Training-Free Bottleneck Width Planning for Convolutional Autoencoders

Autoencoders (AEs) learn low-dimensional representations by mapping data into a latent space while minimizing reconstruction error. Despite their empirical success, theoretical understanding remains limited and largely restricted to linear models or settings without a bottleneck. In this work, we study nonlinear AEs with a fixed finite-dimensional bottleneck in the mean-field (MF) regime. We derive explicit MF learning dynamics for both encoder and decoder, providing a tractable characterization of training in the nonlinear setting. We show that, over finite time horizons, the empirical risk of finite-width networks trained with stochastic gradient descent closely tracks the MF risk trajectory with high probability. At optimality, we further establish that the finite-width risk converges to the MF optimum, demonstrating that finite networks are sufficiently expressive to approximate the infinite-width solution.

Autoencoders typically meet tight latent-memory budgets by making each latent representation smaller, sacrificing representational capacity. We ask a different question: can multiple wider latents be stored together instead? We introduce the Superposed Latent Autoencoder (SLAE), which preserves high-capacity latent representations while sharing storage through learned superposition. SLAE transforms latents into storage-friendly codes, binds them with randomized keys, superposes multiple codes into a single memory tensor, and learns to recover each latent before decoding. Under the same storage budget, SLAE replaces irreversible dimensional bottlenecks with structured interference that can be suppressed. Across CIFAR-10/100, SVHN, STL-10, Tiny ImageNet, and a wide range of memory budgets, SLAE substantially improves the reconstruction--memory tradeoff, reducing reconstruction error by up to 56% over conventional autoencoders at matched storage. Further analysis shows that SLAE's advantage comes from making wider representations usable under the same storage budget. These gains also extend beyond reconstruction: the information preserved by SLAE improves downstream classification by up to 16.79 percentage points under the same memory budget. Our results suggest a new principle for representation compression: instead of making every latent smaller, keep representations wide and let them share memory.

Continuous audio encoders compress audio along feature and time axes through latent width and frame rate, but their joint effect on downstream performance remains unclear. We train sixteen encoders spanning four widths and four frame rates, with downstream adapters and probes, using matched training protocols. Despite generally improved reconstruction at larger widths, automatic speech recognition (ASR) and spoken question answering (SQA) favor moderate widths at higher rates, with the best observed widths shifting toward larger values under stronger temporal compression. Frozen-model PCA interventions reveal distinct reconstruction and recognition sensitivities: removing the trailing half of the components substantially degrades ASR in high-rate 512-dimensional encoders with comparatively small reconstruction penalties, whereas 1024-dimensional encoders largely preserve both. Yet the projected 1024-dimensional model underperforms unmodified narrower models on ASR at 12.5Hz. These findings identify a width--rate interaction in downstream utility and suggest that how representations are organized during training matters beyond reconstruction fidelity and compressibility.