Supervised and unsupervised methods in machine learning and computer vision using data-driven Log-Determinant Divergences in the manifold space of symmetric and positive definite covariance matrices

Abstract

Understanding the structure of data is fundamental to building robust and efficient models in signal processing, image processing, and machine learning. Traditional approaches, such as Sparse Representations and Bag-of-Words, rely on dictionary-based representations, where data is expressed as linear combinations of learned atoms. While these methods were prevalent in early computer vision research, they have been largely overshadowed by Convolutional Neural Networks and Vision Transformer models. However, despite their strong performance, these models often lack interpretability, making it difficult to derive meaningful insights into data structure. To address this challenge, researchers have explored alternative methodologies, such as differential manifolds and graph-based models. Recently, an approach integrating the widely used Symmetric Positive Definite manifold with dictionary learning has been proposed to tackle multi-class machine learning problems through manifold optimization algorithms. This data-driven framework jointly optimizes both the dictionary and the manifold metric to better capture the underlying data structure. This thesis builds upon this methodology and proposes an extension tailored for Writer-Independent Signature Verification. The proposed model refines the dictionary learning process and adapts the manifold metric to effectively differentiate between genuine and forged signatures using either classification or metric learning methodologies.