  High-dimensional datasets present substantial challenges in statistical
modeling across various disciplines, necessitating effective dimensionality
reduction methods. Deep learning approaches, notable for their capacity to
distill essential features from complex data, facilitate modeling,
visualization, and compression through reduced dimensionality latent feature
spaces, have wide applications from bioinformatics to earth sciences. This
study introduces a novel workflow to evaluate the stability of these latent
spaces, ensuring consistency and reliability in subsequent analyses. Stability,
defined as the invariance of latent spaces to minor data, training
realizations, and parameter perturbations, is crucial yet often overlooked.
  Our proposed methodology delineates three stability types, sample,
structural, and inferential, within latent spaces, and introduces a suite of
metrics for comprehensive evaluation. We implement this workflow across 500
autoencoder realizations and three datasets, encompassing both synthetic and
real-world scenarios to explain latent space dynamics. Employing k-means
clustering and the modified Jonker-Volgenant algorithm for class alignment,
alongside anisotropy metrics and convex hull analysis, we introduce adjusted
stress and Jaccard dissimilarity as novel stability indicators.
  Our findings highlight inherent instabilities in latent feature spaces and
demonstrate the workflow's efficacy in quantifying and interpreting these
instabilities. This work advances the understanding of latent feature spaces,
promoting improved model interpretability and quality control for more informed
decision-making for diverse analytical workflows that leverage deep learning.
