Efficient Algorithms for Low-Rank Tensor Decomposition over Finite Fields

Core Concepts

Polynomial-time algorithms exist for finding rank-R decompositions of 3D tensors over finite fields, for R ≤ 4. However, allowing arbitrary values in some tensor cells makes rank-2 decomposition NP-hard over the integers modulo 2.

Abstract

The key insights of this paper are: By constructing a basis of tensor slices, the original tensor decomposition problem can be reduced to solving a linear system of equations over matrix variables, where each matrix has rank at most 1. For R ≤ 4, this linear system can be solved efficiently by carefully analyzing the structure of the coefficient matrix and utilizing matrix rank factorizations. However, when allowing arbitrary values in some tensor cells ("wildcards"), the problem becomes NP-hard for rank-2 decomposition over the integers modulo 2, via a reduction from Not-All-Equal 3SAT. The paper also sketches polynomial-time algorithms for rank-1 decomposition with wildcards over 3D tensors and matrices, over arbitrary finite fields.

Stats

Ti,j,k = ∑R-1_r=0 Ar,i Br,j Cr,k ∀i, j, k Rank-R decomposition of an n × n × n tensor can be found in O(f(|F|, R)n^3) time, for some function f.

Quotes

"Tensor decomposition is at the heart of fast matrix multiplication, a problem that is the primary bottleneck of numerous linear algebra and graph combinatorics algorithms, such as matrix inversion and triangle detection." "All asymptotically fast algorithms for matrix multiplication use a divide-and-conquer scheme, and finding an efficient scheme is equivalent to finding a decomposition of a certain tensor with low rank."

Key Insights Distilled From

Low-Rank Tensor Decomposition over Finite Fields

by Jason Yang at arxiv.org 04-16-2024

https://arxiv.org/pdf/2401.06857.pdf

Deeper Inquiries

How can the algorithms be extended to handle higher-dimensional tensors or tensors with larger ranks?

To extend the algorithms to handle higher-dimensional tensors or tensors with larger ranks, we can follow a few approaches: Higher-Dimensional Tensors: For higher-dimensional tensors, we can generalize the decomposition process by considering higher-order tensors and modifying the algorithm to account for the additional dimensions. This would involve extending the matrix operations to tensor operations, such as tensor contractions and outer products. Larger Ranks: To handle tensors with larger ranks, we can adapt the algorithm to accommodate higher-rank decompositions. This may involve optimizing the factorization process for larger matrices and tensors, as well as considering more complex linear systems for higher-rank decompositions. Efficient Data Structures: Implementing efficient data structures and algorithms specifically designed for higher-dimensional tensors can improve the scalability of the algorithms. This may include utilizing tensor decomposition libraries or frameworks that are optimized for handling large tensors and higher ranks. Parallelization: Leveraging parallel computing techniques can help in processing higher-dimensional tensors and larger ranks more efficiently. By distributing the computations across multiple processors or nodes, we can speed up the decomposition process for complex tensors.

What are the implications of the NP-hardness result for rank-2 decomposition with wildcards over Z/2Z? Are there any approximation algorithms or heuristics that can be developed?

The NP-hardness result for rank-2 decomposition with wildcards over Z/2Z implies that finding a rank-2 decomposition of a tensor with wildcards is computationally challenging and falls into the class of NP-hard problems. This suggests that there may not be a polynomial-time algorithm to solve this problem optimally in all cases. To address this challenge, researchers can explore approximation algorithms or heuristics that provide suboptimal solutions with guaranteed performance bounds. These approximation techniques aim to find near-optimal solutions in a reasonable amount of time, even for NP-hard problems. Some possible approaches include: Greedy Algorithms: Greedy algorithms can be used to iteratively build a decomposition by selecting the best components at each step based on certain criteria. While greedy algorithms do not guarantee optimality, they can provide fast and reasonably good solutions. Randomized Algorithms: Randomized algorithms introduce randomness into the decomposition process to explore different solutions and potentially find better decompositions. Techniques like Monte Carlo methods or randomized rounding can be applied to approximate the rank-2 decomposition. Metaheuristic Algorithms: Metaheuristic algorithms, such as genetic algorithms or simulated annealing, can be employed to search the solution space efficiently and find good approximations for the rank-2 decomposition with wildcards. Developing approximation algorithms and heuristics for the NP-hard rank-2 decomposition problem with wildcards can provide practical solutions for real-world applications where exact solutions are infeasible due to computational complexity.

Can the techniques used in this paper be applied to other tensor-related problems, such as tensor completion or tensor factorization?

Yes, the techniques used in the paper for low-rank tensor decomposition can be applied to other tensor-related problems, such as tensor completion and tensor factorization. Here's how these techniques can be extended: Tensor Completion: In tensor completion, the goal is to fill in missing entries of a partially observed tensor. The algorithms developed for low-rank tensor decomposition can be adapted to handle tensor completion by incorporating constraints on the observed entries and optimizing the reconstruction of the missing values based on the low-rank structure of the tensor. Tensor Factorization: Tensor factorization involves expressing a tensor as a product of lower-dimensional tensors. The decomposition algorithms can be utilized for tensor factorization by decomposing the original tensor into a set of factor matrices or tensors that capture the underlying structure of the data. This can help in dimensionality reduction and feature extraction from high-dimensional tensor data. Sparse Tensor Decomposition: Extending the techniques to sparse tensor decomposition involves considering tensors with sparse structures. By incorporating sparsity constraints and regularization terms into the decomposition process, the algorithms can be adapted to handle sparse tensors and extract meaningful patterns from the data. By applying the principles of low-rank tensor decomposition to these related problems, researchers can develop efficient algorithms for various tensor analysis tasks in fields such as machine learning, signal processing, and data mining.