用ADMM求解非线性矩阵分解，支持多种损失和非线性函数。
Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions
- 基于ADMM框架，将非线性矩阵分解转化为可优化的交替问题。
- 在真实数据上验证高效性，支持ReLU、平方、MinMax等非线性变换。
- 适用于推荐系统、稀疏数据建模等场景，扩展性强。
我们提出一种基于交替方向乘子法（ADMM）的算法，用于求解非线性矩阵分解（NMD）。给定输入矩阵 $X \in \mathbb{R}^{m \times n}$ 和分解秩 $r \ll \min(m, n)$，NMD 寻找矩阵 $W \in \mathbb{R}^{m \times r}$ 与 $H \in \mathbb{R}^{r \times n}$，使得 $X \approx f(WH)$，其中 $f$ 为逐元素非线性函数。我们在多个典型非线性模型上评估该方法：适用于非负稀疏数据近似的修正线性单元激活 $f(x) = \max(0, x)$，适用于概率电路表示的分量平方 $f(x) = x^2$，以及适用于推荐系统的MinMax变换 $f(x) = \min(b, \max(a, x))$。所提框架灵活支持多种损失函数，包括最小二乘、$\ell_1$ 范数及Kullback-Leibler散度，并可方便扩展至其他非线性形式与度量方式。我们在真实世界数据集上展示了该方法的适用性、效率与适应性，凸显其在广泛应用场景中的潜力。
原文摘要 · Abstract (English)
We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications.
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