Neural Operator-Grounded Continuous Tensor Function Representation and Its Applications

Ruoyang Su; Xi-Le Zhao; Sheng Liu; Wei-Hao Wu; Yisi Luo; Michael K. Ng

cs.CVMar 2, 2026

Neural Operator-Grounded Continuous Tensor Function Representation and Its Applications

Authors: Ruoyang Su, Xi-Le Zhao, Sheng Liu, Wei-Hao Wu, Yisi Luo, Michael K. Ng

Organizations: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China · School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China · Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

Abstract

Recently, continuous tensor functions have attracted increasing attention, because they can unifiedly represent data both on mesh grids and beyond mesh grids. However, since mode- $n$ product is essentially discrete and linear, the potential of current continuous tensor function representations is still locked. To break this bottleneck, we suggest neural operator-grounded mode- $n$ operators as a continuous and nonlinear alternative of discrete and linear mode- $n$ product. Instead of mapping the discrete core tensor to the discrete target tensor, proposed mode- $n$ operator directly maps the continuous core tensor function to the continuous target tensor function, which provides a genuine continuous representation of real-world data and can ameliorate discretization artifacts. Empowering with continuous and nonlinear mode- $n$ operators, we propose a neural operator-grounded continuous tensor function representation (abbreviated as NO-CTR), which can more faithfully represent complex real-world data compared with classic discrete tensor representations and continuous tensor function representations. Theoretically, we also prove that any continuous tensor function can be approximated by NO-CTR. To examine the capability of NO-CTR, we suggest an NO-CTR-based multi-dimensional data completion model. Extensive experiments across various data on regular mesh grids (multi-spectral images and color videos), on mesh girds with different resolutions (Sentinel-2 images) and beyond mesh grids (point clouds) demonstrate the superiority of NO-CTR.

Figures & tables

Fig. 1: Discrete and linear mode- $\displaystyle n$ operator and proposed continuous and nonlinear mode- $\displaystyle n$ operator. The basic unit performed upon by discrete and linear mode- $\displaystyle n$ operator is the mode- $\displaystyle n$ fiber vectors of the discrete core tensor $\displaystyle\boldsymbol{\mathcal{G}}$ , which are then mapped to the mode- $\displaystyle n$ fiber vectors of the discrete target tensor $\displaystyle\boldsymbol{\mathcal{X}}$ . The basic unit performed upon by continuous and nonlinear mode- $\displaystyle n$ operator is the mode- $\displaystyle n$ univariate fiber functions of the continuous core tensor function $\displaystyle\boldsymbol{\mathsf{G}}$ , which are then mapped to the mode- $\displaystyle n$ univariate fiber functions of the continuous target tensor function $\displaystyle\boldsymbol{\mathsf{X}}$ .

Panel of Figure 2 — Fig. 2: Visual and quantitative recovery results with discrete and linear mode- $\displaystyle n$ operators (LRTFR [ 34 ] ), continuous and nonlinear mode- $\displaystyle n$ operators (proposed NO-CTR) on the MSI Toy (top) and point cloud Frog (bottom) when the sampling rates are $\displaystyle 10\%$ .

Fig. 3: Flow chart of NO-CTR and the corresponding multi-dimensional data completion model. We first cleverly leverage neural operators to suggest continuous and nonlinear mode- $\displaystyle n$ operators. Specifically, the basic unit performed upon by continuous and nonlinear mode- $\displaystyle n$ operator is the mode- $\displaystyle n$ univariate fiber functions of the continuous core tensor function $\displaystyle\boldsymbol{\mathsf{G}}$ , which are then mapped to the mode- $\displaystyle n$ univariate fiber functions of the continuous target tensor function $\displaystyle\boldsymbol{\mathsf{X}}$ . Empowering with continuous and nonlinear mode- $\displaystyle n$ operators, we propose a neural operator-grounded continuous tensor function representation (NO-CTR), which can faithfully represent continuous target tensor function $\displaystyle\boldsymbol{\mathsf{X}}$ as a continuous core tensor function $\displaystyle\boldsymbol{\mathsf{G}}$ and a series of continuous and nonlinear mode- $\displaystyle n$ operators $\displaystyle\{\mathscr{F}^{\langle n\rangle}\}_{n=1}^{N}$ . To examine the capability of NO-CTR, we suggest an NO-CTR-based multi-dimensional data completion model.

