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Dimensional analysis is a powerful tool that is used in physics and engineering to understand and predict the behavior of physical systems. The basic idea behind dimensional analysis is to express physical quantities in terms of fundamental dimensions such as the mass, length, and time. Derived dimensions like the velocity, acceleration, and force are derived from the combinations of these fundamental dimensions.
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The generalized ratios intrinsic dimension estimator.

Francesco Denti1, Diego Doimo2, Alessandro Laio2,3

  • 1Department of Statistics, Università Cattolica del Sacro Cuore, Milan, Italy. francesco.denti@unicatt.it.

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Summary

Estimating intrinsic dimension (id) is crucial for complex datasets. A new method, Gride, accurately measures id across scales without data decimation, offering robust and efficient analysis.

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Area of Science:

  • Data Science
  • Statistical Modeling
  • Machine Learning

Background:

  • Modern datasets feature complex dependencies requiring dimensionality reduction.
  • Intrinsic dimension (id) estimation is key but scale-dependent and sensitive to noise.
  • Existing methods like dataset decimation introduce statistical errors.

Purpose of the Study:

  • Introduce Gride, a novel statistical method for scale-dependent intrinsic dimension estimation.
  • Enable accurate id assessment across various scales without data decimation.
  • Provide uncertainty quantification for id estimates.

Main Methods:

  • Developed Gride, a method relying on rigorous distributional results.
  • Utilizes only pairwise distances among data points for computation.
  • Avoids dataset decimation, mitigating statistical errors.

Main Results:

  • Gride estimates intrinsic dimension as an explicit function of scale.
  • Demonstrated asymptotic unbiasedness and robustness to short-scale noise via simulations.
  • Achieved comparable performance to state-of-the-art methods.

Conclusions:

  • Gride offers a computationally efficient and statistically rigorous approach to intrinsic dimension estimation.
  • The method provides reliable scale-dependent id analysis, outperforming other likelihood-based techniques in noisy conditions.