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Efficient Tensor Decompositions for Fast Matrix Multiplication


Core Concepts
This paper proposes a new generalized cyclic symmetric structure in the factor matrices of polyadic decompositions of matrix multiplication tensors to reduce the number of variables in the optimization problem and improve convergence.
Abstract
The paper focuses on the fast matrix multiplication (FMM) problem, which aims to rewrite matrix multiplication as a tensor equation. The key insights are: Matrix multiplication can be represented as a tensor equation using the matrix multiplication tensor (MMT). Finding a low-rank polyadic decomposition (PD) of the MMT corresponds to finding an efficient algorithm for matrix multiplication. The paper proposes a new generalized cyclic symmetric (CS) structure in the factor matrices of the PD to reduce the number of variables in the optimization problem and improve convergence. This structure generalizes the known CS structure for the case when the matrix dimensions are equal. Numerical experiments show that the proposed generalized CS structure can lead to the discovery of new practical PDs of MMTs, with the number of practical solutions increasing as the CS rank is increased, up to a certain point. However, the rank of these PDs may not equal the canonical rank. The paper also discusses the properties of MMTs, such as the invariance transformations and the recursive structure of PDs, which are leveraged in the proposed approach.
Stats
The number of arithmetic operations needed to compute two n×n matrices is O(nω), where ω := log_n r and r is the rank of the bilinear equation for matrix multiplication. The rank r is bounded from below by 2 and from above by n^3.
Quotes
"Minimizing the rank of the bilinear equation corresponds to finding a (canonical) polyadic decomposition ((C)PD) of the so called matrix multiplication tensor (MMT)." "Because of this definition, Tmpn is a sparse tensor consisting of mpn ones."

Deeper Inquiries

How can the proposed generalized cyclic symmetric structure be extended to other tensor decomposition problems beyond matrix multiplication

The proposed generalized cyclic symmetric structure can be extended to other tensor decomposition problems by considering the underlying symmetries and patterns present in the tensors. By identifying similar cyclic symmetries in other tensor multiplication operations, such as higher-order tensors or tensors with different dimensions, we can apply a similar approach to reduce the number of variables in the optimization problem. This extension would involve adapting the factor matrices and constraints based on the specific tensor multiplication operation under consideration. Additionally, exploring the relationships between different tensor decomposition methods and identifying common structural properties can help in generalizing the cyclic symmetric structure to a broader range of tensor decomposition problems.

What are the theoretical limitations of the rank of PDs with the proposed structure compared to the canonical rank

The theoretical limitations of the rank of PDs with the proposed structure compared to the canonical rank lie in the trade-off between reducing the number of variables and preserving the accuracy and optimality of the decomposition. While the generalized cyclic symmetric structure can help in reducing the search space and improving convergence in optimization problems, it may not always guarantee the exact canonical rank of the tensor decomposition. The constraints imposed by the cyclic symmetry may restrict the possible solutions, potentially leading to suboptimal ranks compared to the canonical rank. Therefore, there is a balance between leveraging the structural properties for computational efficiency and ensuring the accuracy of the decomposition.

Can the insights from this work on leveraging structural properties of tensors be applied to improve optimization methods for other tensor decomposition problems in machine learning and scientific computing

The insights from leveraging structural properties of tensors, such as cyclic symmetry, can be applied to improve optimization methods for other tensor decomposition problems in machine learning and scientific computing. By incorporating known symmetries and patterns in the tensor data, optimization algorithms can be tailored to exploit these structural properties for more efficient and accurate decompositions. This approach can lead to faster convergence, reduced computational complexity, and improved scalability of tensor decomposition algorithms. Furthermore, by understanding the underlying structures of tensors, researchers can develop specialized optimization techniques that leverage these properties to enhance the performance of tensor decomposition methods in various applications, including image processing, signal processing, and data analysis.
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