揭示多模态信号在低信噪比下的可恢复边界与学习顺序的关键作用
Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model
- 利用矩阵与张量共享低秩结构，设计分步恢复策略
- 先恢复矩阵再引导张量，实现最优弱恢复阈值
- 揭示传统联合优化失效机制，适合高维多模态研究者
我们研究从两个噪声相关、共享低秩结构的模态——一个带突变矩阵和一个带突变张量中恢复多个高维信号。该设置推广了经典的突变矩阵与张量模型，揭示了推断通道间的复杂交互及意外的算法行为。值得注意的是，尽管突变张量模型通常在低信噪比下不可解，但其与矩阵的相关性使得通过贝叶斯近似消息传递（BAMP）能高效恢复，产生类似神经网络的阶梯状相变。相反，联合经验风险最小化失败：张量成分阻碍矩阵有效恢复，联合优化显著降低性能，凸显朴素多模态学习的局限性。我们证明，一种简单的分阶段课程学习策略——先恢复矩阵，再利用其指导张量恢复——可突破此瓶颈并达到最优弱恢复阈值。该策略可用谱方法实现，强调了结构相关性和学习顺序在高维多模态推断中的关键作用。
原文摘要 · Abstract (English)
We study the recovery of multiple high-dimensional signals from two noisy, correlated modalities: a spiked matrix and a spiked tensor sharing a common low-rank structure. This setting generalizes classical spiked matrix and tensor models, unveiling intricate interactions between inference channels and surprising algorithmic behaviors. Notably, while the spiked tensor model is typically intractable at low signal-to-noise ratios, its correlation with the matrix enables efficient recovery via Bayesian Approximate Message Passing, inducing staircase-like phase transitions reminiscent of neural network phenomena. In contrast, empirical risk minimization for joint learning fails: the tensor component obstructs effective matrix recovery, and joint optimization significantly degrades performance, highlighting the limitations of naive multi-modal learning. We show that a simple Sequential Curriculum Learning strategy-first recovering the matrix, then leveraging it to guide tensor recovery-resolves this bottleneck and achieves optimal weak recovery thresholds. This strategy, implementable with spectral methods, emphasizes the critical role of structural correlation and learning order in multi-modal high-dimensional inference.
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