# Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery

arXiv: https://arxiv.org/abs/2212.00262 · HTML: https://mlsanity.com/paper/2212.00262/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 104 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## Builds on (most cited first, 30 of 48)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [pi-GAN: Periodic Implicit Generative Adversarial Networks for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2012.00926/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Implicit Functions in Feature Space for 3D Shape Reconstruction and Completion](https://mlsanity.com/paper/2003.01456/) (2020)
- [Neural RGB-D Surface Reconstruction](https://mlsanity.com/paper/2104.04532/) (2021)
- [State of the Art on Neural Rendering](https://mlsanity.com/paper/2004.03805/) (2020)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [PU-Net: Point Cloud Upsampling Network](https://mlsanity.com/paper/1801.06761/) (2018)
- [Shape As Points: A Differentiable Poisson Solver](https://mlsanity.com/paper/2106.03452/) (2021)
- [Meta-SR: A Magnification-Arbitrary Network for Super-Resolution](https://mlsanity.com/paper/1903.00875/) (2019)
- [PU-GAN: a Point Cloud Upsampling Adversarial Network](https://mlsanity.com/paper/1907.10844/) (2019)
- [SnowflakeNet: Point Cloud Completion by Snowflake Point Deconvolution with Skip-Transformer](https://mlsanity.com/paper/2108.04444/) (2021)
- [Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces](https://mlsanity.com/paper/2011.13495/) (2020)
- [Adversarial Generation of Continuous Images](https://mlsanity.com/paper/2011.12026/) (2020)
- [Morphing and Sampling Network for Dense Point Cloud Completion](https://mlsanity.com/paper/1912.00280/) (2019)
- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Point Cloud Upsampling via Disentangled Refinement](https://mlsanity.com/paper/2106.04779/) (2021)

## Built on by (most cited first, 30 of 46)

- [Revisiting Nonlocal Self-Similarity from Continuous Representation](https://mlsanity.com/paper/2401.00708/) (2024)
- [Functional Bayesian Tucker Decomposition for Continuous-indexed Tensor Data](https://mlsanity.com/paper/2311.04829/) (2023)
- [Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observations](https://mlsanity.com/paper/2505.09284/) (2025)
- [Hyperspectral and multispectral image fusion with arbitrary resolution through self-supervised representations](https://mlsanity.com/paper/2405.17818/) (2024)
- [Spatiotemporal Implicit Neural Representation as a Generalized Traffic Data Learner](https://mlsanity.com/paper/2405.03185/) (2024)
- [NeurTV: Total Variation on the Neural Domain](https://mlsanity.com/paper/2405.17241/) (2024)
- [PromptHSI: Universal Hyperspectral Image Restoration with Vision-Language Modulated Frequency Adaptation](https://mlsanity.com/paper/2411.15922/) (2024)
- [F-INR: Functional Tensor Decomposition for Implicit Neural Representations](https://mlsanity.com/paper/2503.21507/) (2025)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Cross-Frequency Implicit Neural Representation with Self-Evolving Parameters](https://mlsanity.com/paper/2504.10929/) (2025)
- [Calibrated Harmonic Overlaid Implicit Neural Representations for Multi-Dimensional Data](https://mlsanity.com/paper/2606.26763/) (2026)
- [Functional Complexity-adaptive Temporal Tensor Decomposition](https://mlsanity.com/paper/2502.06164/) (2025)
- [It's Not the Target, It's the Background: Rethinking Infrared Small Target Detection via Deep Patch-Free Low-Rank Representations](https://mlsanity.com/paper/2506.10425/) (2025)
- [Low-rank tensor completion via tensor joint rank with logarithmic composite norm](https://mlsanity.com/paper/2309.16208/) (2023)
- [Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration](https://mlsanity.com/paper/2407.08509/) (2024)
- [Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization](https://mlsanity.com/paper/2506.22134/) (2025)
- [Beyond Motion Cues and Structural Sparsity: Revisiting Small Moving Target Detection](https://mlsanity.com/paper/2509.07654/) (2025)
- [When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning](https://mlsanity.com/paper/2512.21486/) (2025)
- [Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2603.01034/) (2026)
- [Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework](https://mlsanity.com/paper/2603.22362/) (2026)
- [A Distributional View for Visual Mechanistic Interpretability: KL-Minimal Soft-Constraint Principle](https://mlsanity.com/paper/2605.17504/) (2026)
- [Collaborative Imputation of Urban Time Series through Cross-city Meta-learning](https://mlsanity.com/paper/2501.11306/) (2025)
- [Dynamic PET Image Reconstruction via Non-negative INR Factorization](https://mlsanity.com/paper/2503.08025/) (2025)
- [Under-Sampled High-Dimensional Data Recovery via Symbiotic Multi-Prior Tensor Reconstruction](https://mlsanity.com/paper/2504.05992/) (2025)
- [Motion-Enhanced Nonlocal Similarity Implicit Neural Representation for Infrared Dim and Small Target Detection](https://mlsanity.com/paper/2504.15665/) (2025)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [Score-Based Model for Low-Rank Tensor Recovery](https://mlsanity.com/paper/2506.22295/) (2025)
- [Mode-Aware Non-Linear Tucker Autoencoder for Tensor-based Unsupervised Learning](https://mlsanity.com/paper/2508.06784/) (2025)
- [Superpixel-informed Continuous Low-Rank Tensor Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2508.12261/) (2025)
- [Robust Spatiotemporally Contiguous Anomaly Detection Using Tensor Decomposition](https://mlsanity.com/paper/2510.00460/) (2025)

## Abstract (arXiv)

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can only represent data on finite meshgrid due to their intrinsical discrete nature, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR), which can continuously represent data beyond meshgrid with infinite resolution. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in the LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.