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

2026-04-27

AI Summary

This work addresses the limited generalization of existing monocular depth estimation methods under viewpoint changes, which stems from their neglect of the algebraic and geometric structures inherent in perspective projection. To overcome this, the paper introduces algebraic geometry into deep learning for the first time, proposing a modeling framework that enforces projective equivariance and topological consistency. The approach achieves geometry-aware multiscale feature fusion through learnable group actions, graded ring homomorphisms, and Čech neural layer architectures. Key components include Group-defined Feature Manifolds (GFM), Ring Convolutional Layers (RCL), and a sheaf-theoretic Sheaf Module (SM). Zero-shot evaluations on KITTI, NYU-Depth V2, and ETH3D demonstrate substantial improvements over state-of-the-art methods in both accuracy and cross-domain generalization.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Deep Neural Architectures and Foundation Models
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Monocular Depth Estimation
• Algebraic Structure
• Geometric Structure
• Perspective Projection
• Depth Estimation

Innovation

• algebraic group
• ring convolution
• sheaf theory
• projective equivariance
• monocular depth estimation