Monocular Depth Estimation via Neural Network with Learnable Algebraic Group and Ring Structures

• URL: http://arxiv.org/abs/2604.24328v1
• Date: Mon, 27 Apr 2026 11:19:39 GMT
• Title: Monocular Depth Estimation via Neural Network with Learnable Algebraic Group and Ring Structures
• Abstract summary: LAGRNet is a novel framework that grounds MDE in algebraic geometry by explicitly embedding learnable group, ring, and sheaf structures into the deep learning pipeline.<n>We show that LAGRNet significantly outperforms state-of-the-art methods in both accuracy and generalization capabilities.
• Score: 17.239765616051507
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Monocular depth estimation (MDE) has witnessed remarkable progress driven by Convolutional Neural Networks and transformer-based architectures. However, these approaches typically treat the problem as a generic image-to-image regression on Euclidean grids, thereby overlooking the intrinsic algebraic and geometric structures induced by perspective projection. To address this limitation, we propose LAGRNet, a novel framework that fundamentally grounds MDE in algebraic geometry by explicitly embedding learnable group, ring, and sheaf structures into the deep learning pipeline. Modeling feature maps as sections of a sheaf over an approximated image manifold, our method first establishes a Group-defined Feature Manifold (GFM) parameterized by a learned algebraic group action to enforce projective equivariance and robustness against view changes. To facilitate algebraically consistent cross-scale interactions, we subsequently introduce a Ring Convolution Layer (RCL) that formulates feature fusion as a graded ring homomorphism. Furthermore, to ensure global topological consistency, a Sheaf-based Module (SM) aggregates local depth cues via Čech nerve on the image topology. Extensive zero-shot evaluations across the KITTI, NYU-Depth V2, and ETH3D benchmarks demonstrate that LAGRNet significantly outperforms state-of-the-art methods in both accuracy and generalization capabilities.

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