Beyond the Simplex: Hadamard-Infused Deep Sparse Representations for Enhanced Similarity Measures

Graphical representations are essential for compre-hending high-dimensional data across diverse fields, yet their construction often presents challenges due to the limitations of traditional methods. This paper introduces a novel methodology, Beyond Simplex Sparse Representation (BSSR), which addresses critical issues such as parameter dependencies, scale inconsistencies, and biased data interpretation in constructing similarity graphs. BSSR leverages the robustness of sparse representation to noise and outliers, while incorporating deep learning techniques to enhance scalability and accuracy. Furthermore, we tackle the optimization of the standard simplex, a pervasive problem, by introducing a transformative approach that converts the constraint into a smooth manifold using the Hadamard parametrization. Our proposed Tangent Perturbed Riemannian Gradient Descent (T-PRGD) algorithm provides an efficient and scalable solution for optimization problems with standard simplex or
$\ell_{1}$
-norm sphere constraints. These contributions, including the BSSR methodology, robustness and scalability through deep representation, shift-invariant sparse representation, and optimization on the unit sphere, represent major advancements in the field. Our work offers novel perspectives on data representation challenges and sets the stage for more accurate analysis in the era of big data.