Depth-Dependent Hidden-State Collapse in Dynamical System Autoencoders for LiDAR Point-Cloud Classification

We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions $d\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstruction-derivative error is $\max\{1-M(M-m)/2,0\}$, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale $0.05$, the mean theoretical bound is $0.155$, about $84\%$ of the mean normalized training error $0.185$ across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below $6\times10^{-6}$.

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.

LiDAR point cloud compression is vital for autonomous systems to handle massive data from high-resolution sensors. While learned entropy modeling built upon octree structures yields high compression gains, it faces two critical bottlenecks: 1) prohibitive latency, particularly during decoding, caused by causal, multi-stage context modeling; and 2) a rigid performance-latency trade-off, preventing a single model from adapting to varying constraints. These limitations stem from the tight coupling between the context aggregation backbone and probability prediction. To address this, we propose PACE, a new framework that reformulates ancestral context aggregation as a non-causal backbone and confines causality to a lightweight, stage-scalable predictor, eliminating repetitive backbone executions and reducing computational overhead. The predictor supports an arbitrary number of prediction stages, enabling seamless adaptation across diverse performance-latency trade-offs without reloading parameters. Experiments demonstrate that PACE sets a new state-of-the-art in compression efficiency, achieving notable BD-BR savings and reducing decoding latency by over 90% in autoregressive mode, making it attractive for practical applications.