用对抗优化实现稀疏PCA，自动调参，无需人工干预。
A Robust Optimization Approach to Sparse Principal Component Analysis
- 通过对抗性扰动优化重构目标，自然诱导稀疏性。
- 推导出闭式解，算法迭代更新编码器与解码器。
- 数据自适应参数化，开箱即用，适合基因组等高维数据。
主成分分析（PCA）是降维的基本工具，但其稠密表示在高维数据中不适用。现有方法通过显式的ℓ₁正则化促进稀疏性，但因任务无监督而难以调参。本文提出对抗性PCA（AdvPCA），利用鲁棒优化，在有界最坏情况潜在空间扰动下优化重构目标，实现稀疏性。该公式可化为闭式解，导出一种交替迭代算法：编码器采用对抗性线性回归式更新，解码器进行正交更新。理论分析给出数据自适应参数化，使算法无需调参即可有效运行。在合成数据和真实基因组数据上的实验验证了其有效性。
原文摘要 · Abstract (English)
While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.
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