Posterior Collapse

A language model $p_θ(y \mid x)$ trained on reasoning tasks learns to solve problems via multiple distinct strategies, yet these strategies are implicit and entangled within the model's response distribution. We study the problem of decomposing the response distribution of a given pretrained language model into a structured, strategy-conditioned representation. Specifically, we learn a latent-variable factorization $p_θ(y \mid x) \leadsto (r_φ(z \mid x), g_φ(y \mid x,z))$, where a router $r$ maps each input to a distribution over latent strategies $z$ and a generator $g$ produces the response conditioned on that strategy. A key challenge is that the generator, initialized from the base model, already represents $p_θ(y \mid x)$ without using $z$. Standard variational inference therefore gives the model no incentive to route information through $z$ and can yield a severe form of posterior collapse. To address this, we propose a variational objective that measures fractional information gain relative to the base model's response loss and concentrates reconstruction pressure on tokens with high base model surprisal, encouraging $z$ to encode strategy-relevant response variation. We introduce a benchmark of multi-strategy algorithmic tasks and show that this objective recovers latent codes aligned with distinct reference strategies while preserving the base model's response distribution.

Variational Autoencoders (VAEs) frequently suffer from posterior collapse, a failure mode in which the approximate posterior converges to the prior, rendering the latent code uninformative. Despite extensive research, a unified account of why collapse occurs has remained an open question. We identify and formalize two logically independent but coupled causes. \emph{Gradient imbalance} occurs when the decoder's reconstruction signal vanishes faster than the $\mathbb{KL}$ regularization pressure as the posterior widens. \emph{Information gap} occurs when the stochastic sampling step discards a substantial fraction of the encoder's computed representation, attenuating decoder sensitivity and making collapse inexpensive. Both causes share the same collapse trajectory, and we show that the information gap is algebraically equivalent to mismatch between the aggregate posterior and the prior, unifying two pathologies. Subsequently, we introduce $λ$-VAE, which resolves both causes through a single modification to the reparameterization step: the sampling noise is scaled by per-dimension exponent, while the $\mathbb{KL}$ penalty retains the original posterior variance. This asymmetry shifts the stable training attractor away from the degenerate collapsed state, driving all latent dimensions toward the same equilibrium -- a mechanism we term \emph{variance equalization}. A closed-form optimal exponent per dimension follows from a net information gain objective, with a single hyperparameter controlling the reconstruction-generation tradeoff. We validate on standard benchmarks (Binary MNIST, Binary Omniglot, CIFAR-10, CelebA-64), showing consistent reductions in collapsed dimensions, information capacity gains of up to $2.8\times$ nats, and reconstruction quality improvements of up to $+0.33$ BPD.

DGPs are probabilistic models with remarkable prediction performance that concatenate GPs across several layers. Exact inference in DGPs is intractable, and variational inference is often used to approximate the posterior with a parametric distribution tuned by minimizing the Kullback-Leibler divergence. Moreover, finding a good VI approximation is challenging. In particular, a problem of VI is posterior collapse, where VI converges to a variational posterior that matches the prior. In variational DGPs, this implies explaining the data as noise. This work studies posterior collapse in DGPs and identifies its connection to the DSVI algorithm and the widely used linear prior mean function employed in all but the last layer. We show that the benefit of the linear prior mean does not arise from avoiding the non-injective pathology in very deep DGPs, as previously believed, but from improving the conditioning of the optimization problem at initialization. Thus, we propose an alternative initialization of a zero prior mean DGP that mimics a DGP with a linear prior mean at initialization. This enables successful training of DGPs without imposing optimization-driven constraints on the prior, allowing to choose the prior based on modeling assumptions rather than optimization convenience. Our analysis considers three common parameterizations of DGPs and shows that not all of them benefit from a linear prior mean. We also explain why a whitened parameterization of the \DGP provides more stable convergence, something often assumed from experience, but lacking a rigorous analysis. Furthermore, we show that this stability is also beneficial to avoid the posterior collapse problem. Extensive experiments validate our findings: the proposed initialization prevents posterior collapse, improves stability, and achieves performance comparable to (and sometimes better than) DGPs with a linear prior mean.

