Multi-task learning has been observed by many researchers, which supposes that different tasks can share a low-rank common yet latent subspace. It means learning multiple tasks jointly is better than learning them independently. In this paper, we propose two novel multi-task learning formulations based on two regularization terms, which can learn the optimal shared latent subspace by minimizing the exactly $k$ minimal singular values. The proposed regularization terms are the more tight approximations of rank minimization than trace norm. But it's an NP-hard problem to solve the exact rank minimization problem. Therefore, we design a novel re-weighted based iterative strategy to solve our models, which can tactically handle the exact rank minimization problem by setting a large penalizing parameter. Experimental results on benchmark datasets demonstrate that our methods can correctly recover the low-rank structure shared across tasks, and outperform related multi-task learning methods.
From Group Sparse Coding to Rank Minimization: A Novel Denoising Model for Low-level Image Restoration
College of Electronic and Optical Engineering & College of Microelectronics, Nanjing University of Posts and Telecommunications（南京邮电大学电子与光学工程学院、微电子学院）；College of Telecommunications & Information Engineering, Nanjing University of Posts and Telecommunications（南京邮电大学通信与信息工程学院）
Equivalence of LP Relaxation and Max-Product for Weighted Matching in General Graphs
LIDS, MIT（麻省理工学院 LIDS）
最大乘积置信传播是一种局部迭代算法，用于找到概率分布的模式/MAP估计。虽然它已被成功地应用于各种各样的应用，有相对较少的理论保证收敛性和正确性的一般圈图，可能有许多短周期。其中，提供确切的"必要和顺从“特征的更少。在本文中，我们研究的问题，使用最大积找到最大的重量匹配的任意图的边重量。这是通过首先构建一个概率分布，其模式对应于最佳匹配，然后运行最大乘积来完成的。加权匹配也可以作为一个整数规划，其中有一个LP松弛。这种放松并不总是紧张的。在本文中，我们表明，\begin{enumerate} \item If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. \item If the LP relaxation is loose, then max-product does not converge. \end{enumerate}这提供了一个精确的，数据依赖的表征最大产品的性能，和精确的连接到LP松弛，这是一个很好的研究优化技术。此外，由于LP松弛是紧的二部图，我们的结果推广了其他最近的结果使用最大产品，以找到加权匹配的二部图。
