  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.
