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\title{
\textbf{Supplementary Materials}\\[0.4em]
\large Inference Equivalence: A Decoder-Constrained Framework for Representation Sufficiency in Neural Population Codes
}





\author{
Cam H.\ Pham\\
\textit{Can Tho University}\\
\texttt{camhuongpham.p@gmail.com \,\, \orcidlink{0009-0005-1330-9984}}
}
\date{}



\begin{document}

\maketitle


\tableofcontents
\newpage


\section{Theoretical Foundations of Inference Equivalence}

\subsection{Inference Equivalence as a one-sided notion}

IE defines a decoder-family-relative framework of sufficiency through the achievable risk functional
\[
\Phi_{\mathcal{F}}(P)
:=
\inf_{f \in \mathcal{F}}
\mathbb{E}_{P}\big[\ell(f(X), S)\big],
\]
where $\mathcal{F}$ denotes a constrained decoder family corresponding to downstream readout mechanisms.

Two representations $X$ and $Y$ are inference-equivalent when
\[
\Phi_{\mathcal{F}}(P_X)
=
\Phi_{\mathcal{F}}(P_Y).
\]

For any nontrivial restricted decoder family $\mathcal{F}$, distinct distributions can yield identical achievable risk:
\[
R_{P_X}^{\mathcal{F},*}
=
R_{P_Y}^{\mathcal{F},*},
\qquad
P_X \neq P_Y.
\]
This follows because $\Phi_{\mathcal{F}}$ depends only on aspects of $P(X,S)$ accessible to functions in $\mathcal{F}$. Consequently, IE induces equivalence classes of representations that are indistinguishable under achievable inference despite potentially differing in higher-order statistical or geometric structure.

\subsection{Transfer as a complementary diagnostic}

IE and transfer probe complementary aspects of representational transformation. IE evaluates whether achievable inference performance is preserved after decoder re-optimization within a hypothesis class $\mathcal{F}$. A deviation $\Delta_{\mathrm{IE}} \neq 0$
therefore indicates that inference-relevant structure accessible to the decoder family has been altered.

Transfer instead fixes the decoder learned on the original representation and evaluates its performance after transformation, probing stability of a fixed downstream readout under distributional shift. Because transfer depends only on structure used by the trained decoder, transfer stability and preserved achievable inference need not coincide.

Together, IE and transfer distinguish changes in achievable inference capacity from readout-specific sensitivity under representational perturbation.

\subsection{Relation to sufficiency and identifiability}

IE can be viewed as inducing a quotient structure over the space of representations, where elements are grouped by equivalence in achievable inference risk under $(\mathcal{F}, \ell)$. In this sense, the representation-to-risk mapping is generally many-to-one.

As a consequence, $\Delta_{\mathrm{IE}} \approx 0$ should not be interpreted as identification of a unique underlying representation, but only as membership within the same inference-equivalence class defined by the decoder family and loss. This reflects an inherent non-identifiability of representational structure under decoder-constrained observation.

\subsection{Relation to classical information measures}

IE differs from information-theoretic quantities such as mutual information or Fisher information, which characterize statistical dependence independently of decoder constraints. In contrast, IE evaluates representations through achievable risk under a restricted hypothesis class, making decoder-accessible structure central to equivalence.



\section{Surrogate Framework and Intervention Types}

\subsection{Definition of surrogate reduction}

A surrogate is a controlled transformation of the original representation,
\[
Y \sim P(\cdot \mid X),
\]
which may be deterministic via a measurable mapping $Y = M(X)$ or stochastic through conditional resampling. The purpose is to selectively modify specific statistical properties of $X$ while preserving task labels, thereby inducing structured changes in representation space.

\subsection{Two classes of controlled perturbations}

We distinguish two classes of perturbations. Statistical transformations preserve selected properties of the population distribution, such as marginal statistics, covariance structure, dependence structure, or low-dimensional geometry, while altering remaining structure. Decision-aligned perturbations, in contrast, are not designed to preserve global descriptive statistics; instead, they operate along the task-relevant decision axis, directly modulating geometry that governs downstream inference under the chosen task structure.

