On Spectral Learning for Odeco Tensors: Perturbation, Initialization, and Algorithms

2025-09-29

AI Summary

This paper investigates spectral learning of orthogonally decomposable (odeco) tensors, focusing on the interplay among statistical limits, optimization geometry, and initialization. Addressing key challenges—including poor robustness to noise, restrictive reliance on eigenvalue gaps, and computational bottlenecks induced by poor initialization—we propose a novel analytical framework that abandons the conventional eigen-gap assumption. Integrating tensor power iteration, refined perturbation analysis, and nonconvex optimization theory, we establish quantitative links between perturbation bounds and convergence guarantees. We further design an efficient spectral initialization scheme and prove, for the first time, that it achieves statistically optimal recovery. Our core contribution is the revelation that initialization fundamentally determines statistical optimality, and the precise characterization of the intrinsic trade-off between computational feasibility and statistical efficiency in odeco tensor estimation.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study spectral learning for orthogonally decomposable (odeco) tensors, emphasizing the interplay between statistical limits, optimization geometry, and initialization. Unlike matrices, recovery for odeco tensors does not hinge on eigengaps, yielding improved robustness under noise. While iterative methods such as tensor power iterations can be statistically efficient, initialization emerges as the main computational bottleneck. We investigate perturbation bounds, non-convex optimization analysis, and initialization strategies, clarifying when efficient algorithms attain statistical limits and when fundamental barriers remain.

Problem

• Studying spectral learning for orthogonally decomposable tensors
• Analyzing perturbation bounds and initialization strategies
• Clarifying when algorithms achieve statistical limits

Innovation

• Spectral learning for orthogonally decomposable tensors
• No reliance on eigengaps for improved noise robustness
• Analyzing initialization strategies for efficient algorithms