Neural Operator-Grounded Continuous Tensor Function Representation and Its Applications

2026-03-02

AI Summary

Existing tensor function representations are limited by discrete and linear mode-n products, which struggle to flexibly model complex continuous data. This work proposes NO-CTR, a neural operator–driven framework for continuous tensor function representation, which introduces neural operators into tensor function modeling for the first time. NO-CTR constructs continuous, nonlinear mode-n operators that directly map continuous core functions to target tensor functions. The framework is theoretically proven to possess universal approximation capability. Extensive experiments on multidimensional data—including multispectral images, color videos, Sentinel-2 imagery, and point clouds—demonstrate that NO-CTR significantly outperforms existing methods in both representational capacity and reconstruction accuracy.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Representation Learning for Vision
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• continuous tensor function
• mode-n product
• discretization artifacts
• nonlinear representation
• real-world data representation

Innovation

• neural operator
• continuous tensor function
• nonlinear mode-n operator
• mesh-free representation
• tensor completion