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This study introduces a renormalization group (RG) approach for analyzing systems with many degrees of freedom, analogous to principal components analysis (PCA). It reveals relevant and irrelevant operators by examining eigenvalue density, with applications in neural and financial data analysis.

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Area of Science:

  • Complex Systems Analysis
  • Statistical Physics
  • Computational Neuroscience

Background:

  • Principal Components Analysis (PCA) is commonly used to reduce dimensionality in systems with many degrees of freedom by analyzing covariance matrix eigenvalues.
  • When eigenvalue spectra are nearly continuous, arbitrary cutoffs in PCA can obscure meaningful distinctions between components.
  • The arbitrary nature of component selection in PCA for continuous spectra necessitates alternative analytical frameworks.

Purpose of the Study:

  • To develop a novel analytical framework for high-dimensional systems with continuous eigenvalue spectra.
  • To draw an analogy between PCA component selection and the renormalization group (RG) in statistical physics.
  • To define relevant and irrelevant operators based on eigenvalue density properties and apply this to real-world data.

Main Methods:

  • The study employs an analogy to the momentum shell renormalization group (RG) to analyze systems characterized by covariance matrices.
  • It defines relevant and irrelevant operators by considering properties of the eigenvalue density, analogous to dimensionality in RG.
  • The framework is applied to analyze neural activity data from vertebrate retinas and financial datasets.

Main Results:

  • The RG analogy provides a method to define relevant and irrelevant operators, overcoming arbitrary cutoffs in PCA for continuous spectra.
  • Analysis of neural activity in the vertebrate retina revealed behavior governed by a nontrivial fixed point.
  • Financial data analysis successfully separated modes dominated by sampling noise from a distinct set of non-Gaussian modes.

Conclusions:

  • The proposed RG-inspired approach offers a principled way to analyze high-dimensional data, particularly when dealing with continuous eigenvalue spectra.
  • This method provides insights into complex systems, as demonstrated by its application to neural and financial data.
  • The findings suggest a robust framework for distinguishing meaningful patterns from noise in diverse datasets.