Tensors in computations (2106.08090v1)

Abstract: The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.

• The paper demonstrates how tensors, via principles like multilinearity and coordinate invariance, simplify complex computational problems.
• It outlines the importance of tensor ranks and norms in reducing computational complexity and ensuring numerical stability across algorithms.
• It connects tensor calculus with modern physics and machine learning, paving the way for advancements in efficient algorithms and quantum computing applications.

Overview of "Tensors in Computations"

This paper by Lek-Heng Lim presents a comprehensive discussion on tensors and their critical role in computations. Tensors, which capture the ideas of equivariance, multilinearity, and separability, are foundational to myriad applications in computational and applied mathematics. The paper elaborates on these concepts through linear algebra, numerical linear algebra, and functional analysis perspectives, offering an insightful narrative into the integral role that tensors play in mathematical computations, scientific simulations, and modern AI applications.

Key Concepts

1. Equivariance and Transformation Rules:
   • Tensors are described via transformation rules—equivariance under coordinate changes, which aligns well with the physical notion that the laws of physics should remain invariant under different reference frames. This concept is pivotal in computational frameworks wherein coordinate changes are frequent, such as in image recognition using neural networks or solving PDEs numerically.
2. Multilinearity of Tensors:
   • Tensors rationalize complex computations by representing them as multilinear maps. This allows operations that are only linear when restricted to each input argument separately. Multilinearity is leveraged in many computational algorithms, providing the foundation for efficient and stable numerical methods.
3. Separation of Variables:
   • Tensors facilitate the separation of variables, enabling simplifications in the solutions of differential equations, image processing tasks, and quantum computations. This approach is encapsulated in many algorithms that benefit from reduced computational complexity through tensor decomposition techniques.

Numerical and Computational Implications

• Tensor Ranks and Computational Complexity:
  • The paper discusses the tensor rank, an extension of the matrix rank, offering insights into the bilinear complexity of products, which directly impacts computational costs. The work of Strassen on matrix multiplication using tensors illustrates significant reductions in computational time and resources.
• Tensor Norms:
  • The paper also outlines tensor norms, which quantify the size of a tensor and are crucial for ensuring numerical stability across computations. Establishing bounds on tensor norms allows for better control of error margins and optimization of algorithms in high-dimensional spaces.
• Extension to Higher Dimensions:
  • In extending analysis to hyperspaces and infinite-dimensional vector spaces, the paper highlights the comprehensive nature of tensor calculus, its applicability to problems in quantum mechanics, and the modeling of high-dimensional data.

Theoretical Implications

• Unified Framework for Multivariable Functions:
  • By generalizing tensor products to function spaces, the paper provides a unified framework for analyzing multivariable functions, essential for many applications in machine learning and artificial intelligence where data dimensionality is inherently high.
• Tensor Fields and Modern Physics:
  • Tensors as fields on manifolds connect linear algebraic concepts with differential geometry, elucidating the behavior of physical phenomena across different coordinate systems.

Future Directions

• The exploration of efficient algorithms for approximating tensor decompositions remains a forefront area in AI research, particularly as data grows in dimension and complexity.
• Tensors underscore potential advancements in quantum computing, promising advances in entanglement theory, and its applications.

Conclusion

Lek-Heng Lim's paper provides an exhaustive account of tensors, placing them at the center of both theoretical and practical advancements in computation. By articulating the relationships between multilinearity, coordinate invariance, and numerical methods, this paper establishes a foundation for future work in leveraging tensors for the development of efficient computational algorithms and further exploration in physics, machine learning, and beyond. This synthesis of algebra and analysis signifies the profound flexibility and capability of tensors in addressing complex scientific problems.