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Solvable Model for the Linear Separability of Structured Data.

Marco Gherardi1,2

  • 1Department of Physics, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy.

Entropy (Basel, Switzerland)
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This study introduces new ways to measure linear separability in machine learning beyond simple binary classification. A combinatorial method quantifies data structure, offering insights into model capacity and performance.

Keywords:
data structurelinear separabilitystorage capacity

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Area of Science:

  • Machine Learning
  • Statistical Physics
  • Data Science

Background:

  • Linear separability is fundamental to supervised machine learning, determining if data is classifiable by a linear model.
  • Existing metrics offer limited insight into the nuances of data structure and separability.
  • Quantifying linear separability requires interpretable models and analytical tractability.

Purpose of the Study:

  • To develop and apply analytical models for quantifying linear separability beyond binary classification.
  • To introduce a combinatorial method for computing descriptors of linear separability.
  • To demonstrate the utility of multiple metrics using synthetic and benchmark datasets.

Main Methods:

  • Utilized a combinatorial approach within a mean-field approximation.
  • Developed interpretable models of data structure parameterized by analytical quantities.
  • Applied methods to a synthetic dataset with controlled correlations and the MNIST dataset.

Main Results:

  • Computed two novel descriptors of linear separability, one related to storage capacity.
  • Showcased that storage capacity alone provides an incomplete picture of linear separability in complex datasets like MNIST.
  • Observed a high degree of universality in analytical results, indicating robustness to data structure parameters.

Conclusions:

  • The proposed methods offer a more comprehensive quantification of linear separability.
  • Multiple metrics are essential for fully understanding data structure and classifier performance.
  • The findings suggest a general applicability of these descriptors across various datasets.