A Simplex Witness Certificate for Constant Collapse in Variational Autoencoders

We study exact constant collapse in variational autoencoders, where the deterministic encoder path becomes independent of the input. The VAE prior is kept as the standard Gaussian. Before VAE training, we construct a single fixed teacher posterior by searching a GMM-based approximation of the data. We then attach a fixed latent-only simplex witness to the encoder mean and compare its output with the teacher. The resulting alignment loss has an exact constant-predictor baseline: if the latent witness beats this baseline, the encoder mean cannot be input-independent constant. The same construction also gives a closed-form latent target that realizes zero teacher-witness alignment error for any full-support teacher posterior. This yields a concrete design principle: choose a teacher with nontrivial information but controlled log-odds energy, fix the witness, train only the encoder and decoder, and certify non-collapse by a positive margin. We present the theory, a minimal training protocol, and preliminary MNIST sanity checks. The analysis targets exact constant collapse. Reconstruction quality, sampling quality, and other collapse modes are evaluated with additional diagnostics rather than folded into the certificate itself.

A Testable Certificate for Constant Collapse in Teacher-Guided VAEs

Posterior collapse in variational autoencoders is often diagnosed by its symptoms: a small KL term, a strong decoder, or weak use of the latent code. These signals are useful, but they do not define a collapse boundary. We study a concrete failure mode, input-independent constant collapse, and show that this case admits an exact threshold. For any fixed nonconstant teacher distribution \(T(\cdot\mid x)\), the best constant student is the dataset-average teacher distribution, and its alignment cost is the teacher mutual information \(I_T(X;T)\). Therefore, if a strictly latent-only raw witness achieves alignment loss below this value, with a safety margin, the witness cannot be constant in the input. This identity turns a qualitative failure mode into a measurable one. In CIFAR-100 experiments with per-seed teacher search, full training stays on the certified side of the boundary, removing alignment drives the raw witness into the constant-student regime, and restarting from a collapsed checkpoint with alignment enabled restores the certificate. Tiny-ImageNet-200 fixed-target runs show the same prevention--collapse--rescue pattern across three independently searched teachers. Standard VAE-style baselines, including methods that preserve reconstruction quality or post-hoc predictability, remain negative under the raw certificate. The guarantee is intentionally narrow: it certifies that the matched nonconstant teacher-relative variation passes through the latent pathway, rather than claiming that all forms of posterior collapse have been ruled out.

Historical Consensus: Preventing Posterior Collapse via Iterative Selection of Gaussian Mixture Priors

Variational autoencoders (VAEs) frequently suffer from posterior collapse, where latent variables become uninformative and the approximate posterior degenerates to the prior. Recent work has characterized this phenomenon as a phase transition governed by the spectral properties of the data covariance matrix. In this paper, we propose a fundamentally different approach: instead of avoiding collapse through architectural constraints or hyperparameter tuning, we eliminate the possibility of collapse altogether by leveraging the multiplicity of Gaussian mixture model (GMM) clusterings. We introduce Historical Consensus Training, an iterative selection procedure that progressively refines a set of candidate GMM priors through alternating optimization and selection. The key insight is that models trained to satisfy multiple distinct clustering constraints develop a historical barrier -- a region in parameter space that remains stable even when subsequently trained with a single objective. We prove that this barrier excludes the collapsed solution, and demonstrate through extensive experiments on synthetic and real-world datasets that our method achieves non-collapsed representations regardless of decoder variance or regularization strength. Our approach requires no explicit stability conditions (e.g., $σ^{\prime 2} < λ_{\max}$) and works with arbitrary neural architectures. The code is available at https://github.com/tsegoochang/historical-consensus-vae.