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.
翻译:由于高阶高压器自然适合在现实世界中代表多维数据,例如彩色图像和视频,因此,低调高压表示器已成为机器学习和计算机视觉中新出现的领域之一,然而,古典低调高压表示器只能代表有限的网格数据,因为其内在的离散性质阻碍了其在网格之外的许多情景中的潜在适用性。为了打破这一屏障,我们建议低调的沙尔功能代表器(LRTFR)能够以无限的分辨率持续代表超出网格的数据。具体地说,拟议的云层代表器功能可以任意地与相应值相协调,从而在无限的实际空间中持续代表数据。与离散式高压代表器平行,我们只能代表有限网格功能的两个基本概念,即:高压功能和低调调调调调调调调调,在LRTFR(LRTFR)中,低调和平稳调调调调调调调调,这可以以无限的分辨率代表方法持续地代表数据。具体地说,拟议的多维数据恢复应用来自图像处理方法(平整平整平整平整平整平整平整平流和平流的平流法),以及机器学习方法(平整平整平整的平整的平整的平整方法(平整的平整的平整的平整的平整的平整),以及平整的平流的平流的平流的平流的平流和平流)。