标题： 一种用于非负或混合输入的半非负矩阵分解的全局最优解析解 显示英文标题

标题： A Globally Optimal Analytic Solution for Semi-Nonnegative Matrix Factorization with Nonnegative or Mixed Inputs

摘要： 半非负矩阵分解（半-NMF）通过允许基矩阵包含正负条目，扩展了经典的非负矩阵分解（NMF），使其适用于具有混合符号的数据分解。 然而，大多数现有的半-NMF算法是迭代的、非凸的，并且容易陷入局部最小值。 在本文中，我们提出了一种新方法，在弗罗贝尼乌斯范数下通过输入数据的散度矩阵得到的正交分解，获得半-NMF问题的全局最优解。 我们严格证明了我们的解达到了重构误差的全局最小值。 此外，我们证明当输入矩阵为非负时，我们的方法通常比标准NMF算法达到更低的重构误差，尽管不幸的是基矩阵可能不满足非负性。 特别是，在低秩情况下，如秩1或秩2，我们的解恰好退化为非负分解，恢复了NMF结构。 我们通过在合成数据和UCI Wine数据集上的实验验证了我们的方法，结果表明我们的方法在重构精度方面始终优于现有的NMF和半-NMF方法。 这些结果证实了我们提出的全局最优、非迭代公式在理论保证和经验优势方面提供了新的视角，对优化和数据分析中的矩阵分解有新的见解。 显示英文摘要

摘要： Semi-Nonnegative Matrix Factorization (semi-NMF) extends classical Nonnegative Matrix Factorization (NMF) by allowing the basis matrix to contain both positive and negative entries, making it suitable for decomposing data with mixed signs. However, most existing semi-NMF algorithms are iterative, non-convex, and prone to local minima. In this paper, we propose a novel method that yields a globally optimal solution to the semi-NMF problem under the Frobenius norm, through an orthogonal decomposition derived from the scatter matrix of the input data. We rigorously prove that our solution attains the global minimum of the reconstruction error. Furthermore, we demonstrate that when the input matrix is nonnegative, our method often achieves lower reconstruction error than standard NMF algorithms, although unfortunately the basis matrix may not satisfy nonnegativity. In particular, in low-rank cases such as rank 1 or 2, our solution reduces exactly to a nonnegative factorization, recovering the NMF structure. We validate our approach through experiments on both synthetic data and the UCI Wine dataset, showing that our method consistently outperforms existing NMF and semi-NMF methods in terms of reconstruction accuracy. These results confirm that our globally optimal, non-iterative formulation offers both theoretical guarantees and empirical advantages, providing a new perspective on matrix factorization in optimization and data analysis.