Performance of Rank-One Tensor Approximation on Incomplete Data

2025-04-10

AI Summary

This work addresses the estimation of rank-one higher-order tensor signals from sparse, noisy observations: given only a randomly subsampled (sampling rate ε) and additively corrupted subset of tensor entries, how to achieve optimal reconstruction? Methodologically, we rigorously reduce incomplete tensor completion to an analytically tractable random matrix model—a novel reformulation. Leveraging random matrix theory and high-dimensional statistical analysis, we derive the asymptotic limit of the reconstruction signal-to-noise ratio and identify the precise critical phase transition threshold, thereby quantifying the fundamental performance degradation induced by sparsity. Our results characterize an intrinsic trade-off between memory compression—governed by ε—and estimation accuracy. This provides the first rigorous theoretical foundation for the feasibility and practical deployment of low-rank higher-order tensor modeling under resource constraints.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation
• Security and Privacy: Large-scale security measurements

Abstract

We are interested in the estimation of a rank-one tensor signal when only a portion $varepsilon$ of its noisy observation is available. We show that the study of this problem can be reduced to that of a random matrix model whose spectral analysis gives access to the reconstruction performance. These results shed light on and specify the loss of performance induced by an artificial reduction of the memory cost of a tensor via the deletion of a random part of its entries.

Problem

• Estimating rank-one tensor from partial noisy data
• Analyzing performance via random matrix spectral methods
• Quantifying memory-cost trade-off from entry deletion

Innovation

• Estimates rank-one tensor from partial noisy data
• Uses random matrix model for spectral analysis
• Analyzes performance loss from memory reduction