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Nyström-aware approximations for matrix-based Rényi's entropy.

Tieliang Gong1, Wen Wen1, Yuxin Dong1

  • 1School of Computer Science of Technology, Xi'an Jiaotong University, Xi'an, China.

Neural Networks : the Official Journal of the International Neural Network Society
|November 13, 2025
PubMed
Summary

We developed efficient Nyström-aware approximations for matrix-based Rényi

Keywords:
EntropyInformation theoryRandomized numerical linear algebraTrace estimation

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Area of Science:

  • Information Theory
  • Machine Learning
  • Numerical Analysis

Background:

  • Matrix-based Rényi's α-order entropy is valuable for information-theoretical learning.
  • Exact computation has prohibitive O(n³) time complexity, limiting its use in large-scale machine learning.
  • Existing approximations like Hutch++ require multiple passes over data.

Purpose of the Study:

  • To propose novel Nyström-aware approximation strategies for matrix-based Rényi's entropy.
  • To reduce computational complexity while maintaining accuracy.
  • To improve upon existing approximation methods like Hutch++.

Main Methods:

  • Developed Nyström-aware sketching techniques for arbitrary α-order matrix-based Rényi's entropy.
  • Reduced time complexity to O(n²s) using s queried randomized vectors.
  • Focused on symmetric positive semi-definite matrices.

Main Results:

  • Achieved significant reduction in computational complexity compared to exact methods.
  • Demonstrated superior efficiency over Hutch++-based approximations with fewer data passes.
  • Query complexity shown to be near-optimal with respect to approximation error.
  • Enabled potential for parallel computation.

Conclusions:

  • Nyström-aware approximations offer a computationally efficient and accurate alternative for matrix-based Rényi's entropy.
  • The proposed methods outperform existing techniques in terms of speed and data efficiency.
  • These approximations are suitable for large-scale machine learning applications.