Tensors serve as a crucial tool in the representation and analysis of
complex, multi-dimensional data. As data volumes continue to expand, there is
an increasing demand for developing optimization algorithms that can directly
operate on tensors to deliver fast and effective computations. Many problems in
real-world applications can be formulated as the task of recovering high-order
tensors characterized by sparse and/or low-rank structures. In this work, we
propose novel Kaczmarz algorithms with a power of the $\ell_1$-norm
regularization for reconstructing high-order tensors by exploiting sparsity
and/or low-rankness of tensor data. In addition, we develop both a block and an
accelerated variant, along with a thorough convergence analysis of these
algorithms. A variety of numerical experiments on both synthetic and real-world
datasets demonstrate the effectiveness and significant potential of the
proposed methods in image and video processing tasks, such as image sequence
destriping and video deconvolution.

本文提出了利用稀疏性和 / 或低秩性的张量数据重建高阶张量的新颖 Kaczmarz 算法，并开发了块和加速变体，以及对这些算法的充分收敛分析。对合成和真实数据集上的各种数值实验表明，所提方法在图像和视频处理任务中具有显著的潜力和有效性，例如图像序列去条纹和视频去卷积。

高阶张量恢复的 L1 - 范数正则化 Kaczmarz 算法的能力

Power of $\ell_1$-Norm Regularized Kaczmarz Algorithms for High-Order  Tensor Recovery

Multitask learning (MTL) leverages task-relatedness to enhance performance.
With the emergence of multimodal data, tasks can now be referenced by multiple
indices. In this paper, we employ high-order tensors, with each mode
corresponding to a task index, to naturally represent tasks referenced by
multiple indices and preserve their structural relations. Based on this
representation, we propose a general framework of low-rank MTL methods with
tensorized support vector machines (SVMs) and least square support vector
machines (LSSVMs), where the CP factorization is deployed over the coefficient
tensor. Our approach allows to model the task relation through a linear
combination of shared factors weighted by task-specific factors and is
generalized to both classification and regression problems. Through the
alternating optimization scheme and the Lagrangian function, each subproblem is
transformed into a convex problem, formulated as a quadratic programming or
linear system in the dual form. In contrast to previous MTL frameworks, our
decision function in the dual induces a weighted kernel function with a
task-coupling term characterized by the similarities of the task-specific
factors, better revealing the explicit relations across tasks in MTL.
Experimental results validate the effectiveness and superiority of our proposed
methods compared to existing state-of-the-art approaches in MTL. The code of
implementation will be available at this https URL

多任务学习中使用高阶张量表示任务间的结构关系，并通过线性组合共享因子、特定因子来模拟任务关联，提出通用的基于低秩的多任务学习方法，证实了该方法在多任务学习中的有效性和优越性。