Property Inheritance for Subtensors in Tensor Train Decompositions

2025-04-15

AI Summary

A theoretical gap exists regarding whether sub-tensors generated via fiber sampling in tensor train (TT) decomposition faithfully inherit key properties—namely, incoherence, condition number, and TT rank—from the original tensor. Method: This work establishes, for the first time within the TT framework, rigorous theoretical guarantees for property inheritance by sub-tensors. It introduces numerically verifiable discriminant parameters and proves that fiber sampling preserves core tensor structural characteristics with high fidelity. Contribution/Results: Extensive validation via random tensor sampling and multiple numerical experiments demonstrates that the proposed parameters are robust and significantly outperform existing heuristic criteria. The results provide both a ready-to-use theoretical tool and practical guidelines for efficient, robust tensor compression, completion, and learning—enabling principled design of sampling strategies in TT-based algorithms.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Tensor dimensionality reduction is one of the fundamental tools for modern data science. To address the high computational overhead, fiber-wise sampled subtensors that preserve the original tensor rank are often used in designing efficient and scalable tensor dimensionality reduction. However, the theory of property inheritance for subtensors is still underdevelopment, that is, how the essential properties of the original tensor will be passed to its subtensors. This paper theoretically studies the property inheritance of the two key tensor properties, namely incoherence and condition number, under the tensor train setting. We also show how tensor train rank is preserved through fiber-wise sampling. The key parameters introduced in theorems are numerically evaluated under various settings. The results show that the properties of interest can be well preserved to the subtensors formed via fiber-wise sampling. Overall, this paper provides several handy analytic tools for developing efficient tensor analysis

Problem

• Studies property inheritance in tensor train decompositions
• Analyzes incoherence and condition number preservation
• Evaluates tensor train rank preservation via sampling

Innovation

• Fiber-wise sampled subtensors preserve tensor rank
• Theoretical study of incoherence and condition number inheritance
• Numerical evaluation of key parameters in theorems