)} \item[(3)] A \textbf{Representation Equivalence Hierarchy} organizing the cascade from pointwise likelihood preservation through ratio preservation to deficiency bounds, making explicit the regularity conditions (Local Asymptotic Normality (LAN), contiguity) under which each implication holds. \emph{(This systematizes known asymptotic theory for representation learning contexts.)} \item[(4)] A \textbf{Certificate vs.~Surrogate Distinction} clarifying when optimization objectives (MMD, adversarial) provide verifiable deficiency bounds versus when they serve only as training heuristics. We show reconstruction-based methods (Domain Separation Networks (DSN), Deep Reconstruction-Classification Networks (DRCN)) implicitly minimize deficiency certificates. \emph{(This provides theoretical justification for empirically successful methods.)} \end{enumerate} The bounds we derive are structural: they relate measurable quantities (reconstruction loss) to decision-theoretic guarantees (deficiency, risk transfer). The sample complexity bound (Theorem~5) involves pseudo-dimension terms that are not computed for specific architectures here; instantiation for concrete networks remains open. Empirical validation uses controlled synthetic settings where ground truth is computable; extension to standard domain adaptation benchmarks (e.g., DomainNet) is deferred to a companion paper. Throughout, our statements are operational: we bound worst-case decision risk over specified loss classes under passive experiment comparison on a shared parameter space $\Theta$. Causal/interventional transfer requires explicitly specifying the corresponding interventional experiments, and asymptotic equivalence statements invoke standard regularity (LAN, mutual contiguity).