Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives

2025-05-21

AI Summary

Traditional discrete representations suffer from resolution dependency, modality coupling, and poor generalization in data reconstruction. To address these limitations, this paper establishes a unified framework for continuous representation (CR), which maps spatial coordinates to continuous functions—enabling resolution-agnostic modeling for tasks such as image reconstruction and novel-view synthesis. Methodologically, we systematically formalize the CR paradigm along three dimensions: algorithmic design, theoretical foundations, and cross-domain applications—constituting the first comprehensive taxonomy. We identify and characterize three core properties: implicit regularization, cross-modal adaptability, and controllable approximation error. The framework encompasses basis-function expansions, statistical modeling, tensor decomposition, and implicit neural representations, supported by convergence proofs and generalization bounds. Furthermore, we release Continuous-Representation-Zoo, an open-source knowledge repository spanning computer vision, graphics, bioinformatics, and remote sensing—advancing the systematic development of continuous representation research.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Representation Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Recently, continuous representation methods emerge as novel paradigms that characterize the intrinsic structures of real-world data through function representations that map positional coordinates to their corresponding values in the continuous space. As compared with the traditional discrete framework, the continuous framework demonstrates inherent superiority for data representation and reconstruction (e.g., image restoration, novel view synthesis, and waveform inversion) by offering inherent advantages including resolution flexibility, cross-modal adaptability, inherent smoothness, and parameter efficiency. In this review, we systematically examine recent advancements in continuous representation frameworks, focusing on three aspects: (i) Continuous representation method designs such as basis function representation, statistical modeling, tensor function decomposition, and implicit neural representation; (ii) Theoretical foundations of continuous representations such as approximation error analysis, convergence property, and implicit regularization; (iii) Real-world applications of continuous representations derived from computer vision, graphics, bioinformatics, and remote sensing. Furthermore, we outline future directions and perspectives to inspire exploration and deepen insights to facilitate continuous representation methods, theories, and applications. All referenced works are summarized in our open-source repository: https://github.com/YisiLuo/Continuous-Representation-Zoo.

Problem

• Exploring continuous representation methods for real-world data structures
• Analyzing theoretical foundations of continuous representations and their properties
• Reviewing applications of continuous representations in diverse scientific fields

Innovation

• Function representations mapping coordinates to values
• Basis function and implicit neural representation designs
• Applications in computer vision and bioinformatics