提出最优谱方法，解决高维多指标模型的样本效率问题。
Optimal Spectral Transitions in High-Dimensional Multi-Index Models
- 基于消息传递线性化设计谱算法
- 达到理论最优重构阈值，实现弱可学习性边界突破
- 适合研究神经网络计算复杂性的学者参考
我们研究高斯多指标模型中，需要多少样本才能弱重建相关指标子空间。尽管该模型日益成为研究神经网络计算复杂性的测试平台，但除单指标情形外，现有结果仍不明确。本文提出一种基于消息传递线性化的谱算法。主要贡献在于证明所提方法达到最优重建阈值。通过高维算法表征分析，发现超过临界阈值后，主特征向量与相关指标子空间呈强相关，此现象类似随机矩阵理论中尖刺模型的Baik-Ben Arous-Peche (BBP) 转变。结合数值实验与严格理论框架，本工作填补了多指标模型弱可学习性计算极限的关键空白。
原文摘要 · Abstract (English)
We consider the problem of how many samples from a Gaussian multi-index model are required to weakly reconstruct the relevant index subspace. Despite its increasing popularity as a testbed for investigating the computational complexity of neural networks, results beyond the single-index setting remain elusive. In this work, we introduce spectral algorithms based on the linearization of a message passing scheme tailored to this problem. Our main contribution is to show that the proposed methods achieve the optimal reconstruction threshold. Leveraging a high-dimensional characterization of the algorithms, we show that above the critical threshold the leading eigenvector correlates with the relevant index subspace, a phenomenon reminiscent of the Baik-Ben Arous-Peche (BBP) transition in spiked models arising in random matrix theory. Supported by numerical experiments and a rigorous theoretical framework, our work bridges critical gaps in the computational limits of weak learnability in multi-index model.
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