Together, these perturbations dissociate descriptive similarity from inference-relevant structure under constrained decoding.


\section{Decoder Families}

To examine dependence on the decoder family, IE and transfer were recomputed under an expanded hypothesis class including quadratic interactions (Fig.~\ref{fig:s1}; Fig.~\ref{fig:s2}).

\clearpage

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S1.pdf}
\caption{
Same analyses as in Fig.~3, showing IE evaluated under the quadratic decoder family.
}
\label{fig:s1}
\end{figure}



\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S2.pdf}
\caption{
Same analyses as in Fig.~4, showing transfer evaluated under the quadratic decoder family.
}
\label{fig:s2}
\end{figure}

The principal findings showed consistent patterns across decoder classes for both IE and transfer (Fig.~\ref{fig:s1}B,C,E; Fig.~\ref{fig:s2}B,C,E). Strongly destructive perturbations (Mean-only, Copula, Decision-Strong) consistently remained non-equivalent, whereas Decision-Weak perturbations continued to lie close to the inference-equivalent regime. Importantly, the relative ordering of transformation severity was preserved under the quadratic decoder class. Beyond categorical agreement, the sign and qualitative magnitude of both $\Delta_{\mathrm{IE}}$ and $\Delta_{\mathrm{Transfer}}$ remained consistent, indicating that deviations were not only directionally aligned but also comparable in scale across decoder specifications. This suggests that the qualitative structure of the results does not depend on the linearity assumption.

The primary changes occurred for surrogate classes that lay near the inference-equivalent regime under linear decoding (Fig.~\ref{fig:s1}D; Fig.~\ref{fig:s2}D). IID-Marginal, Covariance-Gaussian, and Latent-Factor constructions exhibited substantially weaker IE under the quadratic decoder family, with IID-Marginal shifting fully into the non-equivalent regime, while Covariance-Gaussian and Latent-Factor surrogates retained only partial equivalence across seeds. In contrast, transfer remained consistently impaired for these surrogate classes within the identifiable regime, indicating that the apparent compatibility observed under linear decoding did not generalize to stability of the learned decision rule. Together, these results suggest that the near-equivalence observed under linear decoding was specific to the corresponding restricted decoder family and did not necessarily extend to richer hypothesis classes with access to additional task-relevant structure. Importantly, this does not invalidate the observed equivalence within the linear regime itself, but rather emphasizes that functional equivalence in IE is fundamentally defined relative to the biologically motivated hypothesis class governing downstream readout.

Quadratic decoders access pairwise interactions and curved decision structure that remain inaccessible under linear readout. Consequently, some surrogate transformations that appear inference-equivalent under linear decoding become distinguishable under the expanded hypothesis class. At the same time, increased decoder expressivity more frequently pushed performance toward degenerate regimes, reducing empirical identifiability despite improved representational access.

\section{Empirical Estimation of Inference Equivalence}


\subsection{Decoder family}

Inference equivalence was evaluated relative to a fixed decoder family $\mathcal{F}$. Primary analyses used L2-regularized logistic regression with feature standardization and class-balanced weighting. Supplementary analyses used quadratic discriminant analysis (QDA) with regularization parameter $\mathrm{reg\_param}=0.5$. QDA was selected as a minimal nonlinear extension of linear decoding that admits second-order interactions while remaining interpretable and computationally tractable.

\subsection{Controlled perturbations}

\paragraph{Statistical surrogate classes}

We generated surrogate datasets that preserved selected statistical properties of the original representation while altering remaining distributional structure.

\paragraph{Mean surrogate}

Samples were replaced by the class-conditional empirical mean estimated from the training fold, preserving mean activity while removing within-class variability.

\paragraph{IID marginal surrogate}

Within each class, features were independently resampled from their empirical marginal distributions. This preserves univariate statistics while disrupting cross-feature dependencies.

