Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model

2025-06-03

AI Summary

This work addresses the joint recovery of noisy spiked matrices and spiked tensors (bimodal) sharing a common low-rank structure. To overcome the failure of conventional joint optimization under weak signal regimes, we propose a staircase-like phase transition mechanism induced by inter-modal structural correlations and design a sequential curriculum learning strategy for progressive, robust signal recovery. Our method integrates Bayesian approximate message passing (BAMP), spectral initialization, and spiked matrix-tensor theoretical analysis, achieving the optimal weak-recovery threshold with computational efficiency. Theoretically, we establish that both inter-modal coupling strength and learning schedule critically govern the phase transition behavior; predicted thresholds align closely with empirical results.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Recovering high-dimensional signals from noisy correlated matrix and tensor modalities
• Analyzing algorithmic behaviors and phase transitions in multi-modal learning
• Optimizing recovery via Sequential Curriculum Learning to overcome joint learning limitations

Innovation

• Bayesian Approximate Message Passing for recovery
• Sequential Curriculum Learning strategy
• Spectral methods for optimal recovery