Panel of Table 4 — TABLE I: Quantitative Recovery Results of MSI Completion

Panel of Figure 5 — Fig. 4: Visual recovery results of MSI completion at a sampling rate of $\displaystyle 10\%$ .

Panel of Table 6 — TABLE II: Quantitative Recovery Results of Color Video Completion

Panel of Figure 7 — Fig. 5: Visual recovery results of color video completion at a sampling rate of $\displaystyle 10\%$ .

Panel of Table 8 — TABLE III: Quantitative Recovery Results of Sentinel-2 Image Completion

Panel of Figure 9 — Fig. 6: Visual recovery results of Sentinel-2 image completion at a sampling rate of $\displaystyle 10\%$ .

Panel of Table 10 — TABLE IV: Quantitative Recovery Results of Point Cloud Completion

Panel of Figure 11 — Fig. 7: Visual recovery results of point cloud completion at a sampling rate of $\displaystyle 10\%$ .

Sampling Rate		10%		20%
MSI	Mode- $\displaystyle n$ Operators	PSNR	SSIM	PSNR	SSIM
Flowers	No Operator	30.565	0.898	34.367	0.948
	Discrete & Linear	39.175	0.984	43.119	0.984
	Continuous & Nonlinear	42.591	0.994	49.717	0.999
Toy	No Operator	26.190	0.835	29.087	0.895
	Discrete & Linear	38.907	0.982	41.395	0.991

TABLE V: Quantitative Recovery Results of MSI Completion With Different Mode-

\displaystyle n

Operators

Sampling Rate		10%		20%
MSI	Architecture	PSNR	SSIM	PSNR	SSIM
Flowers	No Operator	30.565	0.898	34.367	0.948
	FNO	40.940	0.992	47.446	0.997
	DeepONet	42.591	0.994	49.717	0.999
Toy	No Operator	26.190	0.835	29.087	0.895
	FNO	41.167	0.994	46.076	0.998

TABLE VI: Quantitative Recovery Results of MSI Completion With Different Neural Operators

Sampling Rate		10%		20%
MSI	Core	PSNR	SSIM	PSNR	SSIM
Flowers	LRTFR	39.586	0.983	44.232	0.992
	PE-MLP	42.086	0.993	47.326	0.998
	SIREN	42.591	0.994	49.717	0.999
Toy	LRTFR	37.781	0.980	41.948	0.991
	PE-MLP	40.831	0.993	45.765	0.998

TABLE VII: Quantitative Recovery Results of MSI Completion With Different Representation Methods for the Continuous Core Tensor Function

Sampling Rate		10%			20%
MSI	Method	PSNR	SSIM	Parameters	PSNR	SSIM	Parameters
Flowers	SIREN	38.681	0.977	16 711	46.041	0.995	47 671
	LRTFR	36.879	0.965	130 191	43.249	0.991	331 023
	NO-CTR	42.591	0.994	507 341	49.717	0.999	472 721
Toy	SIREN	37.259	0.974	16 711	45.306	0.995	47 671
	LRTFR	37.661	0.975	130 191	43.099	0.991	595 231

TABLE VIII: Quantitative Recovery Results and Number of Parameters of MSI Completion With Different Continuous Tensor Function Representations

Panel of Figure 16 — Fig. 8: Quantitative recovery results w.r.t. different numbers of sensors on the MSIs Flowers (left) and Toy (right) at a sampling rate of $\displaystyle 10\%$ .

Panel of Figure 17 — Fig. 9: Quantitative recovery results w.r.t. different numbers of branch networks on the MSIs Flowers (left) and Toy (right) at a sampling rate of $\displaystyle 10\%$ .

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Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University · Department of Statistics and Data Science, Southern University of Science and Technology

Tensor Decomposition Tensor Completion

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Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.

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