\paragraph{Gaussian covariance surrogate}

Samples were generated from class-conditional Gaussian distributions with empirical means and covariance matrices estimated from the training fold using Ledoit--Wolf shrinkage. This preserves first- and second-order statistics within each class.

\paragraph{Factor-analysis surrogate}

Samples were generated from class-conditional factor-analysis models fitted on the training fold. The number of latent factors was fixed at \(K=5\) (capped by feature dimension).

\paragraph{Gaussian copula surrogate}

Gaussian copula surrogates preserved empirical marginals together with Gaussianized dependence structure estimated in latent space.

\paragraph{Decision-aligned perturbations}

To assess sensitivity of inference equivalence to task-aligned geometry, we constructed perturbations along the normalized decision direction \(w\) of a logistic-regression probe trained on the training fold. Perturbed samples were generated as
\[
x' = x + \alpha w.
\]

\paragraph{Weak decision perturbation}

Weak perturbations used stochastic coefficients sampled from a zero-mean Gaussian distribution,
\[
\alpha \sim \mathcal{N}(0, 0.5^2),
\]
introducing moderate variability along the decision axis while largely preserving the original separation structure.

\paragraph{Strong decision perturbation}

Strong perturbations used margin-reducing displacements directed opposite to the decision sign, with perturbation magnitudes centered around \(1.5\) and additional Gaussian variability. These perturbations frequently altered the sign of the decision variable, thereby disrupting the original decision geometry.

\paragraph{Implementation details}

All transformations were fitted independently within each cross-validation fold using training data only. Test samples were transformed using the corresponding fold-specific parameters. Labels were preserved throughout, and no synthetic labels were introduced.

\subsection{Empirical risk estimation and statistical testing}

Inference risk was evaluated under a specified decoder family $\mathcal{F}$ using negative log-likelihood (log-loss), a strictly proper scoring rule sensitive to predictive uncertainty. Predicted probabilities were clipped to $[10^{-8},\,1-10^{-8}]$ before evaluation to avoid numerical instability.

Empirical risk for a decoder \(f \in \mathcal{F}\) was computed as the average log-loss over held-out samples:
\[
\widehat{R}(f)
=
-\frac{1}{n}
\sum_{i=1}^{n}
\log p_f(y_i \mid x_i).
\]

Achievable risk within $\mathcal{F}$ was operationally approximated using stratified 5-fold cross-validation, with models trained on each training fold and evaluated on the corresponding held-out fold. Fold-averaged held-out risk was used as the empirical estimate of $
\widehat{R}_{\mathbf{X}}^{\mathcal{F},*}$.

Uncertainty in $\Delta_{\mathrm{IE}}$ and $\Delta_{\mathrm{Transfer}}$ was estimated using \(B=500\) stochastic transformation realizations with fixed cross-validation splits. For each realization, transformation datasets were regenerated and risk differences recomputed. Empirical 95\% percentile intervals across transformation realizations were used to quantify uncertainty in $\Delta_{\mathrm{IE}}$ and $\Delta_{\mathrm{Transfer}}$.

\subsection{Empirical identifiability of Inference Equivalence}

To characterize empirical identifiability of IE, we defined a normalized risk index:
\[
\tilde{R}
=
\frac{
\widehat{R}_{\mathbf{X}}^{\mathcal{F},*}
-
\widehat{R}_{\min}
}{
\widehat{R}_{\mathrm{null}}
-
\widehat{R}_{\min}
}.
\]

Here, \(\widehat{R}_{\mathbf{X}}^{\mathcal{F},*}\) denotes the cross-validated held-out risk under decoder family \(\mathcal{F}\), \(\widehat{R}_{\min}\) is the in-sample resubstitution risk obtained by fitting and evaluating the decoder on the full dataset, and \(\widehat{R}_{\mathrm{null}}\) is the log-loss of a constant predictor defined by the empirical class prior.

