Neural Operator-Grounded Continuous Tensor Function Representation and Its Applications

TL;DR

This paper introduces a neural operator-grounded continuous tensor function representation (NO-CTR) that offers a nonlinear, continuous alternative to traditional discrete tensor methods, improving data representation and reducing artifacts.

Contribution

The paper proposes a novel neural operator-grounded mode-n operator and NO-CTR, enabling continuous, nonlinear tensor representations with theoretical approximation guarantees.

Findings

NO-CTR outperforms classic tensor methods in data completion tasks

Demonstrates effectiveness on regular and irregular grid data

Proves universal approximation capability of NO-CTR

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.