Powers of Tensors and Fast Matrix Multiplication (1401.7714v1)

Abstract: This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound $\omega<2.3728639$ on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

• The paper introduces a convex optimization framework that computes tighter bounds on the matrix multiplication exponent ω.
• It leverages higher powers of Coppersmith-Winograd tensors to improve the upper bound to ω < 2.3728639.
• The study provides both theoretical insights and practical algorithmic advancements for more efficient matrix multiplication.

Powers of Tensors and Fast Matrix Multiplication

The paper "Powers of Tensors and Fast Matrix Multiplication" by François Le Gall provides a novel method for evaluating the asymptotic complexity of matrix multiplication by leveraging trilinear forms, also referred to as tensors. Specifically, this method utilizes convex optimization to ensure polynomial-time tractability, distinguishing it from traditional approaches that often involve more complex and sometimes intractable optimization procedures.

Key Contributions

The primary contribution of the paper is the use of convex optimization to analyze tensor powers and derive tighter upper bounds on the matrix multiplication exponent, denoted by $\omega$. The matrix multiplication exponent $\omega$ is defined as the smallest value such that two $n \times n$ matrices can be multiplied using $O(n^{\omega+\varepsilon})$ arithmetic operations for any $\varepsilon > 0$. There exists a widely held conjecture that $\omega = 2$, yet current bounds have only managed to show $\omega < 2.38$.

The breakthrough achieved in this paper is summarized as follows:

• Framework Development: The paper adapts a framework that allows the reduction of the computation of lower bounds on the tensor value to solving polynomially many instances of convex optimization problems.
• Tensor Analysis: By studying higher powers of Coppersmith-Winograd tensors, the paper derives an improved upper bound on $\omega$. Specifically, it achieves $\omega < 2.3728639$ by examining up to the 32nd power of these tensors.
• Algorithm Description: The proposed algorithms, labeled as Algorithm $\mathcal{A}$ and Algorithm $\mathcal{B}$, systematically approach the problem by optimizing a set of convex functions, enabling both a theoretical and practical analysis of tensor powers.

Methodology

The paper begins by revisiting the foundational work of Coppersmith and Winograd, who initially showed that modifying specific trilinear forms could yield better bounds on $\omega$. Their work involved defining and estimating the value $V_{\rho}(t)$ of trilinear forms and demonstrating how these values can provide upper bounds on $\omega$. Le Gall extends this theory by developing a polynomial-time method to analyze these values using convex optimization techniques.

The analysis proceeds by:

1. Defining the Tensors and Forms: Establishing a decomposition of the tensor and verifying its tight support.
2. Optimization Framework: Introducing Algorithm $\mathcal{A}$, which optimizes the value of a tensor through convex optimization, and Algorithm $\mathcal{B}$, which deals with more complex cases involving additional constraining conditions.
3. Recursive Analysis: Applying these algorithms to recursively analyze higher powers of tensors. He articulates how the bounds on $\omega$ can be successively improved by evaluating these higher powers.

Results

The numerical results achieved in this paper are significant. By using the proposed framework, Le Gall manages to refine the previously known upper bound of $\omega < 2.3728642$ (obtained for the eighth power of the tensor) to $\omega < 2.3728639$ by analyzing the 32nd power.

Practical and Theoretical Implications

Practical Implications

The primary practical implication of this work is in the field of efficient algorithm design for matrix multiplication, a problem of significant interest in both theoretical and applied computer science. Improved bounds on $\omega$ lead directly to more efficient algorithms, which can have substantial impact in fields such as numerical analysis, computer graphics, machine learning, and more.

Theoretical Implications

From a theoretical perspective, this paper advances the understanding of tensor analysis and algebraic complexity. The framework developed lays the groundwork for future research into more efficient algorithms, potentially moving closer to the conjectured bound of $\omega = 2$.

Speculation on Future Developments

While the bounds derived in this paper represent a meaningful advancement, it is likely that further research could continue to refine these upper limits. Future studies might incorporate advanced techniques from optimization theory, machine learning, or quantum computing to explore even higher tensor powers or entirely new classes of tensors.

Moreover, the method of using convex optimization to paper algebraic properties of tensors could be extended to other computational problems beyond matrix multiplication, indicating a rich area for ongoing research.

Conclusion

The work by François Le Gall represents a significant step in the field of fast matrix multiplication by showing a methodology to achieve polynomial-time analysis of tensor powers, leading to improved upper bounds on the matrix multiplication exponent $\omega$. The use of convex optimization in this context not only improves theoretical limits but also provides a more tractable approach for practical algorithm development. This research opens new avenues for further exploration, potentially moving closer to the long-standing conjecture that $\omega = 2$.