用黎曼几何优化低秩矩阵的乘积，提升建模效率与稳定性。
Riemannian Optimization for Hadamard Products of Low-Rank Matrices
- 基于黎曼商流形构建优化框架，解决双因子缩放对称性问题。
- 提出新型块对角度量，保持对称性不变性并支持高效计算。
- 算法线性扩展，适用于大规模观测数据，适合低秩建模任务。
两个低秩矩阵的逐元素哈达玛积可有效建模具有乘法结构的数据，但其学习因因子间耦合的行/列缩放对称性而困难。为此，本文将该模型的学习问题形式化为黎曼商流形上的优化。我们提出一种从弗罗贝尼乌斯内积拉回得到的新颖块对角黎曼度量，该度量在对称变换下保持不变。进一步设计了一种无需调参的高斯-牛顿步长的黎曼梯度下降算法，每轮迭代复杂度随观测条目数线性增长。所提黎曼商优化框架可统一支持一阶与二阶方法，后者通过闭式连接与黎曼海森矩阵实现。在真实与合成数据集上的实验验证了该方法的有效性。
原文摘要 · Abstract (English)
The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.
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