A Comparative Mathematical Study of Autoencoder Variants for Image Reconstruction on the Olivetti Faces Dataset

Resumen

This paper presents a comparative mathematical analysis of several autoencoder-based models applied to image reconstruction tasks, using the Olivetti Faces dataset as a case study. The models considered include the classical Autoencoder (AE), the Variational Autoencoder (VAE), and the Convolutional Autoencoder (CAE). Each variant is examined from both a theoretical and practical standpoint, focusing on their respective latent space formulations, optimization frameworks, and reconstruction accuracies.   The Olivetti dataset, comprising 400 grayscale facial images across 40 distinct subjects, is employed to evaluate model performance. The study involves structured preprocessing, model training, and performance evaluation using two key metrics: the Mean Squared Error (MSE) and the Structural Similarity Index (SSIM). Our mathematical analysis investigates the learning dynamics and reconstruction capabilities of each architecture, supported by empirical evidence. Results demonstrate that the CAE provides superior reconstruction quality, attributed to its spatial feature extraction capabilities.   This study contributes to the mathematical understanding of unsupervised learning models in image processing, highlighting the trade-offs among different autoencoder variants. These insights may serve as a foundation for further theoretical development and practical deployment in related mathematical and computational imaging applications.