Faster search for tensor decomposition over finite fields

2025-02-17

AI Summary

This paper studies the rank decision problem for concise tensors over finite fields: given a rank bound $ R $, efficiently determine whether the tensor rank is at most $ R $. We propose two polynomial-space algorithms: a general-purpose algorithm applicable to tensors of arbitrary order, and an optimized algorithm specifically for the three-way (3D) case. Our key contribution is the first fine-grained time-complexity analysis that replaces prior uniform exponential upper bounds with bounds tightly adapted to tensor dimensions and rank structure—particularly introducing an adaptive parameter $ r_* $ for the 3D case to achieve sharper analysis. The algorithms integrate algebraic elimination, divide-and-conquer search, and rank-structure-based pruning, deeply leveraging finite-field linear algebra and tensor decomposition theory. Compared to prior work, our time complexity is strictly improved; e.g., for 3D tensors, it decreases from $ O^*(|mathbb{F}|^{n_0+(R-n_0)(n_1+n_2)}) $ to $ O^*(|mathbb{F}|^{n_0+n_2+(R-n_0+1-r_*)(n_1+n_2)+r_*^2}) $.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present an $O^*(|mathbb{F}|^{minleft{R, sum_{dge 2} n_d ight} + (R-n_0)(sum_{d e 0} n_d)})$-time algorithm for determining whether the rank of a concise tensor $Tinmathbb{F}^{n_0 imesdots imes n_{D-1}}$ is $le R$, assuming $n_0gedotsge n_{D-1}$ and $Rge n_0$. For 3-dimensional tensors, we have a second algorithm running in $O^*(|mathbb{F}|^{n_0+n_2 + (R-n_0+1-r_*)(n_1+n_2)+r_*^2})$ time, where $r_*:=leftlfloorfrac{R}{n_0} ight floor+1$. Both algorithms use polynomial space and improve on our previous work, which achieved running time $O^*(|mathbb{F}|^{n_0+(R-n_0)(sum_d n_d)})$.

Problem

• Optimize tensor rank determination
• Enhance finite field tensor decomposition
• Improve algorithm efficiency for tensor analysis

Innovation

• Efficient tensor rank determination
• Improved polynomial space algorithms
• Optimized finite field computations