从不完整噪声数据中学习非线性因子模型的未知单调关系
Learning Nonlinear Factor Models with Unknown Monotone Links from Incomplete and Noisy Data
- 用再生核希尔伯特空间建模未知单调链接函数，保持可识别性
- 提出投影块坐标下降算法，在噪声和缺失数据下仍能收敛
- 适用于需要挖掘非线性潜结构的统计建模任务
我们研究了一种非线性因子模型，其中观测响应通过未知单调链接函数依赖于低秩潜因子。该设定因严重的非凸性和可识别性问题而极具挑战，且研究较少。链接函数被假设属于再生核希尔伯特空间（RKHS），可在保持可识别性的前提下实现灵活的非参数建模。我们将问题建模为从可能不完整和噪声化的观测中联合恢复低秩因子、载荷矩阵及非线性链接函数，并提出一种带有显式正则化的投影块坐标下降（BCD）算法，以解决尺度与旋转歧义。在因子满足弱相干性、采样条件标准的前提下，我们建立了无噪声与有噪声情形下的收敛保证，并给出链接函数更新的次线性遗憾界。结果将经典线性因子模型推广至广泛的非线性场景，提供了一个学习非线性潜结构的合理框架。通过受控合成实验评估，表明方法表现良好。
原文摘要 · Abstract (English)
We study a nonlinear factor model in which observed responses depend on low-rank latent factors through an unknown monotone link function. This setting is challenging and largely underexplored due to severe nonconvexity and identifiability issues. The link function is assumed to lie in a reproducing kernel Hilbert space (RKHS), enabling flexible nonparametric modeling while preserving identifiability. We formulate the problem as the joint recovery of the low-rank factors, loadings, and the nonlinear link function from possibly incomplete and noisy observations and propose a projected block coordinate descent (BCD) algorithm with explicit regularization to address scale and rotational ambiguities. Under mild incoherence of factors and standard sampling conditions, we establish convergence guarantees in both noiseless and noisy regimes, along with sublinear regret bounds for the link-function updates. Our results extend classical linear factor models to a broad nonlinear regime and provide a principled framework for learning nonlinear latent structures. We evaluate the proposed approach using controlled synthetic experiments, indicating promising performance.
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