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Hutch++: Optimal Stochastic Trace Estimation.

Raphael A Meyer1, Cameron Musco2, Christopher Musco1

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Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)
|November 1, 2021
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A new randomized algorithm, Hutch++, efficiently estimates matrix traces using fewer matrix-vector products than Hutchinson's estimator. This method offers optimal complexity for matrix trace approximation.

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Area of Science:

  • Numerical Analysis
  • Linear Algebra
  • Computational Mathematics

Background:

  • Estimating the trace of a matrix is crucial in various computational tasks.
  • Existing methods like Hutchinson's estimator require a high number of matrix-vector products, limiting their efficiency.
  • There is a need for more efficient algorithms for trace estimation, especially for large matrices.

Purpose of the Study:

  • To introduce a novel randomized algorithm, Hutch++, for approximating the trace of a matrix.
  • To significantly reduce the number of matrix-vector products required compared to existing methods.
  • To establish the theoretical optimality and practical performance of the proposed algorithm.

Main Methods:

  • Development of Hutch++, a randomized algorithm for trace estimation.
  • Utilizing a low-rank approximation technique to reduce the variance of Hutchinson's estimator.
  • Theoretical analysis of the algorithm's complexity and approximation guarantees.
  • Empirical evaluation and comparison with Hutchinson's estimator.

Main Results:

  • Hutch++ computes a (1 ± ε) approximation to the trace using only O(1/ε) matrix-vector products for positive semidefinite matrices.
  • The algorithm demonstrates a significant improvement over Hutchinson's estimator, which requires O(1/ε^2) products.
  • Theoretical analysis shows Hutch++ has near-optimal complexity among matrix-vector query algorithms.
  • Experimental results confirm Hutch++'s superior performance, with empirical gains extending to non-PSD matrices.

Conclusions:

  • Hutch++ provides a more efficient and theoretically grounded method for matrix trace estimation.
  • The algorithm's simplicity, efficiency, and strong performance make it a valuable tool for computational mathematics.
  • The findings have implications for applications involving large matrices, including network analysis.