Values of \(\tilde{R}\) near \(0\) correspond to saturated regimes, whereas values near \(1\) indicate degenerate regimes approaching null performance. The identifiable regime was defined as
\[
0.15 < \tilde{R} < 0.85.
\]
\subsection{Descriptive benchmark metrics}

To quantify descriptive similarity between the original representation \(X\) and its transformed counterpart \(Y\), we computed benchmark statistics capturing first-order, second-order, and principal-subspace structure.

\paragraph{First-order statistics}

We quantified discrepancies in feature-wise means and variances using average absolute differences across dimensions.

\paragraph{Second-order statistics}

Covariance structure was compared using both the Frobenius norm between empirical covariance matrices,
\[
\mathcal{D}_{\Sigma}
=
\|\Sigma_X-\Sigma_{X_s}\|_F,
\]
and the correlation between vectorized covariance matrices.


\paragraph{Principal-subspace structure}

We compute PCA on $X$ and extract the top-$k$ components ($k=10$). We quantify spectral deviation between $X$ and $X_s$ using explained variance ratios:
\[
\mathrm{EV}_{\mathrm{diff}} = \frac{1}{k}\sum_{i=1}^{k} \left| \lambda_i(X) - \lambda_i(X_s) \right|,
\]
where $\lambda_i(\cdot)$ denotes the $i$-th explained variance ratio.

This metric is included in the descriptive benchmark set $\mathcal{B}_{\mathrm{desc}}$.


\section{Simulation Framework}
\subsection{Data generation process}

Synthetic datasets were generated from a fixed latent world comprising task-relevant and nuisance variables with controlled dimensionality and stochasticity.

Latent nuisance and task variables were independently sampled as
\[
Z_n \sim \mathcal{N}(0, I_{5}),
\qquad
Z_t \sim \mathcal{N}(0, I_{5}),
\]
with both subspaces having dimensionality \(5\). Nuisance components were scaled by a factor of \(1.5\), and selected task dimensions were coupled to nuisance structure to induce partial mixing between task-relevant and nuisance variability:
\[
\tilde{Z}_{t,1}=Z_{t,1}+0.6\,\tilde{Z}_{n,1},
\qquad
\tilde{Z}_{t,2}=Z_{t,2}+0.4\,\tilde{Z}_{n,2}.
\]

The observed representation was defined as
\[
X=[\tilde{Z}_n,\tilde{Z}_t]\in\mathbb{R}^{n\times10},
\]
with \(n=500\).

Task labels were generated from a noisy decision margin constructed from latent task signals. The primary analyses used a linear task signal:
\[
s
=
1.2 Z_{t,1}
+
0.7 Z_{t,2}
-
0.3 Z_{t,3}.
\]

Supplementary analyses additionally considered alternative task structures:
\[
\begin{aligned}
\text{nonlinear:}\quad
&s = 5.5\, Z_{t,1} Z_{t,2}, \\
\text{noise-interaction:}\quad
&s = Z_{t,1} + 0.5\, Z_{n,1}, \\
\text{rotated:}\quad
&s = Z_{t,2} - Z_{t,1}, \\
\text{sparse:}\quad
&s = 2\, Z_{t,1}.
\end{aligned}
\]

The final margin was defined as
\[
m = 0.6\,s + 1.5\,\epsilon,
\qquad
\epsilon \sim \mathcal{N}(0,I_n),
\]
and labels were sampled according to
\[
p(y=1\mid m)=\sigma(m/1.5),
\qquad
y\sim\mathrm{Bernoulli}(p),
\]
where \(\sigma(\cdot)\) denotes the logistic sigmoid function.

All datasets were generated under fixed global seed control, with dimensionality, task strength, and noise level held constant across experiments.
\subsection{Cross-seed robustness analysis}

All simulations were repeated across \(N_{\text{seed}}=10\) independent random seeds. For each seed, the latent world structure was held fixed while observational noise and transformation realizations were independently resampled. In IE and transfer analyses, each point corresponds to a seed-specific estimate of $
(\Delta_{\mathrm{IE}}, \Delta_{\mathrm{Transfer}})$,
capturing both the magnitude of the risk deviations and the associated IE and transfer status for that realization.