We show that posterior collapse in $β$-VAEs implements automatic spectral pruning. A latent mode collapses if its contribution to reconstruction is below the cutoff set by $β$. Equilibrium solutions with different $β$ thus reveal a cascade of collapses as latent modes decouple from least to most useful. We derive this as a consequence of the loss via a Landau stability analysis. We define a latent-rescaling-invariant order parameter that ranks active latent modes and whose collapse thresholds identify which effective variables to inspect first. In the linear Gaussian case, the collapse spectrum, utility spectrum, and normalized PCA spectrum coincide, and each collapse follows a mean-field law. We test these predictions on the WorldClim dataset.

We study exact constant collapse in variational autoencoders: the deterministic encoder mean becomes independent of the input. The prior remains the standard Gaussian. Before VAE training, we select a fixed teacher posterior from a GMM-based view of the data and attach a fixed latent-only simplex witness to the encoder mean. This construction yields two linked objects. The first is a certificate: if the witness prediction improves on the best constant predictor of the teacher, the encoder mean cannot be input-independent constant. The second is a local escape direction: on the collapsed manifold, the teacher residual gives a sample-dependent descent direction for the alignment loss. For any full-support teacher posterior, the same geometry also gives a closed-form latent code with zero teacher-witness alignment error. Its scaled versions trace a margin-energy path from the constant predictor to the exact teacher code, which quantifies non-collapse inside the protected witness subspace. We instantiate the method on MNIST, CIFAR-10, and CIFAR-100. With searched unsupervised PCA-GMM teachers, vanilla VAEs fail the teacher-witness certificate in all five seeds on CIFAR-10 and CIFAR-100, while RST variants pass in all five seeds. Under collapse-stress settings with $β_{\mathrm{KL}}\in\{2,4,8\}$, vanilla VAE again fails in all seeds, whereas RST-alpha-prefit remains certificate-positive. Escape trajectories on both natural-image datasets increase the witness margin from a low-margin initialization and exhibit nonzero teacher-induced gradient norms. The analysis is confined to exact constant collapse of the encoder mean; generation quality, decoder use, and other collapse modes remain separate questions.

Despite their ubiquity, variational autoencoders (VAEs) inherently suffer from posterior collapse, a failure mode in which latent variables are effectively ignored. This failure arises because explicit prior imposition drives optimization toward loss landscape regions corresponding to uninformative latent representations. Here, we introduce Entropic Autoencoders (EAEs), a framework in which reconstruction loss is the only explicit objective, and entropy generates the latent variables' prior implicitly through a free energy-minimizing ensemble of encoders. This ensemble biases learning toward high-volume regions of near-optimal solutions, while decoder updates direct the search trajectories toward informative latent representations. We demonstrate that EAEs mitigate posterior collapse by learning non-Gaussian, multimodal latent distributions that yield diverse, data-consistent generations and preserve different forms of underlying structure in the data. As a proof-of-concept, we show that an EAE captures a superposition of the known low-dimensional dynamics of a reaction-diffusion process. Then, we show that an EAE identifies implicit categorical distinctions in MNIST latent representations, and displays a hierarchical understanding of facial structure on the CelebA dataset, from an "all-human" face to individual-dependent features.

Variational Autoencoders (VAEs) often exhibit a polarised regime in which latent variables separate into active, passive, and mixed subsets. Existing criteria for identifying active dimensions depend on a Gaussian prior, limiting their applicability to variational models and specific priors. We propose a simple information-theoretic classification of the polarised regime based on the entropy of the mean representation. We show theoretically how this entropy couples to KL minimisation through entropy--variance bounds, and we relate the resulting criterion to Bonheme's active/passive conditions. We also clarify a key limitation: entropy of the mean alone cannot reliably distinguish active from mixed dimensions without additional signals from the variance representation. Empirically, we evaluate the entropy criterion on $β$-VAEs, identifiable VAEs, Least-Volume Autoencoders, and L2-regularised autoencoders, and find that it consistently recovers a polarised regime when such a regime is present across the model classes studied. Finally, we show that passive dimensions can yield small but consistent improvements on downstream tasks when latent codes are appropriately normalised, suggesting that collapse is often a matter of scale rather than absolute information removal.

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.