Lost and Found in Translation: Variational Diagnostics for Neural Codebook Channels

TL;DR

This paper introduces a diagnostic tool for VAEs called the neural codebook channel, which assesses whether the decoder correctly interprets the encoder's latent code, addressing a gap in existing VAE diagnostics.

Contribution

It develops a new coupled encoder-decoder diagnostic with an architecture-free Bernoulli-KL certificate to detect mismatched decoding in neural codebooks.

Findings

The certificate bounds the encoder-decoder disagreement with high accuracy on multiple datasets.

The framework detects mismatched decoding as a failure mode in deep generative models.

The package provides an audit-ready reporting unit for assessing codebook compatibility.

Abstract

Classical communication systems fail not only through random noise but also when transmitter and receiver use incompatible operational codebooks. Variational autoencoders (VAEs) train an encoder $q_\phi$ and decoder $p_\theta$ jointly, and practitioners treat the resulting latent space as a discrete code -- for clustering, conditional generation, and mechanistic interpretability. Yet standard VAE diagnostics -- ELBO, active units, mutual information, and code histograms -- certify only whether this code is used, never whether the decoder reads each latent under the encoder's code. We close this gap with the neural codebook channel $K_{e\to d}(j\mid i)$, a coupled encoder-decoder diagnostic whose off-diagonal mass is bounded by an architecture-free Bernoulli-KL certificate $d_{\mathrm{bin}}(1-\mathcal{A} \,\|\, \bar\eta_p) \le \bar\Delta$ controlled by the variational gap. The certificate is the operational specialization of the classical KL chain rule under disintegration to the encoder-decoder disagreement event, complemented by a constructive marginal-impossibility result: no combination of marginal histograms, entropies, active-code counts, or mutual information determines $K_{e\to d}$. We audit the certificate on four sklearn datasets (finite-grid exact, 5/5 seeds, 20/20 pairs satisfy the bound), a 2D model where the bound is non-vacuous at $2.71\times$ the observed disagreement and the four-term identity closes within $10^{-4}$, MNIST under importance-sampling control, and a VQ-VAE attaining the predicted limit $\hat{\mathcal{A}}=1.000$. The package $(K_{e\to d}, \mathcal{A}, R_{\mathrm{eff}}, R, \mathrm{AU})$ is an audit-ready reporting unit. More broadly, the framework makes mismatched decoding -- a failure mode classical communication theory named decades ago -- visible inside a single deep generative model.