一类带基数罚的稀疏回归问题的投影神经网络
Projection Neural Network for a Class of Sparse Regression Problems with Cardinality Penalty
摘要：
本文考虑一类稀疏回归问题，其目标函数是凸损失函数和基数罚函数之和。通过构造基数函数的平滑函数，我们提出了一种投影神经网络，并设计了一种修正方法来解决这个问题。所提出的神经网络的解是唯一的、全局存在的、有界的和全局Lipschitz连续的。此外，我们证明了所提出的神经网络的所有聚集点有一个共同的支持集和一个统一的下界的非零元素。结合所提出的神经网络与校正方法，任何校正的聚点是所考虑的稀疏回归问题的局部极小。此外，我们还分析了所考虑的稀疏回归问题和另一个稀疏回归问题在局部极小点上的等价关系。最后，通过数值实验验证了所提出的神经网络在解决实际稀疏回归问题中的有效性。
英文摘要：
In this paper, we consider a class of sparse regression problems, whose objective function is the summation of a convex loss function and a cardinality penalty. By constructing a smoothing function for the cardinality function, we propose a projected neural network and design a correction method for solving this problem. The solution of the proposed neural network is unique, global existent, bounded and globally Lipschitz continuous. Besides, we prove that all accumulation points of the proposed neural network have a common support set and a unified lower bound for the nonzero entries. Combining the proposed neural network with the correction method, any corrected accumulation point is a local minimizer of the considered sparse regression problem. Moreover, we analyze the equivalent relationship on the local minimizers between the considered sparse regression problem and another regression sparse problem. Finally, some numerical experiments are provided to show the efficiency of the proposed neural networks in solving some sparse regression problems in practice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
