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Fast Approximation of Rotations and Hessian Matrices

Basic Information

  • Title: Fast Approximation of Rotations and Hessian Matrices
  • Authors: Michael Mathieu and Yann LeCun
  • Affiliations: Courant Institute of Mathematical Sciences, New York University
  • Publication Date: April 29, 2014
  • Journal: arXiv:1404.7195v1 [cs.LG]
  • Keywords: Rotation matrices, Hessian matrices, Symmetric matrices, Gaussian models, Optimization, Machine learning

Introduction

The paper introduces a novel method to represent and approximate rotation matrices with linearithmic complexity, designed to expedite computations involving symmetric matrices like covariance and Hessian matrices. This approach is pivotal for tasks such as density models, Bayesian inference, and optimization in machine learning, where direct evaluations are infeasible due to the dimensional growth of these matrices.

Contributions

  • Proposed a method for approximating symmetric matrices, representing them as \(QDQ^T\), where \(Q\) is a rotation matrix approximated by a series of \(n \log(n) / 2\) Givens rotations, and \(D\) is a diagonal matrix.
  • Demonstrated how this approximation can speed up inference in Gaussian models and estimate and track the inverse Hessian of an objective function.
  • Conducted experiments on synthetic matrices, real data covariance matrices, and Hessian matrices of machine learning problem objectives to validate the approach.

Techniques and Method

The technique involves parameterizing the rotation matrix \(Q\) as a product of elementary rotations in an FFT-like fashion, making the computational complexity linearithmic. This setup simplifies the approximation of symmetric matrices, aiding in the efficient computation of matrix-vector products and the inverse of large matrices. The approximation is optimized using stochastic gradient descent.

Experimental Results

  • Experiments on synthetic and real-world matrices, including covariance matrices of the MNIST dataset, showed that the method can achieve meaningful approximations with reduced computational complexity.
  • The approach's validity was further demonstrated through its application in approximating Hessian matrices for optimization problems, showing potential for speeding up learning and optimization processes in machine learning systems.

Limitations and Future Work

The paper identifies the need for further exploration in: - Improving the approximation accuracy for a broader range of matrices. - Extending the method's application to more complex machine learning models and optimization scenarios. - Developing more sophisticated learning algorithms for optimizing the parameters of the approximation.

Conclusion

This work presents a significant advancement in efficiently approximating rotation and Hessian matrices, facilitating faster computations in various machine learning tasks. The method's ability to reduce computational complexity while maintaining approximation accuracy offers promising avenues for enhancing performance in high-dimensional modeling and optimization challenges.