Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model
This addresses the limitations of naive multi-modal learning in high-dimensional inference, with implications for machine learning and signal processing, though it is incremental in extending classical spiked models.
The paper tackles the problem of recovering multiple high-dimensional signals from two noisy, correlated modalities (a spiked matrix and tensor), showing that while naive joint learning fails, a Sequential Curriculum Learning strategy achieves optimal weak recovery thresholds by first recovering the matrix and then using it to guide tensor recovery.
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.