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

TL;DR

This survey addresses the core problem of representing real-world data with continuous function mappings rather than discrete grids, emphasizing resolution flexibility, cross-modal adaptability, smoothness, and parameter efficiency. It systematically categorizes methods into parametric designs (basis functions, INRs, grid-encodings) and structural modeling (tensor decompositions, statistical/Bayesian frameworks, continuous regularization), and then discusses their theoretical underpinnings through approximation theory, NTK-based convergence/generalization, and implicit regularization. The paper surveys diverse real-world applications across vision/graphics, scientific computing, medical imaging, and geosciences, illustrating how continuous representations enable high-fidelity reconstruction, efficient computation, and robust modeling under irregular sampling. It concludes with forward-looking directions, advocating cross-disciplinary integration, meta-learning, physics-informed constraints, and principled theory–algorithm co-design to realize practical impact at scale. Key contributions include organizing a fragmented literature into a cohesive framework, summarizing theoretical advances (NTK and implicit regularization), and outlining concrete application pathways and open challenges for continuous representation methods.

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

Figures (9)

• Figure 1: This work is structured to provide a comprehensive overview of continuous representation methods through three perspectives. First, we systematically review continuous representation methods including parametric model architectures and data structural modeling strategies (Section \ref{['sec:method']}). Second, we review representative theoretical frameworks for formally analyzing the properties and capabilities of continuous representation methods (Section \ref{['sec:theory']}). Third, we review real-world applications of continuous representation methods derived from diverse fields (Section \ref{['sec:applications']}). Finally, we discuss future research directions and perspectives aimed at further enhancing the effectiveness, theoretical insights, and applicability of continuous representation frameworks (Section \ref{['sec:future']}).
• Figure 2: The figure illustrates a comparative analysis between discrete grid modeling and emerging continuous representation modeling approaches. In contrast to conventional grid representations, the novel continuous representation paradigm employs a parametric mapping function $f(\cdot)$ that maps continuous-domain coordinates to their corresponding values, deriving intrinsic advantage such as resolution flexibility, irregular data modeling ability, global characterization of data structures, and interpretable learning theories (such as functional decomposition analysis NeurIPS_FtSVDLRTFR) that extends beyond discrete grids.
• Figure 3: The figure illustrates the structures of the positional-encoding-based INR PE and the multi-resolution grid encoding-based INR InstantNGP. (a) The PE-based INR maps the input coordinate through a Fourier mapping layer before passing through an MLP for effective continuous representation. (b) The multi-resolution grid encoding-based INR (e.g., InstantNGP InstantNGP) constructs learnable feature vectors stored on grids with multiple resolutions. To query the point outside grids, linear interpolation is computed and the interpolated results of different grid resolutions are concatenated and passed through an MLP. Here, the learnable feature vectors are efficiently stored in hash tables.
• Figure 4: The grid encoding-based method InstantNGP InstantNGP achieves faster convergence and captures fine details better as compared with classical INR methods, including positional-encoding (PE) PE and SIREN SIREN. The top row shows image fitting and the bottom shows novel view synthesis using NeRF NeRF of different models with the same iteration step.
• Figure 5: Continuous regularization methods such as NeurTV NeurTV and continuous representation-based nonlocal model (CRNL) CRNL alleviate the overfitting phenomenon of pure INR for image reconstruction tasks including image inpainting (top row) and denoising (bottom row) by incorporating intrinsic priors of data.

Theorems & Definitions (8)

• Theorem 1: Universal approximation Universal_NNUniversal_Ellacott_1994
• Theorem 2: Function class represented by INRDictionary_INR
• Theorem 3: Function class represented by INR using template function convolutions WIRE_theory
• Theorem 4: Functional $t$-SVD NeurIPS_FtSVD
• Theorem 5: Convergence to the NTK with infinite-width at initialization NTK
• Theorem 6: Convergence to the NTK during training NTK
• Theorem 7: Training convergence rate Fine_grained_NTK
• Theorem 8: Implicit regularization of deep matrix factorization Implicit_regularization_NIPS_2019