Empirical Variational Autoencoder

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 generate samples from probabilistic latent representations but do not distinguish uncertainty about the latent location from variability around it. We formulate ELVAE, an evidential learning-based VAE in which each latent coordinate is governed by an input-dependent normal-inverse-gamma posterior. This hierarchy yields an explicit latent-location uncertainty that can be used during generation, not merely reported after inference: low-uncertainty anchors support more reliable synthetic samples, while high-uncertainty anchors can be deliberately exploited for stress testing. The objective is an exact evidence lower bound, and we show that direct regularization of the full hierarchy is required, since the marginalized latent law alone cannot identify the uncertainty decomposition. In an MNIST generation pilot with a frozen external classifier, this uncertainty clearly stratified the semantic reliability of generated digits. A zero-displacement control revealed that most of the effect reflects how reliably an anchor can be re-generated, while a smaller but distinct component is attributable to uncertainty-scaled perturbation itself. The effect holds only under within-class uncertainty ranking, and its magnitude varies across seeds. These findings support the learned latent-location uncertainty as a practical control variable for uncertainty-aware generation, separating anchor reliability from perturbation-induced failure.

Variational Autoencoders (VAEs) commonly assume a standard isotropic Gaussian prior over the latent space, an assumption that often fails to capture the true distribution of latent representations for complex datasets. This mismatch can limit reconstruction accuracy, reduce sample quality, and constrain the expressive power of the learned latent space. We propose the eXact-Prior Variational Autoencoder (X-VAE), a framework that replaces the conventional standard normal prior with a Gaussian prior derived from the latent representations of a pretrained autoencoder (AE). Specifically, the empirical mean and standard deviation of the AE latent codes are used to parameterize a data-adaptive prior that more closely reflects the underlying structure of the training data. During generation, X-VAE introduces a latent scaling factor that enables explicit control over the variance of the sampled latent vectors, providing a simple mechanism for balancing sample diversity and fidelity. This flexibility makes the proposed approach particularly well suited for applications such as industrial and engineering design, where generated solutions must satisfy strict structural or functional constraints while still permitting meaningful design exploration. We present the mathematical formulation of well-suited X-VAE, derive the corresponding KL divergence objective for the proposed prior, and evaluate the method on standard benchmark datasets. Experimental results demonstrate that X-VAE preserves reconstruction quality while producing latent representations that better align with the empirical data distribution, leading to improved controllability and more realistic generated samples.