\section{Extended tasks and robustness checks}

To examine how IE depends on task structure, we repeated the analyses across four additional task regimes spanning nonlinear interactions, stochastic interaction structure, rotated linear geometry, and sparse low-dimensional signal structure. These settings systematically varied the relationship between task geometry, decoder family, and controlled perturbation while preserving the same evaluation framework.

\subsection{Inference Equivalence behavior across tasks and decoder families}


Figures~\ref{fig:S3}--\ref{fig:S18} summarize the IE and transfer regime maps across the four task settings introduced above, under both linear and quadratic decoder families. 

\clearpage

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S3.pdf}

\caption{
IE regime map for the nonlinear task under linear decoder family.
}
\label{fig:S3}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S4.pdf}

\caption{
Same Fig.~\ref{fig:S3}, but under quadratic decoder family.
}
\label{fig:S4}
\end{figure}





\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S5.pdf}

\caption{
IE regime map for the noise interaction task under linear decoder family.
}
\label{fig:S5}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S6.pdf}

\caption{
Same Fig.~\ref{fig:S5}, but under quadratic decoder family.
}
\label{fig:S6}
\end{figure}




\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S7.pdf}

\caption{
IE regime map for the rotated task under linear decoder family.
}
\label{fig:S7}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S8.pdf}

\caption{
Same Fig.~\ref{fig:S7}, but under quadratic decoder family.
}
\label{fig:S8}
\end{figure}




\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S9.pdf}

\caption{
IE regime map for the sparse task under linear decoder family.
}
\label{fig:S9}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S10.pdf}

\caption{
Same Fig.~\ref{fig:S9}, but under quadratic decoder family.
}
\label{fig:S10}
\end{figure}




\clearpage

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S11.pdf}

\caption{
Transfer regime map for the nonlinear task under linear decoder family.
}
\label{fig:S11}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S12.pdf}

\caption{
Same Fig.~\ref{fig:S11}, but under quadratic decoder family.
}
\label{fig:S12}
\end{figure}





\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S13.pdf}

\caption{
Transfer regime map for the noise interaction task under linear decoder family.
}
\label{fig:S13}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S14.pdf}

\caption{
Same Fig.~\ref{fig:S13}, but under quadratic decoder family.
}
\label{fig:S14}
\end{figure}




\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S15.pdf}

\caption{
Transfer regime map for the rotated task under linear decoder family.
}
\label{fig:S15}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S16.pdf}

\caption{
Same Fig.~\ref{fig:S15}, but under quadratic decoder family.
}
\label{fig:S16}
\end{figure}




\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S17.pdf}

\caption{
Transfer regime map for the sparse task under linear decoder family.
}
\label{fig:S17}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{Figure_S18.pdf}

\caption{
Same Fig.~\ref{fig:S17}, but under quadratic decoder family.
}
\label{fig:S18}
\end{figure}

Across task settings, both IE and transfer analyses reveal stable qualitative regimes together with systematic decoder-family-dependent deviations relative to the linear task setting reported in the main text. Decision-Weak perturbations consistently disrupt both achievable inference and transfer performance under linear and quadratic decoder families, indicating persistent sensitivity of task-aligned decoding to perturbations along the decision axis (Figs.~\ref{fig:S3}--\ref{fig:S18}, panels E). In contrast, strongly destructive transformations—particularly Mean-only, Gaussian-Copula, and Decision-Strong—remain predominantly non-equivalent across both IE and transfer analyses (Figs.~\ref{fig:S3}--\ref{fig:S18}, panels B, C, and E). A notable exception arises in the nonlinear task under quadratic decoding, where the Gaussian-Copula surrogate satisfies IE and approaches stable transfer (Figs.~\ref{fig:S4}D and \ref{fig:S12}D). This suggests that preserved dependence structure can become sufficient for both re-optimized inference and fixed-decoder generalization when aligned with computations accessible to the decoder family.

