The need for 3D reconstruction from incomplete data sources leads to difficulties in scenarios where geometric fidelity is crucial. Consequently, the reconstruction from partial or sparse inputs remains a critical challenge in applications such as robotics, autonomous navigation, and cultural heritage digitization, where geometric accuracy and structural consistency are essential. Moreover, existing methods typically rely on Euclidean vector spaces and traditional loss functions, which often fail to capture the full complexity of spatial relationships. Consequently, this work presents a novel approach to 3D reconstruction by introducing Geometric Algebra (GA) as a guiding framework within deep learning models. More in detail, we propose two novel GA-based loss functions, i.e., the MultivectorLoss and the PGALoss, that embed geometric structure directly into the optimization process. This formulation enables the neural network to reason about orientation, scale, and alignment in a more coherent and interpretable way. Preliminary results on the ShapeNet and MVP datasets underscore promising performances when employed in training well-known state-of-the-art models. These findings highlight the potential and innovative use of Geometric Algebra as a powerful and underexplored tool for advancing 3D deep learning tasks.