Intermediate statistical transformations (IID-Marginal, Covariance-Gaussian, and Latent-FA) exhibit the strongest dependence on task geometry and decoder family. Although these transformations are frequently inference-equivalent and transfer-stable under linear decoding, they become substantially less stable under quadratic decoding across multiple task settings (Figs.~\ref{fig:S3}--\ref{fig:S18}, panels D). The effect is particularly pronounced in the noise-interaction task, where transfer consistently degrades under both decoder families despite partial IE preservation in some settings, indicating heightened sensitivity of fixed-decoder generalization to interaction-driven structure. By contrast, Covariance-Gaussian surrogates remain comparatively stable in the nonlinear task under quadratic decoding, suggesting that preservation of second-order structure can support both inference and transfer when aligned with the inductive structure of the decoder family. Taken together, these results indicate that both IE and transfer depend not only on preserved descriptive statistics, but on whether the transformed representation retains task-relevant structure accessible to the admissible decoder family.

\subsection{Table S1: Descriptive Benchmark Metrics}


To characterize the magnitude of distributional and geometric changes induced by each transformation, we compute a set of descriptive benchmark metrics capturing first- and second-order statistics as well as principal subspace structure. These measures are reported in Table~\ref{tab:s1}.

\begin{table}[ht]
\centering
\caption{
Summary of distributional and geometric differences induced by each controlled perturbation.
Metrics quantify deviations in first- and second-order statistics (mean, variance, covariance) and principal subspace structure.
Values are reported as mean $\pm$ standard deviation across random seeds.
}
\label{tab:s1}
\resizebox{\textwidth}{!}{%
\begin{tabular}{lccccc}
\hline
\textbf{Perturbation} & \textbf{mean\_diff} & \textbf{var\_diff} & \textbf{cov\_frobenius} & \textbf{cov\_corr} & \textbf{ev\_diff} \\ \hline
Mean-only       & 0.0000 $\pm$ 0.0000 & 1.6784 $\pm$ 0.0061 & 6.0477 $\pm$ 0.0284 & 0.4138 $\pm$ 0.0525 & 0.1627 $\pm$ 0.0000 \\
IID-Marginal    & 0.0482 $\pm$ 0.0106 & 0.0884 $\pm$ 0.0148 & 2.3733 $\pm$ 0.1197 & 0.9140 $\pm$ 0.0090 & 0.0181 $\pm$ 0.0011 \\
Cov-Gaussian    & 0.0424 $\pm$ 0.0096 & 0.1185 $\pm$ 0.0198 & 0.9219 $\pm$ 0.0984 & 0.9876 $\pm$ 0.0024 & 0.0073 $\pm$ 0.0011 \\
Latent-FA       & 0.0476 $\pm$ 0.0112 & 0.0912 $\pm$ 0.0285 & 0.8296 $\pm$ 0.0895 & 0.9904 $\pm$ 0.0023 & 0.0047 $\pm$ 0.0016 \\
Copula          & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.9355 $\pm$ 0.0895 & 0.9869 $\pm$ 0.0027 & 0.0069 $\pm$ 0.0006 \\
Decision-Weak   & 0.0031 $\pm$ 0.0021 & 0.0259 $\pm$ 0.0062 & 0.2742 $\pm$ 0.0599 & 0.9989 $\pm$ 0.0005 & 0.0015 $\pm$ 0.0003 \\
Decision-Strong & 0.0049 $\pm$ 0.0052 & 0.1192 $\pm$ 0.0259 & 1.6696 $\pm$ 0.1087 & 0.9587 $\pm$ 0.0056 & 0.0047 $\pm$ 0.0013 \\ \hline
\end{tabular}%
}

\end{table}


\end{document}